MATVB_train.json

{
    "Evolute_4": "The normal at point $P$ is the tangent at the curvature center $C$.",
    "Evolute_5": "The evolute of a curve (blue parabola) is the locus of all its centers of curvature (red).",
    "Evolute_1": "The evolute of a curve (blue parabola) is the locus of all its centers of curvature (red).",
    "Evolute_2": "Evolute (red) of an ellipse",
    "Evolute_3": "The evolute of the large nephroid (blue) is the small nephroid (red).",
    "Semi-continuity_2": "A lower semicontinuous function that is not upper semicontinuous at $x_{0}$. The solid blue dot indicates $f\\left(x_{0}\\right).$",
    "Semi-continuity_1": "An upper semicontinuous function that is not lower semicontinuous at $x_{0}$. The solid blue dot indicates $f\\left(x_{0}\\right).$",
    "Envelope_(mathematics)_1": "An astroid as the envelope of the family of lines connecting points $(s, 0)$ and $(0, t)$ with $s^2 + t^2 = 1$.",
    "Envelope_(mathematics)_2": "The orbits' envelope of the projectiles (with constant initial speed) is a concave parabola. The initial speed is $10\\,\\mathrm{m/s}$. We take $g = 10\\,\\mathrm{m/s^2}$.",
    "Line_graph_5": "Construction of the de Bruijn graphs as iterated line digraphs",
    "Line_graph_1": "The diamond graph (left) and its more-symmetric line graph (right), an exception to the strong Whitney theorem",
    "Line_graph_2": "A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles.",
    "Line_graph_3": "Partition of a line graph into cliques",
    "Line_graph_4": "The nine minimal non-line graphs, from Beineke's forbidden-subgraph characterization of line graphs. A graph is a line graph if and only if it does not contain one of these nine graphs as an induced subgraph.",
    "Triangulation_(topology)_8": "A triangulated dolphin shape",
    "Triangulation_(topology)_9": "A 2-dimensional geometric simplicial complex with vertex $V$, $link(V)$, and $star(V)$ are highlighted in red and pink.",
    "Triangulation_(topology)_1": "A triangulated torus",
    "Triangulation_(topology)_2": "Another triangulation of the torus",
    "Triangulation_(topology)_3": "Geometric simplices in dimension 1, 2 and 3",
    "Triangulation_(topology)_4": "The 2-dimensional sphere and a triangulation",
    "Triangulation_(topology)_5": "A two dimensional torus, represented as the gluing of a square via the map $g$, identifying its opposite sites",
    "Triangulation_(topology)_6": "The real projective plane as a simplicial complex and as CW-complex. As CW-complex it can be obtained by gluing first $\\mathbb {D} ^{0}$ and $\\mathbb {D} ^{1}$ to get the 1-sphere and then attaching the disc $\\mathbb {D} ^{2}$ by the map $g:\\mathbb {S} ^{1}\\rightarrow \\mathbb {S} ^{1},e^{ix}\\mapsto e^{2ix}$.",
    "Triangulation_(topology)_7": "One Pachner-move replaces two tetrahedra by three tetrahedra",
    "Implicit_curve_10": "Smooth approximation of 1) one half of a circle, 2) an intersection of two circles",
    "Implicit_curve_1": "Smooth approximation of a convex polygon",
    "Implicit_curve_2": "Implicit curve: $\\sin(x+y)-\\cos(xy)+1=0$",
    "Implicit_curve_3": "Implicit curve $\\sin(x+y)-\\cos(xy)+1=0$ as level curves of the surface $z=\\sin(x+y)-\\cos(xy)+1$",
    "Implicit_curve_4": "Blending curve (red) of two circles",
    "Implicit_curve_5": "to the tracing algorithm: starting points are green",
    "Implicit_curve_6": "Equipotential curves of two point charges at the blue points",
    "Implicit_curve_7": "Example: An illustration of the raster algorithm applied to the implicit curve $F(x,y)=(3x^{2}-y^{2})^{2}y^{2}-(x^{2}+y^{2})^{4}=0$. The curve (red) is what the algorithm is trying to draw. The raster points (black) are used as starting points to find the closest points on the curve (red circles). The spacing between each raster point is exaggerated to show the individual curve points; to more accurately trace the curve, more raster points would be used.",
    "Implicit_curve_8": "Intersection curve between a sphere and a cylinder",
    "Implicit_curve_9": "Cassini ovals: (1) $a=1.1, c=1$ (above), (2) $a=c=1$ (middle), (3) $a=1, c=1.05$ (below)",
    "Pappus_chain_1": "Under a particular inversion centered on $A$, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula $h_n = nd_n$ and the fact that the original points of tangency lie on a common circle.",
    "Abstract_polytope_6": "The Hemicube may be derived from a cube by identifying opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.",
    "Abstract_polytope_7": "A square pyramid and the associated abstract polytope.",
    "Abstract_polytope_8": "A square pyramid and the associated abstract polytope.",
    "Abstract_polytope_1": "The graph (left) and Hasse diagram of a quadrilateral, showing ranks (right)",
    "Abstract_polytope_2": "The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section (red), and a 2-section (green).",
    "Abstract_polytope_3": "The graph (left) and Hasse Diagram of a line segment",
    "Abstract_polytope_4": "The graph (left) and Hasse Diagram of a digon",
    "Abstract_polytope_5": "A hexagonal hosohedron, realized as a spherical polyhedron.",
    "Spiral_of_Theodorus_4": "Philip J. Davis' analytic continuation of the Spiral of Theodorus, including extension in the opposite direction from the origin (negative nodes numbers).",
    "Spiral_of_Theodorus_1": "The spiral of Theodorus up to the triangle with a hypotenuse of ${\\sqrt {17}}$",
    "Spiral_of_Theodorus_2": "Colored extended spiral of Theodorus with 110 triangles",
    "Spiral_of_Theodorus_3": "A triangle or section of spiral",
    "Root_system_6": "Pictures of all the connected Dynkin diagrams",
    "Root_system_1": "The six vectors of the root system $A_2$",
    "Root_system_2": "The Weyl group of the $A_{2}$ root system is the symmetry group of an equilateral triangle",
    "Root_system_3": "The shaded region is the fundamental Weyl chamber for the base $\\{\\alpha_1, \\alpha_2\\}$.",
    "Root_system_4": "Root system $B_3$, $C_3$, and $A_3 = D_3$ as points within a cube and octahedron",
    "Gaussian_process_4": "Gaussian Process Regression (prediction) with a squared exponential kernel. Left plot are draws from the prior function distribution. Middle are draws from the posterior. Right is mean prediction with one standard deviation shaded.",
    "Gaussian_process_1": "The effect of choosing different kernels on the prior function distribution of the Gaussian process. Left is a squared exponential kernel. Middle is Brownian. Right is quadratic.",
    "Gaussian_process_2": "Autocorrelation of a random lacunary Fourier series",
    "Gaussian_process_3": "An example of Gaussian Process Regression (prediction) compared with other regression models.",
    "Method_of_analytic_tableaux_7": "A tableau without unification for $\\{\\forall x.\\,P(x),\\, \\exists x.\\,(\\lnot P(x) \\lor \\lnot P(f(x)))\\}$. For clarity, formulae are numbered on the left and the formula and rule used at each step is on the right.",
    "Method_of_analytic_tableaux_1": "A graphical representation of a partially built propositional tableau",
    "Method_of_analytic_tableaux_2": "$(a \\lor \\lnot b) \\land b$ generates $a \\lor \\lnot b$ and $b$",
    "Method_of_analytic_tableaux_3": "Initial tableau for $\\{(a \\lor \\lnot b) \\land b,\\ \\lnot a\\}$",
    "Method_of_analytic_tableaux_4": "A tableau for the satisfiable set $\\{a \\land c, \\lnot a \\lor b\\}$: all rules have been applied to every formula on every branch, but the tableau is not closed (only the left branch is closed), as expected for satisfiable sets.",
    "Method_of_analytic_tableaux_5": "A first-order tableau with unification for $\\{\\forall x.\\,P(x),\\, \\exists x.\\,(\\lnot P(x) \\lor \\lnot P(f(x)))\\}$. For clarity, formulae are numbered on the left and the formula and rule used at each step is on the right.",
    "Barycentric_coordinate_system_4": "A 3-simplex, with barycentric subdivisions of 1-faces (edges) 2-faces (triangles) and 3-faces (body).",
    "Barycentric_coordinate_system_1": "Barycentric coordinates $(\\lambda _{1},\\lambda _{2},\\lambda _{3})$ on an equilateral triangle and on a right triangle.",
    "Barycentric_coordinate_system_2": "Surface (upper part) obtained from linear interpolation over a given triangular grid (lower part) in the $x, y$ plane. The surface approximates a function $z = f(x, y)$, given only the values of $f$ on the grid's vertices.",
    "Barycentric_coordinate_system_3": "Barycentric coordinates are used for blending three colors over a triangular region evenly in computer graphics.",
    "Quadratrix_3": "Quadratrix of Dinostratus with a central portion flanked by infinite branches",
    "Quadratrix_1": "Quadratrix of Dinostratus (in red)",
    "Quadratrix_2": "Tschirnhaus' quadratrix (red), Hippias quadratrix (dotted)",
    "Orthoptic_(geometry)_7": "Hyperbola Orthoptic of the hyperbola (its director circle) xy-axes and hyperbolic asymptotes",
    "Orthoptic_(geometry)_1": "Parabola Orthoptic of the parabola (its directrix)",
    "Orthoptic_(geometry)_2": "Orthoptic (purple) of an astroid",
    "Orthoptic_(geometry)_3": "Isoptics (purple) of a parabola for angles $80^\\circ$ and $100^\\circ$",
    "Orthoptic_(geometry)_4": "Isoptics (purple) of an ellipse for angles $80^\\circ$ and $100^\\circ$",
    "Orthoptic_(geometry)_5": "Isoptics (purple) of a hyperbola for angles $80^\\circ$ and $100^\\circ$",
    "Orthoptic_(geometry)_6": "Orthoptic (purple) of an astroid",
    "List_of_trigonometric_identities_4": "Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity $1+\\cot ^{2}\\theta =\\csc ^{2}\\theta$, and the red triangle shows that $\\tan ^{2}\\theta +1=\\sec ^{2}\\theta$.",
    "List_of_trigonometric_identities_5": "Transformation of coordinates $(a,b)$ when shifting the angle $\\theta$ in increments of ${\\frac {\\pi}{2}}$",
    "List_of_trigonometric_identities_1": "Transformation of coordinates $(a,b)$ when shifting the reflection angle $\\alpha$ in increments of ${\\frac {\\pi}{4}}$",
    "List_of_trigonometric_identities_2": "Geometric construction to derive angle sum trigonometric identities.",
    "List_of_trigonometric_identities_3": "Diagram showing the angle difference identities for $\\sin(\\alpha -\\beta)$ and $\\cos(\\alpha -\\beta )$",
    "Nephroid_1": "nephroid: tangents as chords of a circle, principle",
    "Nephroid_2": "nephroid: tangents as chords of a circle",
    "Nephroid_3": "nephroid as caustic of a circle: principle",
    "Nephroid_4": "nephroide as caustic of one half of a circle",
    "Nephroid_5": "nephroid and its evolute magenta: point with osculating circle and center of curvature",
    "Nephroid_6": "inversion (green) of a nephroid (red) across the blue circle",
    "Nephroid_7": "Nephroid: definition",
    "Solution_of_triangles_15": "Three angles given",
    "Solution_of_triangles_1": "Standard notation for a triangle",
    "Solution_of_triangles_2": "Overview of particular steps and tools used when solving plane triangles",
    "Solution_of_triangles_3": "Three sides given",
    "Solution_of_triangles_4": "Two sides and the included angle given",
    "Solution_of_triangles_5": "Two sides and a non-included angle given",
    "Solution_of_triangles_6": "Two solutions for the triangle",
    "Solution_of_triangles_7": "One side and two adjacent angles given",
    "Solution_of_triangles_8": "Three sides given",
    "Solution_of_triangles_9": "Two sides and the included angle given",
    "Solution_of_triangles_10": "Two sides and a non-included angle given",
    "Solution_of_triangles_11": "One side and two adjacent angles given",
    "Solution_of_triangles_12": "One side, one adjacent angle and the opposite angle given",
    "Solution_of_triangles_13": "Distance measurement by triangulation",
    "Cartesian_tensor_1": "Two different 3d orthonormal bases: each basis consists of unit vectors that are mutually perpendicular.",
    "Cartesian_tensor_2": "The same position vector $x$ represented in two 3d rectangular coordinate systems each with an orthonormal basis, the cuboids illustrate the parallelogram law for adding vector components.",
    "Finite_subdivision_rule_10": "Subdivisions of the subdivision complex for the Borromean rings complement.",
    "Finite_subdivision_rule_1": "The binary subdivision rule",
    "Finite_subdivision_rule_2": "We start with a complex with four quadrilaterals and subdivide twice. All squares are type $A$ tiles.",
    "Finite_subdivision_rule_3": "Trefoil subdivision rule",
    "Finite_subdivision_rule_4": "Borromean subdivision rule",
    "Finite_subdivision_rule_5": "Subdivisions of the subdivision complex for the trefoil complement.",
    "Finite_subdivision_rule_6": "A subdivision rule for the four-torus. The faces of the $B$ tiles that subdivide can only touch $C$ tiles, and the faces of the $B$ tiles that don't only touch $A$ tiles.",
    "Finite_subdivision_rule_7": "The history graph of the middle thirds subdivision rule.",
    "Finite_subdivision_rule_8": "Subdivision rule used to create the small scale girih pattern from the large scale pattern.",
    "Finite_subdivision_rule_9": "Catmull-Clark subdivision of a cube",
    "Cubic_plane_curve_7": "Lucas Cubic of triangle $\\triangle ABC$: The locus of a point $X$ such that the cevian triangle of $X$ is the pedal triangle of some point $X'$; the point $X'$ lies on the Darboux cubic.",
    "Cubic_plane_curve_1": "Singular cubic $y^2 = x^2 \\cdot (x + 1)$. A parametrization is given by $t \\mapsto (t^2 - 1,\\, t \\cdot (t^2 - 1))$.",
    "Cubic_plane_curve_2": "Neuberg cubic of triangle $\\triangle ABC$: The locus of $X$ such that, if $X_A, X_B, X_C$ are the reflections of $X$ in the sidelines $BC$, $CA$, $AB$, then the lines $A X_A$, $B X_B$, $C X_C$ are concurrent.",
    "Cubic_plane_curve_3": "Example of Thomson cubic (black curve). $X$ is on the cubic, such that the isogonal conjugate of $X$ ($X'$) lies on the line $X_{(2)}X$.",
    "Cubic_plane_curve_4": "Darboux cubic of triangle $\\triangle ABC$: The locus of $X$ such that if $D$, $E$, $F$ are the feet of the perpendiculars from $X$ to the sidelines $BC$, $CA$, $AB$, then the lines $AD$, $BE$, and $CF$ are concurrent.",
    "Cubic_plane_curve_5": "First Brocard Cubic: It is the locus of $X$ such that the intersections of $XA'$, $XB'$, $XC'$ with the sidelines $BC$, $CA$, $AB$, where $\\triangle A'B'C'$ is the first Brocard triangle of triangle $\\triangle ABC$, are collinear. In the figure, $\\Omega$ and $\\Omega'$ are the first and second Brocard points.",
    "Cubic_plane_curve_6": "First equal area cubic of triangle $\\triangle ABC$: The locus of a point $X$ such that the area of the cevian triangle of $X$ equals the area of the cevian triangle of $X^*$.",
    "Factorial_1": "Comparison of the factorial, Stirling's approximation, and the simpler approximation $(n/e)^{n}$, on a doubly logarithmic scale",
    "Factorial_2": "The gamma function (shifted one unit left to match the factorials) continuously interpolates the factorial to non-integer values",
    "Factorial_3": "Absolute values of the complex gamma function, showing poles at non-positive integers",
    "Unit_hyperbola_1": "The unit hyperbola is blue, its conjugate is green, and the asymptotes are red.",
    "Parallel_axis_theorem_3": "Polar moment of inertia of a body around a point can be determined from its polar moment of inertia around the center of mass.",
    "Parallel_axis_theorem_1": "The mass moment of inertia of a body around an axis can be determined from the mass moment of inertia around a parallel axis through the center of mass.",
    "Parallel_axis_theorem_2": "Parallel axes rule for area moment of inertia",
    "Pinwheel_tiling_2": "A pinwheel tiling: tiles can be grouped in sets of five (thick lines) to form a new pinwheel tiling (up to rescaling)",
    "Pinwheel_tiling_1": "Conway's triangle decomposition into smaller similar triangles.",
    "Steiner_conic_7": "example of a Steiner generation of a dual conic",
    "Steiner_conic_1": "Definition of the Steiner generation of a conic section",
    "Steiner_conic_2": "Perspective mapping between lines",
    "Steiner_conic_3": "Example of a Steiner generation: generation of a point",
    "Steiner_conic_4": "Steiner generation of a dual conic",
    "Steiner_conic_5": "definition of a perspective mapping",
    "Steiner_conic_6": "Dual Steiner conic defined by two perspectivities $\\pi _{A},\\pi _{B}$",
    "Uniform_polyhedron_4": "Wythoff constructions for the $(6 3 2)$ triangle group.",
    "Uniform_polyhedron_1": "The great dirhombicosidodecahedron, the only non-Wythoffian uniform polyhedron",
    "Uniform_polyhedron_2": "Example forms from the cube and octahedron",
    "Uniform_polyhedron_3": "Small stellated dodecahedron.",
    "Douady_rabbit_4": "Douady rabbit in an exponential family",
    "Douady_rabbit_5": "Representation of the dynamics inside the rabbit",
    "Douady_rabbit_6": "An example of a Douady rabbit. The colors show the number of iterations required to escape.",
    "Douady_rabbit_1": "Lamination of the rabbit Julia set",
    "Douady_rabbit_2": "Douady rabbit for $\\gamma =2.55268-0.959456i$ or $\\mu =0.122565-0.744864i$",
    "Douady_rabbit_3": "Douady rabbit for $\\gamma =-0.55268+0.959456i$ or $\\mu =0.122565-0.744864i$",
    "Lagrange_multiplier_8": "A paraboloid constrained along two intersecting lines.",
    "Lagrange_multiplier_1": "The red curve shows the constraint $g(x, y) = c$. The blue curves are contours of $f(x, y)$. The point where the red constraint tangentially touches a blue contour is the maximum of $f(x, y)$ along the constraint, since $d_1 > d_2$ .",
    "Lagrange_multiplier_2": "A paraboloid constrained along two intersecting lines.",
    "Lagrange_multiplier_3": "Lagrange multipliers cause the critical points to occur at saddle points.",
    "Lagrange_multiplier_4": "The magnitude of the gradient can be used to force the critical points to occur at local minima.",
    "Lagrange_multiplier_5": "The red curve shows the constraint $g(x, y) = c$. The blue curves are contours of $f(x, y)$. The point where the red constraint tangentially touches a blue contour is the maximum of $f(x, y)$ along the constraint, since $d_1 > d_2$.",
    "Lagrange_multiplier_6": "Lagrange multipliers cause the critical points to occur at saddle points",
    "Lagrange_multiplier_7": "The magnitude of the gradient can be used to force the critical points to occur at local minima",
    "Block-stacking_problem_3": "The first nine blocks in the solution to the single-wide block-stacking problem with the overhangs indicated",
    "Block-stacking_problem_1": "Mike Paterson's method to increase the overhang of $16$ blocks of unit width and breadth $b$ to $2\\sqrt{1 + b^2}$ by offsetting the blocks perpendicular to their lengths in a diamond formation.",
    "Numerical_integration_4": "Numerical integration is used to calculate a numerical approximation for the value $S$, the area under the curve defined by $f(x)$.",
    "Numerical_integration_5": "Illustration of Simpson's rule.",
    "Numerical_integration_6": "The area of a segment of a parabola",
    "Numerical_integration_1": "Antique method to find the Geometric mean",
    "Numerical_integration_2": "Illustration of the rectangle rule.",
    "Numerical_integration_3": "Illustration of the trapezoidal rule.",
    "Hyperoctahedral_group_1": "8 of the 48 permutations of a cube, forming $O_h$",
    "Hyperoctahedral_group_2": "Tetrahedral symmetry in three dimensions, order 24",
    "Hyperoctahedral_group_3": "Pyritohedral symmetry in three dimensions, order 24",
    "Hyperoctahedral_group_4": "Octahedral symmetry in three dimensions, order 24",
    "Hyperoctahedral_group_5": "The $C_2$ group has order 8 as shown on this circle",
    "Hyperoctahedral_group_6": "The $C_3 (O_h)$ group has order 48 as shown by these spherical triangle reflection domains.",
    "Generalized_helicoid_9": "tangent developable type: definition (left) and example",
    "Generalized_helicoid_1": "generalized helicoid: meridian is a parabola.",
    "Generalized_helicoid_2": "screw motion of a point green: pitch, blue: screw axis",
    "Generalized_helicoid_3": "right ruled generalized helicoid: closed (left) and open (right)",
    "Generalized_helicoid_4": "oblique types: closed (left) and open (right)",
    "Generalized_helicoid_5": "on the selfintersection of closed ruled generalized helicoids",
    "Generalized_helicoid_6": "tangent developable: regular parts (green and blue) and the directrix (purple)",
    "Generalized_helicoid_7": "meridian is a circle",
    "Generalized_helicoid_8": "profile curve is a horizontal circle",
    "Sperner's_lemma_3": "A simple two-dimensional triangulation of the example figure, colored and named in accordance with the assumptions of Sperner's Lemma",
    "Sperner's_lemma_1": "The two-dimensional case of Sperner's lemma: a Sperner coloring, with its 3-colored triangles shaded",
    "Sperner's_lemma_2": "One-dimensional case example",
    "Method_of_averaging_4": "Solution to perturbed logistic growth equation ${\\dot {x}}=\\varepsilon (x(1-x)+\\sin {t})~x\\in \\mathbb {R} ,~\\varepsilon =0.05$ (blue solid line) and the averaged equation ${\\dot {y}}=\\varepsilon y(1-y),~y\\in \\mathbb {R}$ (orange solid line).",
    "Method_of_averaging_1": "A simple harmonic oscillator with small periodic damping term given by ${\\ddot {z}}+4\\varepsilon \\cos ^{2}{(t)}{\\dot {z}}+z=0,~z(0)=0,~{\\dot {z}}(0)=1;~\\varepsilon =0.05$.The numerical simulation of the original equation (blue solid line) is compared with averaging system (orange dashed line) and the crude averaged system (green dash-dotted line). The left plot displays the solution evolved in time and the right plot represents on the phase space.  We note that the crude averaging disagrees with the expected solution.",
    "Method_of_averaging_2": "Phase space of a Van der Pol oscillator with $\\varepsilon =0.1$. The stable limit cycle (orange solid line) in the system is captured correctly by the qualitative analysis of the averaged system. For two different initial conditions ( black dots ) we observe the trajectories.(dashed blue line) converging to the periodic orbit.",
    "Method_of_averaging_3": "The plot depicts two fundamental quantities the average technique is based on: the bounded and connected region $D$ of the phase space and how long (defined by the constant $c$) the averaged solution is valid. For this case, $\\textstyle {\\ddot {z}}+z=8\\varepsilon \\cos {(t)}{\\dot {z}}^{2},~z(0)=0,~{\\dot {z}}(0)=1;~8\\varepsilon ={\\frac {2}{15}}$. Note that both solutions blow up in finite time. Hence, $D$ has been chosen accordingly in order to maintain the boundedness of the solution and the time interval of validity of the approximation is $0\\leq \\varepsilon t<L<{\\frac {1}{3}}$.",
    "Convex_cone_1": "A convex cone (light blue). Inside of it, the light red convex cone consists of all points $\\alpha x + \\beta y$ with $\\alpha, \\beta > 0$, for the depicted $x$ and $y$. The curves on the upper right symbolize that the regions are infinite in extent.",
    "Convex_cone_2": "Convex cone generated by the conic combination of the three black vectors.",
    "Convex_cone_3": "A cone (the union of two rays) that is not a convex cone.",
    "Hypergraph_3": "Alternative representation of the hypergraph reported in the figure above, called PAOH. Edges are vertical lines connecting vertices. V7 is an isolated vertex. Vertices are aligned to the left. The legend on the right shows the names of the edges.",
    "Hypergraph_4": "An example of an undirected hypergraph, with $X=\\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\}$ and $E=\\{e_{1},e_{2},e_{3},e_{4}\\}=\\{\\{v_{1},v_{2},v_{3}\\}, \\{v_{2},v_{3}\\},\\{v_{3},v_{5},v_{6}\\},\\{v_{4}\\}\\}$. This hypergraph has order 7 and size 4. Here, edges do not just connect two vertices but several, and are represented by colors.",
    "Hypergraph_1": "An example of a directed hypergraph, with $X=\\{1,2,3,4,5,6\\}$ and $E=\\{a_{1},a_{2},a_{3},a_{4},a_{5}\\}=\\{(\\{1\\},\\{2\\}),(\\{2\\},\\{3\\}),(\\{3\\},\\{1\\}),(\\{2,3\\},\\{4,5\\}),(\\{3,5\\},\\{6\\})\\}$.",
    "Hypergraph_2": "An order-4 Venn diagram, which can be interpreted as a subdivision drawing of a hypergraph with 15 vertices (the 15 colored regions) and 4 hyperedges (the 4 ellipses).",
    "Hexagonal_prism_2": "Hexagonal prismatic honeycomb",
    "Hexagonal_prism_1": "3D model of a uniform hexagonal prism.",
    "Convex_analysis_3": "A graph of the bivariate convex function $x^{2}+xy+y^{2}.$",
    "Convex_analysis_4": "A function (in black) is convex if and only if its epigraph, which is the region above its graph (in green), is a convex set.",
    "Convex_analysis_1": "Convex function on an interval.",
    "Convex_analysis_2": "A 3-dimensional convex polytope. Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces.",
    "Regular_4-polytope_3": "A subset of relations among 8 forms from the 120-cell, polydodecahedron (pD). The three operations {a,g,s} are commutable, defining a cubic framework. There are 7 densities seen in vertical positioning, with 2 dual forms having the same density.",
    "Regular_4-polytope_1": "This shows the relationships among the four-dimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a cuboctahedron.",
    "Regular_4-polytope_2": "The tesseract is one of 6 convex regular 4-polytopes.",
    "Multimodal_distribution_3": "A bimodal distribution.",
    "Multimodal_distribution_1": "A simple bimodal distribution, in this case a mixture of two normal distributions with the same variance but different means. The figure shows the probability density function (p.d.f.), which is an equally-weighted average of the bell-shaped p.d.f.s of the two normal distributions. If the weights were not equal, the resulting distribution could still be bimodal but with peaks of different heights.",
    "Multimodal_distribution_2": "A non-example: a unimodal distribution, that would become multimodal if conditioned on either $x$ or $y$.",
    "Hyperbola_15": "Hyperbola: definition by the distances of points to two fixed points (foci)",
    "Hyperbola_1": "Hyperbola as affine image of $y = \\frac{1}{x}$",
    "Hyperbola_2": "A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.",
    "Hyperbola_3": "Hyperbola (red): features",
    "Hyperbola_4": "Hyperbola: definition with circular directrix",
    "Hyperbola_5": "Rotating the coordinate system in order to describe a rectangular hyperbola as graph of a function",
    "Hyperbola_6": "Hyperbola: directrix property",
    "Hyperbola_7": "Hyperbola: definition with directrix property",
    "Hyperbola_8": "Hyperbola: construction of a directrix",
    "Hyperbola_9": "Hyperbola (red): two views of a cone and two Dandelin spheres $d_1$, $d_2$",
    "Hyperbola_10": "Hyperbola: Pin and string construction",
    "Hyperbola_11": "Hyperbola: Steiner generation",
    "Hyperbola_12": "Hyperbola as an affine image of the unit hyperbola",
    "Hyperbola_13": "Three rectangular hyperbolas $y=A/x$ with the coordinate axes as asymptotes red: $A = 1$; magenta: $A = 4$; blue: $A = 9$",
    "Initial_topology_1": "Characteristic property of the initial topology",
    "Dot_product_1": "Distributive law for the dot product",
    "Dot_product_2": "Vector components in an orthonormal basis",
    "Four-spiral_semigroup_2": "The structure of the four-spiral semigroup Sp4. The set of idempotents (red coloured points) and the subsemigroups $A$, $B$, $C$, $D$, $E$ are shown.",
    "Four-spiral_semigroup_1": "The spiral structure of idempotents in the four-spiral semigroup Sp4. In this diagram, elements in the same row are R-related, elements in the same column are L-related, and the order proceeds down the four diagonals (away from the center).",
    "Pythagorean_theorem_1": "Proof using similar triangles",
    "Pythagorean_theorem_2": "Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof.",
    "Pythagorean_theorem_3": "Proof in Euclid's Elements",
    "Pythagorean_theorem_4": "Illustration including the new lines",
    "Pythagorean_theorem_5": "Showing the two congruent triangles of half the area of rectangle $BDLK$ and square $BAGF$",
    "Pythagorean_theorem_6": "Proof using an elaborate rearrangement",
    "Pythagorean_theorem_7": "The spiral of Theodorus: A construction for line segments with lengths whose ratios are the square root of a positive integer",
    "Pythagorean_theorem_8": "The absolute value of a complex number $z$ is the distance $r$ from $z$ to the origin.",
    "Pythagorean_theorem_9": "Distance between infinitesimally separated points in Cartesian coordinates (top) and polar coordinates (bottom), as given by Pythagoras' theorem",
    "Pythagorean_theorem_10": "Pythagoras' theorem in three dimensions relates the diagonal $AD$ to the three sides.",
    "Pythagorean_theorem_11": "Generalization of Pythagoras' theorem by Tâbit ibn Qorra. Lower panel: reflection of triangle $CAD$ (top) to form triangle $DAC$, similar to triangle $ABC$ (top).",
    "Pythagorean_theorem_12": "Similar right triangles showing sine and cosine of angle $\\theta$",
    "Pythagorean_theorem_13": "Diagram of the two algebraic proofs",
    "Directed_graph_5": "A directed graph with vertices labeled (indegree, outdegree)",
    "Directed_graph_1": "A simple directed graph",
    "Directed_graph_2": "A simple directed acyclic graph",
    "Directed_graph_3": "A tournament on 4 vertices",
    "Directed_graph_4": "Oriented graph with corresponding incidence matrix",
    "Fourier_transform_3": "Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.",
    "Fourier_transform_4": "Original function, which has a strong 3 Hz component. Real and imaginary parts of the integrand of its Fourier transform at +3 Hz.",
    "Fourier_transform_1": "Real and imaginary parts of the integrand for its Fourier transform at +5 Hz.",
    "Fourier_transform_2": "Magnitude of its Fourier transform, with +3 and +5 Hz labeled.",
    "Malfatti_circles_4": "Steiner's construction of the Malfatti circles using bitangents",
    "Malfatti_circles_1": "In an equilateral triangle the area of the Malfatti circles (left) is approximately $1\\%$ smaller than the three area-maximizing circles (right).",
    "Malfatti_circles_2": "Steiner's construction of the Malfatti circles using bitangents",
    "Malfatti_circles_3": "First Ajima-Malfatti point",
    "Hanner_polytope_1": "Schlegel diagram of the octahedral prism",
    "Exponential_distribution_4": "Cumulative distribution function",
    "Exponential_distribution_1": "The mean is the probability mass centre, that is, the first moment.",
    "Exponential_distribution_2": "The median is the preimage $F^{-1}(1/2)$.",
    "Exponential_distribution_3": "plot of the probability density function of the exponential distribution",
    "Lemniscate_4": "The lemniscate of Bernoulli and its two foci",
    "Lemniscate_1": "Lemniscate of Booth",
    "Lemniscate_2": "Lemniscate of Bernoulli",
    "Lemniscate_3": "Lemniscate of Gerono: solution set of $x^4-x^2+y^2=0$",
    "Prismatoid_2": "Prismatoid with parallel faces $A_1$ and $A_3$, midway cross-section $A_2$, and height $h$",
    "Prismatoid_1": "A tetrahedral-cuboctahedral cupola.",
    "Reuleaux_triangle_4": "Triquetra interlaced to form a trefoil knot",
    "Reuleaux_triangle_1": "The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.",
    "Reuleaux_triangle_2": "Centrally symmetric shapes inside and outside a Reuleaux triangle, used to measure its asymmetry",
    "Reuleaux_triangle_3": "The Reuleaux triangle as the central bubble in a mathematical model of a four-bubble planar soap bubble cluster",
    "Bridge_(graph_theory)_1": "An undirected connected graph with no bridge edges",
    "Bridge_(graph_theory)_2": "A graph with 16 vertices and six bridges (highlighted in red)",
    "Archimedean_spiral_1": "Archimedean spiral represented on a polar graph",
    "Archimedean_spiral_2": "Three $360^\\circ$ loops of one arm of an Archimedean spiral",
    "Archimedean_spiral_3": "The $xy$ plane rotates to an angle $\\omega t$ (anticlockwise) about the origin in time $t$. $(c, 0)$ is the position of the object at $t = 0$. $P$ is the position of the object at time $t$, at a distance of $R = vt + c$.",
    "Dodecagon_3": "The symmetries of a regular dodecagon as shown with colors on edges and vertices. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. He gives $d$ (diagonal, diasymmetry) with mirror lines through vertices, $p$ with mirror lines through edges (perpendicular, persymmetry) $i$ with mirror lines through both vertices and edges (isosymmetry), and $g$ for rotational (gyrosymmetry). $a1$ labels asymmetry. These lower symmetries allows degrees of freedoms in defining irregular dodecagons.",
    "Dodecagon_1": "A regular skew dodecagon seen as zig-zagging edges of a hexagonal antiprism.",
    "Convex_function_3": "Convex function on an interval.",
    "Convex_function_1": "A function (in black) is convex if and only if the region above its graph (in green) is a convex set.",
    "Convex_function_2": "A graph of the bivariate convex function $x^2 + xy + y^2$.",
    "Devil's_curve_2": "Devil's curve with a $a$ ranging from 0 to 1 and $b = 1$ (with the curve colour going from blue to red).",
    "Devil's_curve_1": "Devil's curve for $a = 0.8$ and $b = 1$.",
    "Cyclic_quadrilateral_4": "$ABCD$ is a cyclic quadrilateral. $E$ is the point of intersection of the diagonals and $F$ is the point of intersection of the extensions of sides $BC$ and $AD$. $\\omega$ is a circle whose diameter is the segment, $EF$. $P$ and $Q$ are Pascal points formed by the circle $\\omega$. Triangles $FAB$ and $FCD$ are similar.",
    "Cyclic_quadrilateral_1": "Examples of cyclic quadrilaterals",
    "Cyclic_quadrilateral_2": "A cyclic quadrilateral $ABCD$",
    "Cyclic_quadrilateral_3": "$ABCD$ is a cyclic quadrilateral. $EFG$ is the diagonal triangle of $ABCD$. The point $T$ of intersection of the bimedians of $ABCD$ belongs to the nine-point circle of $EFG$.",
    "Equivalence_class_2": "Graph of an example equivalence with 7 classes",
    "Equivalence_class_1": "Congruence is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.",
    "Spherical_segment_4": "Geometric parameters for spherical segment.",
    "Spherical_segment_1": "A spherical segment",
    "Spherical_segment_2": "Pair of parallel planes intersecting a sphere forming a spherical segment (i.e., a spherical frustum)",
    "Spherical_segment_3": "Terminology for spherical segments.",
    "Amoeba_(mathematics)_1": "The amoeba of $P(z,w)=w-2z-1.$",
    "Amoeba_(mathematics)_2": "The amoeba of $P(z,w)=3z^{2}+5zw+w^{3}+1.$ Notice the \"vacuole\" in the middle of the amoeba.",
    "Amoeba_(mathematics)_3": "The amoeba of $P(z,w)=1+z+z^{2}+z^{3}+z^{2}w^{3}+10zw+12z^{2}w+10z^{2}w^{2}.$",
    "Amoeba_(mathematics)_4": "Points in the amoeba of $P(x,y,z)=x+y+z-1.$ Note that the amoeba is actually 3-dimensional, and not a surface (this is not entirely evident from the image).",
    "Medial_triangle_1": "The red triangle is the medial triangle of the black. The endpoints of the red triangle coincide with the midpoints of the black triangle.",
    "Tutte_theorem_2": "Example of a graph and one of its perfect matchings (in red).",
    "Tutte_theorem_1": "A graph (or a component) with an odd number of vertices cannot have a perfect matching, since there will always be a vertex left alone.",
    "Fagnano's_problem_2": "Triangle $abc$ is the orthic triangle of triangle $ABC$",
    "Fagnano's_problem_1": "orthic triangle: $\\triangle DEF$ inscribed triangles: $\\triangle DEF\\,,\\triangle GHI$ $|DE|+|EF|+|FD|\\leq |GH|+|HI|+|IG|$",
    "Vector_bundle_9": "A vector bundle $E$ over a base $M$. A point $m_1$ in $M$ ($= X$) corresponds to the origin in a fibre $E_{m_1}$ of the vector bundle $E$, and this fibre is mapped down to the point $m_1$ by the projection $\\pi : E \\to M$.",
    "Vector_bundle_10": "Two trivial vector bundles over open sets $U_{\\alpha }$ and $U_{\\beta }$ may be glued over the intersection $U_{\\alpha \\beta }$ by transition functions $g_{\\alpha \\beta }$ which serve to stick the shaded grey regions together after applying a linear transformation to the fibres (note the transformation of the blue quadrilateral under the effect of $g_{\\alpha \\beta }$). Different choices of transition functions may result in different vector bundles which are non-trivial after gluing is complete.",
    "Vector_bundle_11": "A line subbundle $L$ of a trivial rank 2 vector bundle $E$ over a one-dimensional manifold $M$.",
    "Vector_bundle_1": "A vector bundle $E$ over a base $M$. A point $m_{1}$ in $M(=X)$ corresponds to the origin in a fibre $E_{m_{1}}$ of the vector bundle $E$, and this fibre is mapped down to the point $m_{1}$ by the projection $\\pi :E\\to M$.",
    "Vector_bundle_2": "A vector bundle $E$ over a base $M$ with section $s$.",
    "Vector_bundle_3": "The map associating a normal to each point on a surface can be thought of as a section. The surface is the space $X$, and at each point x there is a vector in the vector space attached at $x$.",
    "Vector_bundle_4": "The regularity of transition functions describing a vector bundle determines the type of the vector bundle. If the continuous transition functions $g_{UV}$ are used, the resulting vector bundle $E$ is only continuous but not smooth. If the smooth transition functions $h_{UV}$ are used, then the resulting vector bundle $F$ is a smooth vector bundle.",
    "Vector_bundle_5": "The map associating a normal to each point on a surface can be thought of as a section. The surface is the space $X$, and at each point $x$ there is a vector in the vector space attached at $x$.",
    "Vector_bundle_6": "A vector bundle $E$ over a base $M$ with section $s$.",
    "Vector_bundle_7": "A line subbundle $L$ of a trivial rank 2 vector bundle $E$ over a one-dimensional manifold $M$.",
    "Vector_bundle_8": "Two trivial vector bundles over open sets $U_{\\alpha}$ and $U_{\\beta}$ may be glued over the intersection $U_{\\alpha \\beta}$ by transition functions $g_{\\alpha \\beta}$ which serve to stick the shaded grey regions together after applying a linear transformation to the fibres (note the transformation of the blue quadrilateral under the effect of $g_{\\alpha \\beta}$). Different choices of transition functions may result in different vector bundles which are non-trivial after gluing is complete.",
    "Quartic_plane_curve_9": "Ampersand curve",
    "Quartic_plane_curve_10": "Bean curve",
    "Quartic_plane_curve_1": "Illustration of the inverse Pythagorean and regular Pythagorean theorems",
    "Quartic_plane_curve_2": "Three-leaved clover in polar coordinates",
    "Quartic_plane_curve_3": "Spiric section",
    "Quartic_plane_curve_4": "Cruciform curve with parameters $(b,a)$ being $(1,1)$ in red; $(2,1)$ in green; $(3,1)$ in blue.",
    "Quartic_plane_curve_5": "Cruciform curve with parameters $(b,a)$ being $(1,1)$ in red; $(2,2)$ in green; $(3,3)$ in blue.",
    "Quartic_plane_curve_6": "Bow curve",
    "Quartic_plane_curve_7": "Bicuspid curve",
    "Arrangement_of_lines_1": "A simplicial line arrangement (left) and a simple line arrangement (right).",
    "Arrangement_of_lines_2": "Kobon triangles in an arrangement of 17 lines",
    "Arrangement_of_lines_3": "An arrangement with the minimum number of triangles according to Roberts's triangle theorem",
    "Arrangement_of_lines_4": "A non-stretchable pseudoline arrangement of nine pseudolines. (All arrangements of fewer than nine pseudolines are stretchable.) Per Pappus's hexagon theorem, this arrangement cannot be realized in a projective plane over any field.",
    "Arrangement_of_lines_5": "A hyperbolic line arrangement combinatorially equivalent to a chord diagram used by Ageev (1996) to show that triangle-free circle graphs may sometimes need 5 colors.",
    "Parametric_family_3": "Graphs of several quadratic polynomials, varying each of the three coefficients independently.",
    "Parametric_family_1": "A graph of the probability density functions of several normal distributions (from the same parametric family).",
    "Parametric_family_2": "A three-dimensional graph of a Cobb-Douglas production function.",
    "Sphere_1": "Two orthogonal radii of a sphere",
    "Sphere_2": "Sphere and circumscribed cylinder",
    "Sphere_3": "A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius: the radius of the sphere. This means that every point on the sphere will be an umbilical point.",
    "Sphere_4": "Great circle on a sphere",
    "Sphere_5": "Plane section of a sphere: one circle",
    "Sphere_6": "Coaxial intersection of a sphere and a cylinder: two circles",
    "Sphere_7": "Clelia spiral with $c = 8$",
    "Monomorphism_1": "pullback of monomorphism with itself",
    "Monomorphism_2": "Diagram in the definition of a strong monomorphism.",
    "Cross_product_9": "The cross product with respect to a right-handed coordinate system",
    "Cross_product_1": "Finding the direction of the cross product by the right-hand rule",
    "Cross_product_2": "The two nonequivalent triple cross products of three vectors $a$, $b$, $c$. In each case, two vectors define a plane, the other is out of the plane and can be split into parallel and perpendicular components to the cross product of the vectors defining the plane. These components can be found by vector projection and rejection. The triple product is in the plane and is rotated as shown.",
    "Cross_product_3": "Cross product distributivity over vector addition. Left: The vectors $b$ and $c$ are resolved into parallel and perpendicular components to $a$. Right: The parallel components vanish in the cross product, only the perpendicular components shown in the plane perpendicular to $a$ remain.",
    "Cross_product_4": "Cross product scalar multiplication. Left: Decomposition of $b$ into components parallel and perpendicular to $a$. Right: Scaling of the perpendicular components by a positive real number $r$ (if negative, $b$ and the cross product are reversed).",
    "Cross_product_5": "The area of a parallelogram as the magnitude of a cross product",
    "Cross_product_6": "Standard basis vectors $i$, $j$, $k$ and vector components of $a$, denoted here $a_x$, $a_y$, $a_z$",
    "Cross_product_7": "According to Sarrus's rule, the determinant of a $3 \\times 3$ matrix involves multiplications between matrix elements identified by crossed diagonals",
    "Cross_product_8": "Use of Sarrus's rule to find the cross product of $a$ and $b$",
    "Bipyramid_4": "The octahedron is dual to the cube",
    "Bipyramid_5": "Examples of rhombic bipyramids",
    "Bipyramid_6": "Example: ditetragonal bipyramid ($2n = 2\\times 4$)",
    "Bipyramid_1": "Examples of disphenoids and of an 8-faced scalenohedron",
    "Bipyramid_2": "A rhombus is a 2-dimensional analog of a right symmetric bipyramid",
    "Bipyramid_3": "Example: ditrigonal scalenohedron ($2n = 2\\times 3$)",
    "Tangential_quadrilateral_6": "A tangential quadrilateral (in blue) and its contact quadrilateral (in green) joining the four contact points between the incircle and the sides. Also shown are the tangency chords joining opposite contact points (in red) and the tangent lengths on the sides",
    "Tangential_quadrilateral_7": "A tangential quadrilateral (in blue) with its incircle (dashed line) and the four externally tangent circles (in red), each tangent to a given side and the extensions of the adjacent sides.",
    "Tangential_quadrilateral_1": "Tangential quadrilateral with inradius $r$",
    "Tangential_quadrilateral_2": "Construction of the Newton line (in red) of a tangential quadrilateral (in blue), showing the alignment of the incenter I, the midpoints of the diagonals $M_1$ and $M_2$ and the middle $M_3$ of the segment $JK$ (in green) joining the intersection of opposing sides.",
    "Tangential_quadrilateral_3": "A tangential quadrilateral is partitioned in four triangles meeting at its incenter $I$, their orthocenters (purple) and the intersection of the diagonals $P$ (in green) are all colinear,.",
    "Tangential_quadrilateral_4": "Chao and Simeonov's characterization in terms of the radii of circles within each of four triangles",
    "Binomial_coefficient_4": "Visualisation of binomial expansion up to the 4th power",
    "Binomial_coefficient_1": "The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above.",
    "Binomial_coefficient_2": "1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients, right-aligned. The left boundary of the image corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence.",
    "Binomial_coefficient_3": "Binomial coefficients $C(n, k)$ extended for negative and fractional $n$, illustrated with a simple binomial. It can be observed that Pascal's triangle is rotated and alternate terms are negated. The case $n = -1$ gives Grandi's series.",
    "Ellipse_32": "Construction of a directrix",
    "Ellipse_1": "An ellipse (red) obtained as the intersection of a cone with an inclined plane.",
    "Ellipse_2": "Ellipses: examples with increasing eccentricity",
    "Ellipse_3": "Notable points and line segments in an ellipse.",
    "Ellipse_4": "Ellipse: definition by sum of distances to foci",
    "Ellipse_5": "Ellipse: definition by focus and circular directrix",
    "Ellipse_6": "Shape parameters: a: semi-major axis, b: semi-minor axis, c: linear eccentricity, p: semi-latus rectum (usually $\\ell$).",
    "Ellipse_7": "A general ellipse in the plane can be uniquely described as a bivariate quadratic equation of Cartesian coordinates, or using center, semi-major and semi-minor axes, and angle",
    "Ellipse_8": "The construction of points based on the parametric equation and the interpretation of parameter $t$, which is due to de la Hire",
    "Ellipse_9": "Ellipse points calculated by the rational representation with equally spaced parameters ($\\Delta u=0.2$).",
    "Ellipse_10": "Ellipse as an affine image of the unit circle",
    "Ellipse_11": "Whirls: nested, scaled and rotated ellipses. The spiral is not drawn: we see it as the locus of points where the ellipses are especially close to each other.",
    "Ellipse_12": "Polar coordinates centered at the center.",
    "Ellipse_13": "Polar coordinates centered at focus.",
    "Ellipse_14": "Ellipse: directrix property",
    "Ellipse_15": "Pencil of conics with a common vertex and common semi-latus rectum",
    "Ellipse_16": "Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.",
    "Ellipse_17": "Rays from one focus reflect off the ellipse to pass through the other focus.",
    "Ellipse_18": "Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords.",
    "Ellipse_19": "Theorem of Apollonios",
    "Ellipse_20": "For the alternative area formula",
    "Ellipse_21": "Ellipse with its orthoptic",
    "Ellipse_22": "Central projection of circles (gate)",
    "Ellipse_23": "Ellipse: gardener's method",
    "Ellipse_24": "Ellipse construction: paper strip method 2",
    "Ellipse_25": "Approximation of an ellipse with osculating circles",
    "Ellipse_26": "Ellipse: Steiner generation",
    "Ellipse_27": "Circle: inscribed angle theorem",
    "Ellipse_28": "Inscribed angle theorem for an ellipse",
    "Ellipse_29": "The area enclosed by a tilted ellipse is $\\pi \\;y_{\\text{int}}\\,x_{\\text{max}}$.",
    "Ellipse_30": "Ellipses with same circumference",
    "Homothetic_center_8": "Family of tangent circles for the external homothetic center",
    "Homothetic_center_9": "Two circles have both types of homothetic centers, internal ($I$) and external ($E$). The radii of the circles $(r_1, r_2)$ are proportional to the distance (d) from each homothetic center. The points $A_1$, $A_2$ are homologous, as are the points $B_1$, $B_2$.",
    "Homothetic_center_10": "In a three circle configuration, three homothetic centers (one for each pair of circles) lie on a single line",
    "Homothetic_center_11": "The point $O$ is an external homothetic center for the two triangles. The size of each figure is proportional to its distance from the homothetic center.",
    "Homothetic_center_12": "Family of tangent circles for the internal homothetic center",
    "Homothetic_center_1": "Two geometric figures related by an external homothetic center $S$. The angles at corresponding points are the same and have the same sense; for example, the angles $\\angle ABC$, $\\angle A'B'C'$ are both clockwise and equal in magnitude.",
    "Homothetic_center_2": "The external (above) and internal (below) homothetic centers of the two circles (red) are shown as black points.",
    "Homothetic_center_3": "Lines through corresponding antihomologous points intersect on the radical axis of the two given circles (green and blue). The points $Q$, $P'$ are antihomologous, as are $S$, $R'$. These four points lie on a circle that intersects the two given circles; the lines through the intersection points of the new circle with the two given circles must intersect at the radical center $G$ of the three circles, which lies on the radical axis of the two given circles.",
    "Homothetic_center_4": "Every circle which is tangent to two given circles touches them at a pair of antihomologous points",
    "Homothetic_center_5": "The radical axis of tangent circles passes through the radical center",
    "Homothetic_center_6": "All six homothetic centers (dots) of three circles lie on four lines (thick lines)",
    "Homothetic_center_7": "The blue line is the radical axis of the two tangent circles $C_1$, $C_2$ (pink). Each pair of given circles has a homothetic center which belongs to the radical axis of the two tangent circles. Since the radical axis is a line this means that the three homothetic centers are collinear",
    "Tangent_lines_to_circles_10": "Inner tangent. The external tangent lines pass through the internal homothetic center.",
    "Tangent_lines_to_circles_11": "Tangents through a point",
    "Tangent_lines_to_circles_12": "Outer tangents to two circles",
    "Tangent_lines_to_circles_1": "The concept of a tangent line and tangent point can be generalized to a pole point $Q$ and its corresponding polar line $q$. The points $P$ and $Q$ are inverses of each other with respect to the circle.",
    "Tangent_lines_to_circles_2": "Finding outer tangent. Circle tangents.",
    "Tangent_lines_to_circles_3": "Construction of the inner tangent",
    "Tangent_lines_to_circles_4": "Construction of the outer tangent",
    "Tangent_lines_to_circles_5": "Finding outer tangent. Two circles' outer tangents.",
    "Tangent_lines_to_circles_6": "The external (above) and internal (below) homothetic center $S$ of the two circles.",
    "Tangent_lines_to_circles_7": "Construction of tangent lines to a circle ($C$) from a given exterior point ($P$).",
    "Tangent_lines_to_circles_8": "The angle $\\theta$ between a chord and a tangent is half the arc belonging to the chord.",
    "Tangent_lines_to_circles_9": "By the power-of-a-point theorem, the product of lengths $\\overline{PM} \\cdot \\overline{PN}$ for any ray $\\overline{PMN}$ equals the square of $\\overline{PT}$, the length of the tangent line segment (red).",
    "Spearman's_rank_correlation_coefficient_1": "A Spearman correlation of $1$ results when the two variables being compared are monotonically related, even if their relationship is not linear. This means that all data points with greater $\\textstyle x$ values than that of a given data point will have greater $\\textstyle y$ values as well. In contrast, this does not give a perfect Pearson correlation.",
    "Spearman's_rank_correlation_coefficient_2": "When the data are roughly elliptically distributed and there are no prominent outliers, the Spearman correlation and Pearson correlation give similar values.",
    "Spearman's_rank_correlation_coefficient_3": "The Spearman correlation is less sensitive than the Pearson correlation to strong outliers that are in the tails of both samples. That is because Spearman's $\\rho$ limits the outlier to the value of its rank.",
    "Spearman's_rank_correlation_coefficient_4": "A positive Spearman correlation coefficient corresponds to an increasing monotonic trend between $X$ and $Y$.",
    "Triangular_function_2": "Alternative triangle function",
    "Triangular_function_1": "Exemplary triangular function",
    "Problem_of_Apollonius_1": "A solution (in purple) to Apollonius's problem. The given circles are shown in black.",
    "Problem_of_Apollonius_2": "A conjugate pair of solutions to Apollonius's problem (pink circles), with given circles in black.",
    "Problem_of_Apollonius_3": "Two given circles (black) and a circle tangent to both (pink). The center-to-center distances $d_1$ and $d_2$ equal $r_1 + r_s$ and $r_2 + r_s$, respectively, so their difference is independent of $r_s$.",
    "Problem_of_Apollonius_4": "Four complementary pairs of solutions to Apollonius's problem; the given circles are black.",
    "Problem_of_Apollonius_5": "The set of points with a constant ratio of distances $d_1/d_2$ to two fixed points is a circle.",
    "Problem_of_Apollonius_6": "Inversion in a circle. The point $P\\prime$ is the inverse of point $P$ with respect to the circle.",
    "Problem_of_Apollonius_7": "A solution circle (pink) in the first family lies between concentric given circles (black). Twice the solution radius $r_s$ equals the difference $r_{outer} - r_{inner}$ of the inner and outer radii, while twice its center distance $d_s$ equals their sum.",
    "Problem_of_Apollonius_8": "The two tangent lines of the two tangent points of a given circle intersect on the radical axis $R$ (red line) of the two solution circles (pink). The three points of intersection on $R$ are the poles of the lines connecting the blue tangent points in each given circle (black).",
    "Problem_of_Apollonius_9": "The poles (red points) of the radical axis $R$ in the three given circles (black) lie on the green lines connecting the tangent points. These lines may be constructed from the poles and the radical center (orange).",
    "Problem_of_Apollonius_10": "An Apollonius problem with no solutions. A solution circle (pink) must cross the dashed given circle (black) to touch both of the other given circles (also black).",
    "Problem_of_Apollonius_11": "The two solutions (red) to Apollonius' problem with mutually tangent given circles (black), labeled by their curvatures.",
    "Problem_of_Apollonius_12": "A symmetrical Apollonian gasket, also called the Leibniz packing, after its inventor Gottfried Leibniz.",
    "Radial_basis_function_1": "Gaussian function for several choices of $\\varepsilon$",
    "Radial_basis_function_2": "Two unnormalized Gaussian radial basis functions in one input dimension. The basis function centers are located at $\\textstyle x_{1}=0.75$ and $\\textstyle x_{2}=3.25$.",
    "Compact_group_3": "The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication.",
    "Compact_group_1": "Example of the weights of a representation of the group SU(3)",
    "Compact_group_2": "Black dots indicate the dominant integral elements for the group SU(3)",
    "Inversive_geometry_7": "The inverse, with respect to the red circle, of a circle going through $O$ (blue) is a line not going through $O$ (green), and vice versa.",
    "Inversive_geometry_8": "Examples of inversion of circles $A$ to $J$ with respect to the red circle at $O$. Circles $A$ to $F$, which pass through $O$, map to straight lines. Circles $G$ to $J$, which do not, map to other circles. The reference circle and line $L$ map to themselves. Circles intersect their inverses, if any, on the reference circle.",
    "Inversive_geometry_9": "$P'$ is the inverse of $P$ with respect to the circle.",
    "Inversive_geometry_1": "To construct the inverse $P'$ of a point $P$ outside a circle $\\O$: Let $r$ be the radius of $\\O$. Right triangles $OPN$ and $ONP'$ are similar. $OP$ is to $r$ as $r$ is to $OP'$.",
    "Inversive_geometry_2": "The polar line $q$ to a point $Q$ with respect to a circle of radius $r$ centered on the point $O$. The point $P$ is the inversion point of $Q$; the polar is the line through $P$ that is perpendicular to the line containing $O$, $P$ and $Q$.",
    "Inversive_geometry_3": "Inversion of a sphere at the red sphere",
    "Inversive_geometry_4": "Inversion of a spheroid (at the red sphere)",
    "Inversive_geometry_5": "Inversion of a hyperboloid of one sheet",
    "Inversive_geometry_6": "Stereographic projection as an inversion of a sphere",
    "Limiting_parallel_1": "The two lines through a given point $P$ and limiting parallel to line $R$.",
    "End_(graph_theory)_1": "Part of an infinite grid graph, with vertices at the points where two grid lines meet. Despite having many different rays, it has only one end.",
    "Triangular_bipyramid_3": "The Goldner-Harary graph represents a triangular bipyramid, augmented by tetrahedra.",
    "Triangular_bipyramid_1": "Graph of a triangular bipyramid",
    "Triangular_bipyramid_2": "The known solution of the Thomson problem, with one a triangular bipyramid",
    "Mathematics_of_paper_folding_8": "$BQ$ is always rational if $AP$ is.",
    "Mathematics_of_paper_folding_1": "Crease pattern for a Miura fold. The parallelograms of this example have $84^\\circ$ and $96^\\circ$ angles.",
    "Mathematics_of_paper_folding_2": "Doubling the cube: $P_B/P_A = $ cube root of 2",
    "Mathematics_of_paper_folding_3": "The Beloch fold",
    "Mathematics_of_paper_folding_4": "A diagram showing the first and last step of how origami can double the cube",
    "Mathematics_of_paper_folding_5": "Angles around a vertex",
    "Mathematics_of_paper_folding_6": "Trisecting the angle $CAB$",
    "Mathematics_of_paper_folding_7": "Doubling the cube: $PB/PA =$ cube root of 2",
    "Signed_distance_function_3": "A more complicated set (top) and the graph of its signed distance function (bottom, in red).",
    "Signed_distance_function_1": "The graph (bottom, in red) of the signed distance between the points on the xy plane (in blue) and a fixed disk (also represented on top, in gray)",
    "Signed_distance_function_2": "Signed distance fields stored as raster images can be used to represent shapes.",
    "Exponential_integral_7": "$E_{1}(ix)$ against $x$; real part black, imaginary part red.",
    "Exponential_integral_1": "Plot of the exponential integral function $E_n(z)$ with $n=2$ in the complex plane from $-2-2i$ to $2+2i$ with colors created with Mathematica 13.1 function ComplexPlot3D",
    "Exponential_integral_2": "Plot of the exponential integral function $Ei(z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors created with Mathematica 13.1 function ComplexPlot3D",
    "Exponential_integral_3": "Plot of $E_{1}$ function (top) and $\\operatorname {Ei}$ function (bottom).",
    "Exponential_integral_4": "Relative error of the asymptotic approximation for different number $~N~$ of terms in the truncated sum",
    "Exponential_integral_5": "Bracketing of $E_{1}$ by elementary functions",
    "Exponential_integral_6": "Plot of the exponential integral function $E$ $n(z)$ with $n=2$ in the complex plane from $-2-2i$ to $2+2i$ with colors created with Mathematica 13.1 function ComplexPlot3D",
    "Pascal's_simplex_5": "First four components of Pascal's 4-simplex.",
    "Pascal's_simplex_1": "The first five layers of Pascal's 3-simplex (Pascal's pyramid). Each face (orange grid) is Pascal's 2-simplex (Pascal's triangle). Arrows show derivation of two example terms.",
    "Pascal's_simplex_2": "First four components of Pascal's line.",
    "Pascal's_simplex_3": "First four components of Pascal's triangle.",
    "Euler_angles_4": "Tait–Bryan angles: $z$–$x'$–$y''$ sequence (intrinsic rotations; $N$ coincides with $x'$)",
    "Euler_angles_1": "Classic Euler angles geometrical definition. Fixed coordinate system $(x, y, z)$ Rotated coordinate system $(X, Y, Z)$ Line of nodes $(N)$",
    "Euler_angles_2": "Projections of $x$-axis after three Tait–Bryan rotations. Notice that $\theta$ is a negative rotation around the axis $y'$.",
    "Euler_angles_3": "Tait–Bryan angles: $z$–$y'$–$x''$ sequence (intrinsic rotations; $N$ coincides with $y'$). The angle rotation sequence is $\\psi$, $\\theta$, $\\phi$. Note that in this case $\\psi > 90^\\circ$ and $\\theta$ is a negative angle.",
    "External_ray_7": "Dynamic external rays landing on repelling period 3 cycle and 3 internal rays landing on fixed point $\\alpha _{c}$",
    "External_ray_1": "Polar coordinate system and $\\psi _{c}$ for $c=-2$",
    "External_ray_2": "Boundary of Mandelbrot set as an image of unit circle under $\\Psi _{M}\\,$",
    "External_ray_3": "Uniformization of complement (exterior) of Mandelbrot set",
    "External_ray_4": "Julia set for $f_{c}(z)=z^{2}-1$ with 2 external ray landing on repelling fixed point alpha",
    "External_ray_5": "Julia set and 3 external rays landing on fixed point $\\alpha _{c}$",
    "External_ray_6": "Julia set with external rays landing on period 3 orbit",
    "Brownian_excursion_1": "A realization of Brownian Excursion.",
    "Petersen_graph_7": "A 4-coloring of the Petersen graph's edges",
    "Petersen_graph_1": "Petersen graph as Kneser graph $KG_{5,2}$",
    "Petersen_graph_2": "The Petersen graph has crossing number 2 and is 1-planar.",
    "Petersen_graph_3": "The Petersen graph is a unit distance graph: it can be drawn in the plane with each edge having unit length.",
    "Petersen_graph_4": "The Petersen graph and associated map embedded in the projective plane. Opposite points on the circle are identified, yielding a closed surface of non-orientable genus 1.",
    "Petersen_graph_5": "The Petersen graph is hypo-Hamiltonian: by deleting any vertex, such as the center vertex in the drawing, the remaining graph is Hamiltonian. This drawing with order-3 symmetry is the one given by Kempe (1886).",
    "Petersen_graph_6": "The Petersen family.",
    "Absolute_value_4": "Composition of absolute value with a cubic function in different orders",
    "Absolute_value_5": "The absolute value of a number may be thought of as its distance from zero.",
    "Absolute_value_1": "The graph of the absolute value function for real numbers",
    "Absolute_value_2": "The graph of the absolute value function for real numbers",
    "Absolute_value_3": "The absolute value of a complex number $z$ is the distance $r$ of $z$ from the origin. It is also seen in the picture that $z$ and its complex conjugate $\\bar {z}$ have the same absolute value.",
    "Logarithm_7": "The graph of the logarithm base 2 crosses the x-axis at $x = 1$ and passes through the points $(2, 1)$, $(4, 2)$, and $(8, 3)$, depicting, e.g., $\\log_2(8) = 3$ and $2^3 = 8$. The graph gets arbitrarily close to the y-axis, but does not meet it.",
    "Logarithm_1": "Plots of logarithm functions, with three commonly used bases. The special points $\\log_b b = 1$ are indicated by dotted lines, and all curves intersect in $\\log_b 1 = 0$.",
    "Logarithm_2": "Overlaid graphs of the logarithm for bases $1/2$, $2$, and $e$",
    "Logarithm_3": "The graph of the logarithm function $\\log_b(x)$ (blue) is obtained by reflecting the graph of the function $b^x$ (red) at the diagonal line $x = y$.",
    "Logarithm_4": "The graph of the natural logarithm (green) and its tangent at $x = 1.5$ (black)",
    "Logarithm_5": "The natural logarithm of $t$ is the shaded area underneath the graph of the function $f(x) = \\frac{1}{x}$.",
    "Logarithm_6": "A visual proof of the product formula of the natural logarithm",
    "Elliptic_integral_1": "Plot of the complete elliptic integral of the first kind $K(k)$",
    "Elliptic_integral_2": "Plot of the complete elliptic integral of the second kind $E(k)$",
    "Countable_set_2": "Enumeration for countable number of countable sets",
    "Countable_set_3": "Bijective mapping from integer to even numbers",
    "Countable_set_1": "The Cantor pairing function assigns one natural number to each pair of natural numbers",
    "Dihedral_group_of_order_6_5": "Positions of the six elements in the Cayley tableOnly the neutral elements are symmetric to the main diagonal, so this group is not abelian.",
    "Dihedral_group_of_order_6_1": "Cayley graph with permutations of a triangle",
    "Dihedral_group_of_order_6_2": "Cycle graph with permutation matrices of 3 elements(The generators a and b are the same as in the Cayley graph shown above.)",
    "Dihedral_group_of_order_6_3": "Cayley table as multiplication table of the permutation matrices",
    "Dihedral_group_of_order_6_4": "The orbits of 30 evenly spaced points on a circle under the group action of $D_3$",
    "Sign_(mathematics)_4": "Real sign function $y = \\operatorname{sgn}(x)$",
    "Sign_(mathematics)_1": "Real sign function $y = \\text{sgn}(x)$",
    "Sign_(mathematics)_2": "Measuring from the $x$-axis, angles on the unit circle count as positive in the counterclockwise direction, and negative in the clockwise direction.",
    "Sign_(mathematics)_3": "A number line with some integers labeled",
    "Normal_probability_plot_6": "A normal probability plot",
    "Normal_probability_plot_7": "Histogram of a sample from a uniform distribution - it looks multimodal and supposedly roughly symmetric.",
    "Normal_probability_plot_1": "Normal probability plot of a sample from a uniform distribution - it has an S shape.",
    "Normal_probability_plot_2": "Histogram of a sample from a right-skewed distribution - it looks unimodal and skewed right.",
    "Normal_probability_plot_3": "Normal probability plot of a sample from a right-skewed distribution - it has an inverted C shape.",
    "Normal_probability_plot_4": "Normal probability plot of a sample from a normal distribution - it looks fairly straight, at least when the few large and small values are ignored.",
    "Normal_probability_plot_5": "Histogram of a sample from a normal distribution - it looks fairly symmetric and unimodal.",
    "Relation_(mathematics)_1": "Illustration of an example relation on a set $A = \\{a, b, c, d\\}$. An arrow from $x$ to $y$ indicates that the relation holds between $x$ and $y$. The relation is represented by the set $\\{(a,a), (a,b), (a,d), (b,a), (b,d), (c,b), (d,c), (d,d)\\}$ of ordered pairs.",
    "Relation_(mathematics)_2": "Representation $R_{div}$ as Hasse diagram (black lines) and directed graph (all lines)",
    "Relation_(mathematics)_3": "The representation of the relation $\\mathrm{Rel} = { (x,y) \\in \\mathbb{R} \\times \\mathbb{R} \\mid x^{2} + xy + y^{2} = 1 }$ as a 2D-plot yields an ellipse.",
    "Inversion_(discrete_mathematics)_1": "The six possible inversions of a 4-element permutation",
    "Inversion_(discrete_mathematics)_2": "Permutohedron of the symmetric group $S_4$",
    "Surface_integral_5": "A curved surface $S$ with a vector field $F$ passing through it. The red arrows (vectors) represent the magnitude and direction of the field at various points on the surface",
    "Surface_integral_6": "The total flux through the surface is found by adding up $\\mathbf{F} \\cdot \\mathbf{n} \\; dS$ for each patch. In the limit as the patches become infinitesimally small, this is the surface integral $\\displaystyle \\iint_{S} \\mathbf{F} \\cdot \\mathbf{n} \\; dS$.",
    "Surface_integral_1": "The definition of the surface integral relies on splitting the surface into small surface elements.",
    "Surface_integral_2": "An illustration of a single surface element. These elements are made infinitesimally small, by the limiting process, so as to approximate the surface.",
    "Surface_integral_3": "Surface divided into small patches $dS = du\\,dv$ by a parameterization of the surface $[u(\\mathbf{x}), v(\\mathbf{x})]$.",
    "Surface_integral_4": "The flux through each patch is equal to the normal (perpendicular) component of the field $F_n(\\mathbf{x}) = F(\\mathbf{x}) \\cos \\theta$ at the patch's location $\\mathbf{x}$, multiplied by the area $dS$. The normal component is equal to the dot product of $\\mathbf{F}(\\mathbf{x})$ with the unit normal vector $\\mathbf{n}(\\mathbf{x})$ (blue arrows).",
    "Convex_uniform_honeycomb_11": "The alternated cubic honeycomb is one of 28 space-filling uniform tessellations in Euclidean 3-space, composed of alternating yellow tetrahedra and red octahedra.",
    "Convex_uniform_honeycomb_1": "The paracompact hexagonal tiling honeycomb, $\\{6,3,3\\}$, in perspective",
    "Convex_uniform_honeycomb_2": "The order-4 dodecahedral honeycomb, $\\{5,3,4\\}$ in perspective",
    "Convex_uniform_honeycomb_3": "Runcinated cubic honeycomb: Cubic cells become purple cubes. Faces become blue cubes. Edges become red cubes. Vertices become hidden (back) cube.",
    "Convex_uniform_honeycomb_4": "Elongated alternated cubic honeycomb",
    "Convex_uniform_honeycomb_5": "Gyroelongated triangular prismatic honeycomb",
    "Convex_uniform_honeycomb_6": "Gyrated triangular prismatic honeycomb",
    "Convex_uniform_honeycomb_7": "Honeycomb of triangular cupola and octahedra",
    "Convex_uniform_honeycomb_8": "Vertex uniform honeycomb, square cupola and tetrahedra",
    "Convex_uniform_honeycomb_9": "scaliform honeycomb with cube, square pyramid, and tetrahedra",
    "Riemann_surface_2": "The Riemann sphere and stereographic projection.",
    "Riemann_surface_1": "The Riemann surface for the multivalued complex function $f(z) = \\log(z)$ in a neighborhood of the origin. The $(x, y)$ coordinates are the coordinates of $z$ in the complex plane; the vertical coordinate represents the imaginary part of $f(z)$, whose real part is represented by hue.",
    "Transversal_(geometry)_3": "One pair of alternate angles. With parallel lines, they are congruent.",
    "Transversal_(geometry)_1": "One pair of corresponding angles. With parallel lines, they are congruent.",
    "Transversal_(geometry)_2": "One pair of consecutive angles. With parallel lines, they add up to two right angles",
    "Universal_property_8": "The most important arrow here is $u:F(A)\\to X$ which establishes the universal property.",
    "Universal_property_9": "The typical diagram of the definition of a universal morphism.",
    "Universal_property_1": "A morphism in the comma category is given by the morphism $h:B\\to B'$ which also makes the diagram commute.",
    "Universal_property_2": "This demonstrates the connection between a universal diagram being an initial object in a comma category.",
    "Universal_property_3": "This simply demonstrates the definition of a morphism in a comma category.",
    "Universal_property_4": "This shows that a terminal object in a specific comma category corresponds to a universal morphism.",
    "Universal_property_5": "Commutative diagram showing how products have a universal property.",
    "Universal_property_6": "Universal morphisms can behave like a natural transformation between functors under suitable conditions.",
    "Universal_property_7": "The unit and counit of an adjunction, which are natural transformations between functors, are an important example of universal morphisms.",
    "Padua_points_2": "Padua points of the first family and of degree 6, plotted with their generating curve.",
    "Padua_points_1": "Padua points of the first family and of degree 5, plotted with their generating curve.",
    "Regular_map_(graph_theory)_4": "The hexagonal hosohedron, a regular map on the sphere with two vertices, six edges, six faces, and 24 flags.",
    "Regular_map_(graph_theory)_5": "Regular map $\\{6,4\\}_{3}$ on the $\\{6,4\\}$ hyperbolic tiling, with element labels.",
    "Regular_map_(graph_theory)_1": "The regular map $\\{6,3\\}_{4,0}$ on the torus with 16 faces, 32 vertices and 48 edges.",
    "Regular_map_(graph_theory)_2": "The hemicube, a regular map.",
    "Regular_map_(graph_theory)_3": "Branko Grünbaum identified a double-covered cube $\\{8/2,3\\}$, with 6 octagonal faces, double wrapped, needing 24 edges, and 16 vertices. It can be seen as the regular map $\\{8,3\\}_{2,0}$ on a hyperbolic plane with 6 colored octagons.",
    "Branch-decomposition_1": "The four forbidden minors for graphs of branchwidth three.",
    "Windmill_graph_2": "Small windmill graphs.",
    "Windmill_graph_1": "The Windmill graph $Wd(5,4)$.",
    "Heilbronn_triangle_problem_1": "Six points in the unit square, with the smallest triangles (red) having area $\\frac{1}{8}$, the optimal area for this number of points. Other larger triangles are colored blue. These points are an affine transformation of a regular hexagon, but for larger numbers of points the optimal solution does not form a convex polygon.",
    "Heilbronn_triangle_problem_2": "Solutions to the no-three-in-line problem, large sets of grid points with no three collinear points, can be scaled into a unit square with minimum triangle area $\\Omega(1/n^2)$.",
    "Heilbronn_triangle_problem_3": "12 points in a square, 3 of 20 minimal triangles shaded",
    "Heilbronn_triangle_problem_4": "11 points in a square, 8 of 28 minimal triangles shaded",
    "Heilbronn_triangle_problem_5": "9 points in a square, 6 of 11 minimal triangles shaded",
    "Heilbronn_triangle_problem_6": "7 points in a square, all 8 minimal triangles shaded",
    "Heilbronn_triangle_problem_7": "Solutions to the no-three-in-line problem, large sets of grid points with no three collinear points, can be scaled into a unit square with minimum triangle area $\\Omega (1/n^{2})$.",
    "Heilbronn_triangle_problem_8": "Six points in the unit square, with the smallest triangles (red) having area $1/8$, the optimal area for this number of points. Other larger triangles are colored blue. These points are an affine transformation of a regular hexagon, but for larger numbers of points the optimal solution does not form a convex polygon.",
    "Sobolev_inequality_2": "Graphical representation of the embedding conditions. The space $W^{3,p}$, represented by a blue dot at the point $(1/p, 3)$, embeds into the spaces indicated by red dots, all lying on a line with slope $n$. The white circle at $(0,0)$ indicates the impossibility of optimal embeddings into $L^{\\infty}$.",
    "Sobolev_inequality_1": "If the line from the picture above intersects the y-axis at $s = r + \\alpha$, the embedding into a H\\\"older space $C^{r, \\alpha}$ (red) holds. White circles indicate intersection points at which optimal embeddings are not valid.",
    "Implicit_surface_6": "POV-Ray image (central projection) of an approximation of three tori.",
    "Implicit_surface_7": "Approximation of three tori (parallel projection)",
    "Implicit_surface_1": "Implicit surface of genus 2.",
    "Implicit_surface_2": "Implicit non-algebraic surface (wineglass).",
    "Implicit_surface_3": "POV-Ray image: metamorphoses between a sphere and a constant distance product surface (6 points).",
    "Implicit_surface_4": "Implicit surface torus ($R = 40, a = 15$).",
    "Dupin_cyclide_16": "Cyclide with two touching spheres with centers on the directrix hyperbola",
    "Dupin_cyclide_1": "top: ring cyclide with $x_{1}=3,x_{2}=1,x_{3}=-3,x_{4}=-7$ middle: horn cyclide with $x_{1}=1,x_{2}=3,x_{3}=-3,x_{4}=-7$ bottom: spindel cyclide with $x_{1}=3,x_{2}=-3,x_{3}=1,x_{4}=-7$",
    "Dupin_cyclide_2": "Meanings of the design parameters $a,b,c,d$: $d$ is the radius of the generating sphere at the co-vertices of the ellipse The two circles in the x-z-plane with centers $(\\pm a,0,0)$ have radii $d\\mp c$. Here: $a=1,\\;b=0.98\\to c=0.199$ and $d=0.3$",
    "Dupin_cyclide_3": "parabolic cyclide with its directrices (focal parabolas)",
    "Dupin_cyclide_4": "Dupin cyclide as channel surface (envelope of a family of spheres)",
    "Dupin_cyclide_5": "Maxwell: Property of the foci of the directrix ellipse. The ellipse is the equidistant set of the blue and purple circles.",
    "Dupin_cyclide_6": "Determining the design parameters a,b,c,d belonging to the given reals $x_{1},x_{2},x_{3},x_{4}$",
    "Dupin_cyclide_7": "Parallel surfaces of a ring cyclide with parameters $a=1,b=0.98,\\ d=0.30,0.45,0.60$",
    "Dupin_cyclide_8": "Maxwell-property for the hyperbola case. The hyperbola arm $H_{+}$ is an equidistant curve of both the grey circles.",
    "Dupin_cyclide_9": "Generation of a parabolic cyclide as channel surface",
    "Dupin_cyclide_10": "ring cyclide generated by an inversion of a cylinder at a sphere (magenta)",
    "Dupin_cyclide_11": "parabolic ring cyclide generated by an inversion of a cylinder containing the origin",
    "Dupin_cyclide_12": "horn cyclide generated by an inversion of a cone",
    "Dupin_cyclide_13": "ring cyclide generated by an inversion of a torus",
    "Dupin_cyclide_14": "Ring cyclide with Villarceau-circles",
    "Dupin_cyclide_15": "Cyclide (blue) as image by an inversion of a torus (black) at the unit sphere (red)",
    "Linking_number_4": "The Milnor invariants generalize linking number to links with three or more components, allowing one to prove that the Borromean rings are linked, though any two components have linking number 0.",
    "Linking_number_1": "The two curves of this (2, 8)-torus link have linking number four.",
    "Linking_number_2": "With six positive crossings and two negative crossings, these curves have linking number two.",
    "Linking_number_3": "The two curves of the Whitehead link have linking number zero.",
    "Uniform_tiling_1": "The elongated triangular tiling, the only non-Wythoffian convex uniform tiling",
    "Uniform_tiling_2": "The vertex figures for the six tilings with convex regular polygon and apeirogon faces. (The Wythoff symbol is given in red.)",
    "Uniform_tiling_3": "Vertex figures for 21 uniform tilings.",
    "Uniform_tiling_4": "This example, $4.8^* _{\\frac{\\pi}{4}} . 4.4^{**} \\frac{\\pi}{2} . 8^* \\frac{\\pi}{4}$, is considered not edge-to-edge due to the large square, although the latter can be interpreted as a star polygon with four pairs of collinear edges.",
    "Hyperbolic_functions_6": "The hyperbolic sine (red), hyperbolic cosine (green) and hyperbolic tangent (blue) graphed on the same axes.",
    "Hyperbolic_functions_1": "A ray through the unit hyperbola $x^2 - y^2 = 1$ at the point $(\\cosh a, \\sinh a)$, where $a$ is twice the area between the ray, the hyperbola, and the $x$-axis. For points on the hyperbola below the $x$-axis, the area is considered negative.",
    "Hyperbolic_functions_2": "$\\sinh, \\cosh, \\tanh$",
    "Hyperbolic_functions_3": "$\\sinh x$ is half the difference of $e^x$ and $e^{-x}$",
    "Hyperbolic_functions_4": "$\\cosh x$ is half the difference of $e^x$ and $e^{-x}$",
    "Injective_function_1": "The composition of two injective functions is injective.",
    "Injective_function_2": "An injective non-surjective function (injection, not a bijection)",
    "Injective_function_3": "An injective surjective function (bijection)",
    "Injective_function_4": "A non-injective surjective function (surjection, not a bijection)",
    "Injective_function_5": "A non-injective non-surjective function (also not a bijection)",
    "Geodesic_1": "Klein quartic with 28 geodesics (marked by 7 colors and 4 patterns)",
    "Geodesic_2": "A geodesic on a triaxial ellipsoid.",
    "Mohr's_circle_6": "Mohr's circle for a three-dimensional state of stress",
    "Mohr's_circle_7": "Stress components at a plane passing through a point in a continuum under plane stress conditions.",
    "Mohr's_circle_1": "Mohr's circles for a three-dimensional state of stress",
    "Mohr's_circle_2": "Stress in a loaded deformable material body assumed as a continuum.",
    "Mohr's_circle_3": "Stress transformation at a point in a continuum under plane stress conditions.",
    "Mohr's_circle_4": "Mohr's circle for plane stress and plane strain conditions (double angle approach). After a stress analysis, the stress components $\\sigma _{x}$, $\\sigma _{y}$, and $\\tau _{xy}$ at a material point $P$ are known. These stress components act on two perpendicular planes $A$ and $B$ passing through $P$. The coordinates of point $A$ and $B$ on the Mohr circle are the stress components acting on the planes $A$ and $B$ of the material element, respectively. The Mohr circle is then used to find the stress components $\\sigma _{\\mathrm {n} }$ and  $\\tau _{\\mathrm {n} }$, i.e., coordinates of any stress point $D$ on the circle, acting on any other plane $D$ passing through $P$. The angle between the lines ${\\overline {OB}}$ and ${\\overline {OD}}$ is double the angle  $\\theta$ between the normal vectors of planes $B$ and $D$ passing through $P$.",
    "Mohr's_circle_5": "Mohr's circle for plane stress and plane strain conditions (Pole approach). Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line.",
    "Median_1": "Mode, median and mean (expected value) of a probability density function",
    "Melnikov_distance_1": "Homoclinic manifolds $W^{s}(\\gamma (t))$ and $W^{u}(\\gamma (t))$indicated by $\\Gamma _{\\gamma }.$ The lines on $\\Gamma _{\\gamma }$ represent a typical trajectory of the system 4.",
    "Melnikov_distance_2": "Normal vector $\\pi _{p}$ to $\\Gamma _{\\gamma }$.",
    "Melnikov_distance_3": "Splitting of the manifolds giving $W^{s}(\\gamma _{\\epsilon }(t))$ and $W^{u}(\\gamma _{\\epsilon }(t))$ as projections in $\\Sigma ^{\\phi _{0}}.$",
    "Melnikov_distance_4": "Geometrical representation with respect to the crossing of the manifolds to the normal vector $\\pi _{p}.$",
    "Projective_linear_group_5": "The subgroups of the stabilizer of $\\{0, 1, \\infty\\}$ further stabilize the points $\\{-1, \\tfrac{1}{2}, 2\\}$ and $\\{\\zeta^{-}, \\zeta^{+}\\}$.",
    "Projective_linear_group_1": "Relation between the projective special linear group PSL and the projective general linear group PGL; each row and column is a short exact sequence. The set $(F^*)^n$ here is the set of $n$th powers of the multiplicative group of $F$.",
    "Projective_linear_group_2": "Rotation about the $z$ axes rotates the projective plane, while the projectivization of rotation about lines parallel to the $x$ or $y$ axes yield projective rotations of the plane.",
    "Projective_linear_group_3": "Some PSL groups arise as automorphism groups of Hurwitz surfaces, i.e., as quotients of the $(2,3,7)$ triangle group, which is the symmetries of the order-3 bisected heptagonal tiling.",
    "Projective_linear_group_4": "A projective representation of $G$ can be pulled back to a linear representation of a central extension $C$ of $G$. $K^* = K^\\times$.",
    "Regular_icosahedron_3": "Regular icosahedron and its non-convex variant, which differs from Jessen's icosahedron in having different vertex positions and non-right-angled dihedrals",
    "Regular_icosahedron_4": "Three mutually perpendicular golden ratio rectangles, with edges connecting their corners, form a regular icosahedron.",
    "Regular_icosahedron_1": "3D model of a regular icosahedron",
    "Regular_icosahedron_2": "Illustration of a icosahedral symmetry. The five-fold, three-fold, and two-fold are labeled in blue, red, and magenta respectively. The mirror planes are the cyan great circle.",
    "Dihedral_group_8": "The composition of these two reflections is a rotation.",
    "Dihedral_group_1": "The symmetry group of a snowflake is D6, a dihedral symmetry, the same as for a regular hexagon.",
    "Dihedral_group_2": "The six axes of reflection of a regular hexagon",
    "Dihedral_group_3": "The lines of reflection labelled $S_0$, $S_1$, and $S_2$ remain fixed in space (on the page) and do not themselves move when a symmetry operation (rotation or reflection) is done on the triangle (this matters when doing compositions of symmetries).",
    "Dihedral_group_4": "The symmetries of this pentagon are linear transformations of the plane as a vector space.",
    "Dihedral_group_5": "Example subgroups from a hexagonal dihedral symmetry",
    "Dihedral_group_6": "The four elements of $D_2$ (x-axis is vertical here)",
    "Dihedral_group_7": "$D_4$ is nonabelian (x-axis is vertical here).",
    "Tangential_polygon_1": "A tangential trapezoid",
    "Cumulative_distribution_function_4": "Cumulative distribution function for the normal distribution",
    "Cumulative_distribution_function_5": "CDF plot with two red rectangles, illustrating two inequalities",
    "Cumulative_distribution_function_1": "From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part.",
    "Cumulative_distribution_function_2": "Example of a cumulative distribution function with a countably infinite set of discontinuities.",
    "Cumulative_distribution_function_3": "Example of the folded cumulative distribution for a normal distribution function with an expected value of 0 and a standard deviation of 1.",
    "Student's_t-distribution_1": "Plot of the cumulative distribution functions of several members of the Student t-family of probability distributions.",
    "Isosceles_trapezoid_1": "Special cases of isosceles trapezoids",
    "Isosceles_trapezoid_2": "A kite and its dual isosceles trapezoid",
    "Hyperbolic_geometry_6": "Poincaré disk model with truncated triheptagonal tiling",
    "Hyperbolic_geometry_1": "Hypercycle and pseudogon in the Poincare disk model",
    "Hyperbolic_geometry_2": "An apeirogon and circumscribed horocycle in the Poincaré disk model.",
    "Hyperbolic_geometry_3": "Rhombitriheptagonal tiling of the hyperbolic plane, seen in the Poincaré disk model",
    "Hyperbolic_geometry_4": "A collection of crocheted hyperbolic planes, in imitation of a coral reef, by Institute For Figuring",
    "Hyperbolic_geometry_5": "Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions",
    "Logistic_regression_3": "Graph of a logistic regression curve fitted to the $(x_m,y_m)$ data. The curve shows the probability of passing an exam versus hours studying.",
    "Logistic_regression_1": "Comparison of logistic function with a scaled inverse probit function (i.e. the CDF of the normal distribution), comparing $\\sigma(x)$ vs. $\\Phi ({\\sqrt{\\frac{\\pi}{8}}}x)$, which makes the slopes the same at the origin. This shows the heavier tails of the logistic distribution.",
    "Logistic_regression_2": "The standard logistic function $\\sigma(t)$; $\\sigma(t) \\in (0,1)$ for all $t$.",
    "Generalized_Pareto_distribution_3": "The variance of the $exGPD(\\sigma ,\\xi )$ as a function of $\\xi$. Note that the variance only depends on $\\xi$. The red dotted line represents the variance evaluated at $\\xi =0$, that is, $\\psi '(1)=\\pi ^{2}/6$.",
    "Generalized_Pareto_distribution_1": "The pdf of the $exGPD(\\sigma ,\\xi )$ (exponentiated generalized Pareto distribution) for different values $\\sigma$ and $\\xi$.",
    "Generalized_Pareto_distribution_2": "GPD distribution functions for $\\mu = 0$ and different values of $\\sigma$ and $\\xi$",
    "Bilinear_interpolation_1": "The four red dots show the data points and the green dot is the point at which we want to interpolate.",
    "Bilinear_interpolation_2": "Example of bilinear interpolation in grayscale values",
    "Bilinear_interpolation_3": "A geometric visualisation of bilinear interpolation. The product of the value at the desired point (black) and the entire area is equal to the sum of the products of the value at each corner and the partial area diagonally opposite the corner (corresponding colours).",
    "Limit_of_a_sequence_3": "The plot of a convergent sequence $\\{a_n\\}$ is shown in blue. Here, one can see that the sequence is converging to the limit 0 as $n$ increases.",
    "Limit_of_a_sequence_1": "The plot of a Cauchy sequence $(x_n)$, shown in blue, as $x_{n}$ versus $n$. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as $n$ increases. In the real numbers every Cauchy sequence converges to some limit.",
    "Silver_ratio_8": "Silver harmonics: the rectangle and its coloured subzones have areas in ratios $7\\sigma + 3 : \\sigma^3 : \\sigma^2 : \\sigma : 1$.",
    "Silver_ratio_9": "A whirl of silver rectangles.",
    "Silver_ratio_1": "Silver rectangle in a regular octagon.",
    "Silver_ratio_2": "Rectangles with aspect ratios related to s tile the square.",
    "Silver_ratio_3": "Silver triangle and whirling gnomons.",
    "Silver_ratio_4": "Origami construction of a silver rectangle, with creases in green.",
    "Silver_ratio_5": "Powers of $\\sigma$ within a silver rectangle.",
    "Silver_ratio_6": "Diagonal segments of the silver rectangle measure the silver triangle. The ratio $AB : AS$ is $\\sigma$.",
    "Silver_ratio_7": "Silver spirals with different initial angles on a $\\sigma$-rectangle.",
    "Metric_space_1": "One possible approximation for the arc length of a curve. The approximation is never longer than the arc length, justifying the definition of arc length as a supremum.",
    "Metric_space_2": "The plane (a set of points) can be equipped with different metrics. In the taxicab metric the red, yellow and blue paths have the same length (12), and are all shortest paths. In the Euclidean metric, the green path has length $6{\\sqrt {2}}\\approx 8.49$, and is the unique shortest path, whereas the red, yellow, and blue paths still have length 12.",
    "Metric_space_3": "Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard",
    "Metric_space_4": "Diameter of a set.",
    "Metric_space_5": "Euler diagram of types of functions between metric spaces.",
    "Non-Euclidean_geometry_3": "Lambert quadrilateral in hyperbolic geometry",
    "Non-Euclidean_geometry_4": "Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions",
    "Non-Euclidean_geometry_1": "Behavior of lines with a common perpendicular in each of the three types of geometry",
    "Non-Euclidean_geometry_2": "Saccheri quadrilaterals in the three geometries",
    "Coxeter_graph_7": "in octagonmirror symmetry",
    "Coxeter_graph_1": "in 24-gon3-fold rotational symmetry",
    "Coxeter_graph_2": "3-fold dihedral symmetry(compare variant with Fano plane vertices)",
    "Coxeter_graph_3": "The graph obtained by any edge excision from the Coxeter is Hamilton-connected.",
    "Coxeter_graph_4": "The chromatic number of the Coxeter graph is 3.",
    "Coxeter_graph_5": "The rectilinear crossing number of the Coxeter graph is 11.",
    "Coxeter_graph_6": "The Coxeter graph is 1-planar.",
    "Monotonic_function_7": "With the nonmonotonic function \"if $a$ then both $b$ and $c$\", false nodes appear above true nodes.",
    "Monotonic_function_1": "A monotonically non-decreasing function",
    "Monotonic_function_2": "A monotonically non-increasing function",
    "Monotonic_function_3": "A function that is not monotonic",
    "Monotonic_function_4": "Monotonic function with a dense set of jump discontinuities",
    "Monotonic_function_5": "Plots of 6 monotonic growth functions",
    "Monotonic_function_6": "Hasse diagram of the monotonic function \"at least two of $a$, $b$, $c$ hold\". Colors indicate function output values.",
    "Cayley's_nodal_cubic_surface_1": "Real points of the Cayley surface",
    "Binomial_distribution_2": "The extracted formulas are: 1. $p = 0.5$ 2. $n$ and $k$ as in Pascal's triangle 3. $n = 8$ 4. $k = 4$ 5. $70/256$",
    "Binomial_distribution_1": "Binomial probability mass function and normal probability density function for $n = 6$ and $p = 0.5$",
    "Parabola_24": "Parabola: pole-polar relation",
    "Parabola_1": "Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.",
    "Parabola_2": "Parabola with axis parallel to y-axis; $p$ is the semi-latus rectum",
    "Parabola_3": "Parabola: general position",
    "Parabola_4": "When the parabola $\\color {blue}{y=2x^{2}}$ is uniformly scaled by factor 2, the result is the parabola $\\color {red}{y=x^{2}}$",
    "Parabola_5": "Pencil of conics with a common focus",
    "Parabola_6": "Pencil of conics with a common vertex",
    "Parabola_7": "Cone with cross-sections",
    "Parabola_8": "Parabola $y=ax^{2}$",
    "Parabola_9": "Parabola (red): side projection view and top projection view of a cone with a Dandelin sphere",
    "Parabola_10": "Reflective property of a parabola",
    "Parabola_11": "Parabola and tangent",
    "Parabola_12": "Parabola: pin string construction",
    "Parabola_13": "4-points property of a parabola",
    "Parabola_14": "3-points-1-tangent property",
    "Parabola_15": "2-points-2-tangents property",
    "Parabola_16": "Construction of the axis direction",
    "Parabola_17": "Steiner generation of a parabola",
    "Parabola_18": "Simpson's rule: the graph of a function is replaced by an arc of a parabola",
    "Parabola_19": "Parabola as an affine image of the unit parabola",
    "Parabola_20": "Midpoints of parallel chords",
    "Parabola_21": "Parabola (magenta) and line (lower light blue) including a chord (blue). The area enclosed between them is in pink. The chord itself ends at the points where the line intersects the parabola.",
    "Parabola_22": "Inscribed angles of a parabola",
    "Witch_of_Agnesi_1": "Selected witch of Agnesi curves (green), and the circles they are constructed from (blue), with radius parameters $a=1$, $a=2$, $a=4$, and $a=8$.",
    "Witch_of_Agnesi_2": "Agnesi's 1748 illustration of the curve and its construction",
    "Sigmoid_function_4": "Some sigmoid functions compared. In the drawing all functions are normalized in such a way that their slope at the origin is 1.",
    "Sigmoid_function_1": "Plot of the error function",
    "Sigmoid_function_2": "Inverted logistic S-curve to model the relation between wheat yield and soil salinity",
    "Napoleon_points_1": "First Napoleon point",
    "Napoleon_points_2": "Second Napoleon point",
    "Napoleon_points_3": "A point on the Kiepert hyperbola.",
    "Napoleon_points_4": "Generalization of Napoleon point: A special case",
    "Napoleon_points_5": "A generalization of Napoleon point",
    "Pole_and_polar_1": "If a point $A$ lies on the polar line $q$ of another point $Q$, then $Q$ lies on the polar line $a$ of $A$. This is sometimes referred to as La Hire's theorem. More generally, the polars of all the points on the line $q$ must pass through its pole $Q$.",
    "Pole_and_polar_2": "Line $p$ is the polar line to point $P$, $l$ to $L$ and $m$ to $M$",
    "Pole_and_polar_3": "$p$ is the polar line to point $P$; $m$ is the polar line to $M$",
    "Pole_and_polar_4": "The polar line $q$ to a point $Q$ with respect to a circle of radius $r$ centered on the point $O$. The point $P$ is the inversion point of $Q$; the polar is the line through $P$ that is perpendicular to the line containing $O$, $P$ and $Q$.",
    "Normal_(geometry)_5": "A polygon and its two normal vectors",
    "Normal_(geometry)_1": "A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.",
    "Normal_(geometry)_2": "Normal direction (in red) to a curve (in black).",
    "Normal_(geometry)_3": "Plane equation in normal form",
    "Normal_(geometry)_4": "A curved surface showing the unit normal vectors (blue arrows) to the surface",
    "Mixed_graph_1": "Example of a mixed graph",
    "Graph_coloring_1": "A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible.",
    "Graph_coloring_2": "This graph can be 3-colored in 12 different ways.",
    "Graph_coloring_3": "Two greedy colorings of the same graph using different vertex orders. The right example generalizes to 2-colorable graphs with $n$ vertices, where the greedy algorithm expends $n/2$ colors.",
    "Golomb_ruler_2": "Example of a conference room with proportions of a $[0, 2, 7, 8, 11]$ Golomb ruler, making it configurable to 10 different sizes.",
    "Golomb_ruler_3": "The perfect circular Golomb rulers (also called difference sets) with the specified order. (This preview should show multiple concentric circles. If not, click to view a larger version.)",
    "Golomb_ruler_1": "Golomb ruler of order 4 and length 6. This ruler is both optimal and perfect.",
    "Bessel_function_1": "Plot of Bessel function of the first kind, $J_{\\alpha }(x)$, for integer orders $\\alpha =0,1,2$.",
    "Bessel_function_2": "Plot of Bessel function of the second kind, $Y_{\\alpha }(x)$, for integer orders $\\alpha =0,1,2$",
    "Bessel_function_3": "Plot of the Bessel function of the second kind $Y_{\\alpha }(z)$ with $\\alpha =0.5$ in the complex plane from $-2-2i$ to $2+2i$.",
    "Perrin_number_1": "Spiral of equilateral triangles with side lengths equal to Perrin numbers.",
    "Perrin_number_2": "The Perrin function extends the sequence to real numbers.",
    "Supersilver_ratio_6": "The Pilgrim: a supersilver Rauzy fractal of type $a \\mapsto aba$. The three subtiles have areas in ratio $\\varsigma$.",
    "Supersilver_ratio_7": "Rectangles with aspect ratios related to powers of $\\varsigma$ tile the square.",
    "Supersilver_ratio_1": "Supersilver spirals with different initial angles on a $\\varsigma^-$ rectangle.",
    "Supersilver_ratio_2": "Powers of $\\varsigma$ within a supersilver rectangle.",
    "Supersilver_ratio_3": "If the area of the original rectangle is $\\varsigma$, the yellow, red, and blue copies have areas $\\frac{1}{\\varsigma}$, $\\frac{\\varsigma - 1}{\\varsigma}$, and $\\frac{(\\varsigma - 1)^2}{\\varsigma}$. The areas of the red and green rectangles are equal. $\\varsigma$ is the real solution of the cubic equation $x^3 = 2x^2 + 1$.",
    "Supersilver_ratio_4": "Hop o' my Thumb: a supersilver Rauzy fractal of type $a \\mapsto baa$. The fractal boundary has box-counting dimension $1.22$.",
    "Supersilver_ratio_5": "A supersilver Rauzy fractal of type $c \\mapsto bca$, with areas in the ratios $\\varsigma^2 + 1 : \\varsigma (\\varsigma - 1) : \\varsigma : 1$.",
    "Uniform_convergence_3": "The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions $f_{n}(x)=\\sin ^{n}(x)$ (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).",
    "Uniform_convergence_1": "A sequence of functions $(f_{n})$ converges uniformly to $f$ when for arbitrary small $\\varepsilon$ there is an index $N$ such that the graph of $f_{n}$ is in the $\\varepsilon$-tube around f whenever $n\\geq N.$",
    "Uniform_convergence_2": "Counterexample to a strengthening of the uniform convergence theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions $\\sin ^{n}(x)$ converge to the non-continuous red function. This can happen only if convergence is not uniform.",
    "British_flag_theorem_2": "According to the British flag theorem, the red squares have the same total area as the blue squares",
    "British_flag_theorem_1": "The British flag theorem in space, the red squares have the same total area as the blue squares",
    "Euler_brick_3": "Euler brick with edges $a, b, c$ and face diagonals $d, e, f$",
    "Euler_brick_1": "All five primitive Euler bricks with dimensions under 1000",
    "Euler_brick_2": "Euler brick with edges $a, b, c$ and face diagonals $d, e, f$",
    "Cardioid_12": "Cardioid generated by the inversion of a parabola across the unit circle (dashed)",
    "Cardioid_1": "Generation of a cardioid and the coordinate system used",
    "Cardioid_2": "Chords of a cardioid",
    "Cardioid_3": "Cardioid as envelope of a pencil of circles",
    "Cardioid_4": "Cardioid as envelope of a pencil of lines",
    "Cardioid_5": "Cremona's generation of a cardioid",
    "Cardioid_6": "Cardioid as caustic: light source $Z$, light ray ${\\vec {s}}$, reflected ray ${\\vec {r}}$",
    "Cardioid_7": "Cardioid as caustic of a circle with light source (right) on the perimeter",
    "Cardioid_8": "Point of cardioid is foot of dropped perpendicular on tangent of circle",
    "Cardioid_9": "A cardioid Evolute of the cardioid One point $P$; its centre of curvature $M$; and its osculating circle.",
    "Cardioid_10": "4 cardioids in polar representation and their position in the coordinate system",
    "Cardioid_11": "Boundary of the central, period 1, region of the Mandelbrot set is a precise cardioid.",
    "Graph_homomorphism_3": "$K_7$, the complete graph with 7 vertices, is a core.",
    "Graph_homomorphism_1": "The Grötzsch graph, incomparable to $K_3$",
    "Graph_homomorphism_2": "A homomorphism from the flower snark $J_5$ into the cycle graph $C_5$. It is also a retraction onto the subgraph on the central five vertices. Thus $J_5$ is in fact homomorphically equivalent to the core $C_5$.",
    "Vector_space_11": "Unit \"spheres\" in $\\mathbf {R} ^{2}$ consist of plane vectors of norm 1. Depicted are the unit spheres in different $p$-norms, for $p=1,2,$ and $\\infty .$ The bigger diamond depicts points of 1-norm equal to 2.",
    "Vector_space_1": "Addition of functions: the sum of the sine and the exponential function is $\\sin +\\exp :\\mathbb {R} \\to \\mathbb {R}$ with $(\\sin +\\exp )(x)=\\sin(x)+\\exp(x)$.",
    "Vector_space_2": "Describing an arrow vector $v$ by its coordinates $x$ and $y$ yields an isomorphism of vector spaces.",
    "Vector_space_3": "A typical matrix",
    "Vector_space_4": "The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors $r_1$, $r_2$, and $r_3$.",
    "Vector_space_5": "A line passing through the origin (blue, thick) in $R^3$ is a linear subspace. It is the intersection of two planes (green and yellow).",
    "Vector_space_6": "Commutative diagram depicting the universal property of the tensor product",
    "Vector_space_7": "An affine plane (light blue) in $R^3$. It is a two-dimensional subspace shifted by a vector $x$ (red).",
    "Vector_space_8": "Vector addition and scalar multiplication: a vector $\\mathbf{v}$ (blue) is added to another vector $\\mathbf{w}$ (red, upper illustration). Below, $\\mathbf{w}$ is stretched by a factor of 2, yielding the sum $\\mathbf{v} + 2\\mathbf{w}$.",
    "Vector_space_9": "A vector $\\mathbf{v}$ in $\\mathbb{R}^2$ (blue) expressed in terms of different bases: using the standard basis of $\\mathbb{R}^2$: $\\mathbf{v} = x\\mathbf{e}_1 + y\\mathbf{e}_2$ (black), and using a different, non-orthogonal basis: $\\mathbf{v} = \\mathbf{f}_1 + \\mathbf{f}_2$ (red).",
    "Vector_space_10": "A Möbius strip. Locally, it looks like $U \\times \\mathbf{R}$.",
    "Differential_calculus_5": "The graph of an arbitrary function $y=f(x)$. The orange line is tangent to $x=a$, meaning at that exact point, the slope of the curve and the straight line are the same.",
    "Differential_calculus_6": "The graph of $y=-2x+13$",
    "Differential_calculus_1": "The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.",
    "Differential_calculus_2": "The graph of $y=x^{2}$, with a straight line that is tangent to $(2,4)$. The slope of the tangent line is equal to $4$. (The axes of the graph do not use a 1:1 scale.)",
    "Differential_calculus_3": "The dotted line goes through the points $(2,4)$ and $(3,9)$, which both lie on the curve $y=x^{2}$. Because these two points are fairly close together, the dotted line and tangent line have a similar slope. As the two points become closer together, the error produced by the secant line becomes vanishingly small.",
    "Differential_calculus_4": "The mean value theorem: For each differentiable function $f:[a,b]\\to \\mathbb {R}$ with $a<b$ there is a $c\\in (a,b)$ with $f'(c)={\\tfrac {f(b)-f(a)}{b-a}}$.",
    "Structure_tensor_10": "The corresponding structure tensor ellipsoid.",
    "Structure_tensor_1": "Ellipsoidal representation of the 3D structure tensor.",
    "Structure_tensor_2": "The structure tensor ellipsoid of a surface-like neighborhood (\"surfel\"), where $\\lambda _{1}>\\!>\\lambda _{2}\\approx \\lambda _{3}$.",
    "Structure_tensor_3": "A 3D window straddling a smooth boundary surface between two uniform regions of a 3D image.",
    "Structure_tensor_4": "The corresponding structure tensor ellipsoid.",
    "Structure_tensor_5": "The structure tensor of a line-like neighborhood (\"curvel\"), where $\\lambda _{1}\\approx \\lambda _{2}>\\!>\\lambda _{3}$.",
    "Structure_tensor_6": "A 3D window straddling a line-like feature of a 3D image.",
    "Structure_tensor_7": "The corresponding structure tensor ellipsoid.",
    "Structure_tensor_8": "The structure tensor in an isotropic neighborhood, where $\\lambda _{1}\\approx \\lambda _{2}\\approx \\lambda _{3}$.",
    "Structure_tensor_9": "A 3D window containing a spherical feature of a 3D image.",
    "Conditional_probability_3": "A geometric visualization of Bayes' theorem. In the table, the values 2, 3, 6 and 9 give the relative weights of each corresponding condition and case. The figures denote the cells of the table involved in each metric, the probability being the fraction of each figure that is shaded. This shows that $P(A|B)P(B)=P(B|A)P(A)$ i.e. $P(A|B)={\\frac {P(B|A)}{P(A)|P(B)}}$ . Similar reasoning can be used to show that $P({\\bar {A}}|B)={\\frac {P(B|{\\bar {A}})P({\\bar {A}})}{P(B)}}$ etc.",
    "Conditional_probability_4": "Venn Pie Chart describing conditional probabilities",
    "Conditional_probability_1": "On a tree diagram, branch probabilities are conditional on the event associated with the parent node. (Here, the overbars indicate that the event does not occur.)",
    "Conditional_probability_2": "Illustration of conditional probabilities with an Euler diagram. The unconditional probability $P(A) = 0.30 + 0.10 + 0.12 = 0.52$. However, the conditional probabilities are $P(A \\mid B_1) = 1$, $P(A \\mid B_2) = \\frac{0.12}{0.12 + 0.04} = 0.75$, and $P(A \\mid B_3) = 0$.",
    "Gaussian_q-distribution_2": "The Gaussian q-density.",
    "Gaussian_q-distribution_1": "The Cumulative Gaussian q-distribution.",
    "Continuous_function_9": "Plot of the signum function. It shows that $\\lim_{n \\to \\infty} \\operatorname{sgn}\\left(\\tfrac{1}{n}\\right) \\neq \\operatorname{sgn}\\left(\\lim_{n \\to \\infty} \\tfrac{1}{n}\\right)$. Thus, the signum function is discontinuous at $0$.",
    "Continuous_function_10": "The function $f(x)={\\tfrac {1}{x}}$ is continuous on its domain ( $\\mathbb {R} \\setminus \\{0\\}$), but is discontinuous at $x=0,$ when considered as a partial function defined on the reals.",
    "Continuous_function_1": "The sequence $\\exp(1/n)$ converges to $\\exp(0) = 1$.",
    "Continuous_function_2": "Illustration of the $\\varepsilon$-$\\delta$-definition: at $x = 2$, any value $\\delta \\leq 0.5$ satisfies the condition of the definition for $\\varepsilon = 0.5$.",
    "Continuous_function_3": "The failure of a function to be continuous at a point is quantified by its oscillation.",
    "Continuous_function_4": "The graph of a cubic function has no jumps or holes. The function is continuous.",
    "Continuous_function_5": "The graph of a continuous rational function. The function is not defined for $x = - 2$. The vertical and horizontal lines are asymptotes.",
    "Continuous_function_6": "The sinc and the cos functions.",
    "Continuous_function_7": "For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone.",
    "Continuous_function_8": "Continuity at a point: For every neighborhood $V$ of $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U) \\subseteq V$.",
    "Jaccard_index_5": "The intersection of two sets in a Venn diagram",
    "Jaccard_index_1": "The probability Jaccard index can be interpreted as intersections of simplices.",
    "Jaccard_index_2": "A visual proof of the optimality of the Probability Jaccard Index on three element distributions.",
    "Jaccard_index_3": "The union of two sets in a Venn diagram",
    "Jaccard_index_4": "A visual equation for Intersection over Union (Jaccard Index).",
    "E_(mathematical_constant)_1": "Graph of the equation $y = \\frac{1}{x}$. Here, $e$ is the unique number larger than $1$ that makes the shaded area under the curve equal to $1$.",
    "E_(mathematical_constant)_2": "The graphs of the functions $x \\mapsto a^x$ are shown for $a = 2$ (dotted), $a = e$ (blue), and $a = 4$ (dashed). They all pass through the point $(0,1)$, but the red line (which has slope $1$) is tangent to only $e^x$ there.",
    "E_(mathematical_constant)_3": "The value of the natural logarithm function for argument $e$, i.e., $\\ln e$, equals $1$.",
    "E_(mathematical_constant)_4": "The global maximum of $x \\sqrt{x}$ occurs at $x = e$.",
    "Rhombicuboctahedron_2": "The graph of a rhombicuboctahedron",
    "Rhombicuboctahedron_1": "3D model of a rhombicuboctahedron",
    "Circles_of_Apollonius_4": "Proof of Apollonius' definition of a circle",
    "Circles_of_Apollonius_1": "Apollonius' definition of a circle.",
    "Circles_of_Apollonius_2": "Apollonius' problem may have up to eight solutions. The three given circles are shown in black, whereas the solution circles are colored.",
    "Circles_of_Apollonius_3": "A set of Apollonian circles. Every blue circle intersects every red circle at a right angle, and vice versa.",
    "Inellipse_7": "Determination of the inellipse by solving the problem for a hyperbola in an $\\xi$-$\\eta$ plane and transforming the solution into the $x$-$y$ plane. $M$ is the center of the sought inellipse, and $D_1D_2$, $E_1E_2$ are two conjugate diameters. In both planes, the essential points are denoted by the same symbols. $g_{\\infty}$ is the line at infinity in the $x$-$y$ plane.",
    "Inellipse_8": "Three mutually touching inellipses of a triangle",
    "Inellipse_1": "An inellipse of a triangle is uniquely determined by the vertices of the triangle and two points of contact $U,V$.",
    "Inellipse_2": "Brianchon point $K$",
    "Inellipse_3": "Mandart inellipse",
    "Inellipse_4": "Brocard inellipse",
    "Inellipse_5": "Incircle of a triangle",
    "Kepler_orbit_1": "A diagram of the various forms of the Kepler Orbit and their eccentricities. Blue is a hyperbolic trajectory $(e > 1)$. Green is a parabolic trajectory $(e = 1)$. Red is an elliptical orbit $(0 < e < 1)$. Grey is a circular orbit $(e = 0)$.",
    "Kepler_orbit_2": "An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation $r= \\frac {p}{1+e\\cdot \\cos \\theta}$",
    "Poisson_wavelet_5": "Members of the family of Poisson wavelets corresponding to $n = 1, 2, 3, 4$.",
    "Poisson_wavelet_1": "Image of the wavelet associated with the Poisson kernel.",
    "Poisson_wavelet_2": "Image of the Fourier transform of the wavelet associated with the Poisson kernel.",
    "Poisson_wavelet_3": "The graphs of the real parts of the Poisson wavelet $\\psi _{n}(t)$ for $n=1,2,3,4$.",
    "Poisson_wavelet_4": "The graphs of the imaginary parts of the Poisson wavelet $\\psi _{n}(t)$ for $n=1,2,3,4$.",
    "Uniform_4-polytope_4": "Like the 3-dimensional snub cube, an alternation removes half the vertices, in two chiral sets of vertices from the ringed form, however the uniform solution requires the vertex positions be adjusted for equal lengths. In four dimensions, this adjustment is only possible for 2 alternated figures, while the rest only exist as nonequilateral alternated figures.",
    "Uniform_4-polytope_1": "Schlegel diagram for the truncated 120-cell with tetrahedral cells visible",
    "Uniform_4-polytope_2": "Orthographic projection of the truncated 120-cell, in the H3 Coxeter plane (D10 symmetry). Only vertices and edges are drawn.",
    "Uniform_4-polytope_3": "The simplest of the duoprisms, the 3,3-duoprism, in Schlegel diagram, one of 6 triangular prism cells shown.",
    "Regression_analysis_1": "Regression line for 50 random points in a Gaussian distribution around the line $y=1.5x+2.$",
    "Regression_analysis_2": "Illustration of linear regression on a data set",
    "T-norm_5": "Graph of the drastic t-norm. The function is discontinuous at the lines $0 < x = 1$ and $0 < y = 1$.",
    "T-norm_1": "Graph of the minimum t-norm (3D and contours)",
    "T-norm_2": "Graph of the product t-norm",
    "T-norm_3": "Graph of the Hamacher product",
    "T-norm_4": "Graph of the nilpotent minimum. The function is discontinuous at the line $0 < x = 1 - y < 1$.",
    "Necklace_(combinatorics)_1": "The 11 bracelets with 2 red, 2 yellow and 2 green beads. The leftmost one and the four rightmost ones are chiral, so there are 16 necklaces.Compare box(6,7) in the triangle.",
    "Necklace_(combinatorics)_2": "16 tiles from the game Tantrix, corresponding to the 16 necklaces with 2 red, 2 yellow and 2 green beads.",
    "Necklace_(combinatorics)_3": "Possible patterns of bracelets of length ncorresponding to the k-th integer partition(set partitions up to rotation and reflection)",
    "Karnaugh_map_5": "Race hazards are present in this diagram.",
    "Karnaugh_map_1": "An example Karnaugh map.",
    "Karnaugh_map_2": "K-map drawn on a torus, and in a plane. The dot-marked cells are adjacent.",
    "Karnaugh_map_3": "Diagram showing two K-maps. The K-map for the function $f(A, B, C, D)$ is shown as colored rectangles which correspond to minterms. The brown region is an overlap of the red $2 \\times 2$ square and the green $4 \\times 1$ rectangle. The K-map for the inverse of $f$ is shown as gray rectangles, which correspond to maxterms.",
    "Karnaugh_map_4": "The value for $f(A, B, C, D)$ for $ABCD = 1111$ is replaced by a don't care. This removes the green term completely and allows the red term to be larger. It also allows blue inverse term to shift and become larger",
    "Conic_section_1": "The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone.",
    "Conic_section_2": "Conic sections visualized with torch light",
    "Conic_section_3": "This diagram clarifies the different angles of the cutting planes that result in the different properties of the three types of conic section.",
    "Conic_section_4": "Types of conic sections: 1: Circle 2: Ellipse 3: Parabola 4: Hyperbola",
    "Conic_section_5": "Conic parameters in the case of an ellipse",
    "Linear_function_(calculus)_1": "Graph of the linear function: $y(x)=-x+2$",
    "Linear_function_(calculus)_2": "The slope of a line is the ratio ${\\tfrac {\\Delta y}{\\Delta x}}$ between a change in $x$, denoted $\\Delta x$, and the corresponding change in y, denoted $\\Delta y$",
    "Pentagonal_icositetrahedron_6": "Spherical pentagonal icositetrahedron",
    "Pentagonal_icositetrahedron_1": "Pentagonal icositetrahedron, anticlockwise twist",
    "Pentagonal_icositetrahedron_2": "Combining an octahedron and snub cube to form the Pentagonal Icositetrahedron",
    "Pentagonal_icositetrahedron_3": "A geometric construction of the Tribonacci constant ($AC$), with compass and marked ruler, according to the method described by Xerardo Neira.",
    "Pentagonal_icositetrahedron_4": "Snub cube with augmented pyramids and merged faces",
    "Pentagonal_icositetrahedron_5": "3d model of a pentagonal icositetrahedron",
    "Mean_value_theorem_3": "Geometrical meaning of Cauchy's theorem",
    "Mean_value_theorem_1": "The function $f$ attains the slope of the secant between $a$ and $b$ as the derivative at the point $\\xi \\in (a,b)$.",
    "Mean_value_theorem_2": "It is also possible that there are multiple tangents parallel to the secant.",
    "Parallel_curve_9": "Normal fan for defining parallel curves around a sharp corner",
    "Parallel_curve_1": "Parallel curves of the graph of $y=1.5\\sin(x)$ for distances $d=0.25,\\dots ,1.5$",
    "Parallel_curve_2": "Two definitions of a parallel curve: 1) envelope of a family of congruent circles, 2) by a fixed normal distance",
    "Parallel_curve_3": "The parallel curves of a circle (red) are circles, too",
    "Parallel_curve_4": "Parallel curves of the implicit curve (red) with equation $x^{4}+y^{4}-1=0$",
    "Parallel_curve_5": "Involutes of a circle",
    "Parallel_curve_6": "Parallel curves to a curve with a discontinuous normal around a corner",
    "Parallel_curve_7": "Offset surface of a complex irregular shape",
    "Parallel_curve_8": "An envelope of ellipses forming two general offset curves above and below a given curve",
    "Viviani's_theorem_2": "For any interior point $P$, the sum of the lengths of the perpendiculars $s + t + u$ equals the height of the equilateral triangle.",
    "Viviani's_theorem_1": "Visual proof of Viviani's theorem 1. Shortest distances from point $P$ to sides of equilateral triangle $ABC$ are shown. 2. Lines $DE$, $FG$, and $HI$ parallel to $AB$, $BC$ and $CA$, respectively, and passing through $P$ define similar triangles $PHE$, $PFI$ and $PDG$. 3. As these triangles are equilateral, their altitudes can be rotated to be vertical. 4. As $PGCH$ is a parallelogram, triangle $PHE$ can be slid up to show that the altitudes sum to that of triangle $ABC$.",
    "Steiner_inellipse_3": "Equilateral triangle $\\triangle ABC$ Steiner inellipse Steiner ellipse",
    "Steiner_inellipse_1": "The Steiner inellipse. According to Marden's theorem, given the triangle with vertices $(1, 7)$, $(7, 5)$, $(3, 1)$, the foci of the inellipse are $(3, 5)$ and $(\\frac{13}{3}, \\frac{11}{3})$, since ${\\begin{aligned}&D_{x}(1+7i-x)(7+5i-x)(3+i-x)\\\\&=-3\\left({\\tfrac {13}{3}}+{\\tfrac {11}{3}}i-x\\right)(3+5i-x)\\end{aligned}}$",
    "Steiner_inellipse_2": "Arbitrary triangle $\\triangle ABC$ Steiner inellipse Steiner ellipse Major and minor axes",
    "Square_1": "Partitions of a square into three similar rectangles",
    "Square_2": "Among rectangles (top row), the square is the shape with equal sides (blue, middle). Among rhombuses (bottom row), the square is the shape with right angles (blue, middle).",
    "Square_3": "The area of a square is the product of the lengths of its sides.",
    "Square_4": "The axes of reflection symmetry and centers of rotation symmetry of a square (top), rectangle and rhombus (center), isosceles trapezoid, kite, and parallelogram (bottom)",
    "Square_5": "Three-point perspective of a cube, showing perspective transformations of its six square faces into six different quadrilaterals",
    "Square_6": "The inscribed circle (purple) and circumscribed circle (red) of a square (black)",
    "Square_7": "$|x|+|y|=2$ plotted on Cartesian coordinates.",
    "Square_8": "A square formed by multiplying the complex number $p$ by powers of $i$, and its translation obtained by adding another complex number $c$. The background grid shows the Gaussian integers.",
    "Square_9": "The Calabi triangle and the three placements of its largest square. The placement on the long side of the triangle is inscribed; the other two are not.",
    "Square_10": "A circle and square with the same area",
    "Square_11": "The smallest known square that can contain 11 unit squares has side length approximately 3.877084.",
    "Square_12": "Partitions of a square into three similar rectangles",
    "Square_13": "Metric circles using Chebyshev, Euclidean, and taxicab distance functions",
    "Square_14": "The Pythagorean theorem: the two smaller squares on the sides of a right triangle have equal total area to the larger square on the hypotenuse.",
    "Menelaus's_theorem_3": "Menelaus's theorem, case 2: line $DEF$ is entirely outside triangle $\\triangle ABC$",
    "Menelaus's_theorem_1": "Menelaus's theorem, case 1: line $DEF$ passes inside triangle $\\triangle ABC$",
    "Menelaus's_theorem_2": "Homothetie centers $D$, $E$ $F$ are colinear iff the composition is identity.",
    "Fundamental_pair_of_periods_3": "A lattice spanned by periods $\\omega_1$ and $\\omega_2$, showing an equivalent pair of periods $\\alpha_1$ and $\\alpha_2$.",
    "Fundamental_pair_of_periods_1": "Fundamental parallelogram defined by a pair of vectors in the complex plane.",
    "Fundamental_pair_of_periods_2": "The grey depicts the canonical fundamental domain.",
    "Arbelos_3": "An arbelos (grey region)",
    "Arbelos_1": "Some special points on the arbelos.",
    "Arbelos_2": "example of an f-belos",
    "Miquel's_theorem_1": "A diagram showing circles passing through the vertices of a triangle $ABC$ and points $A'$, $B'$ and $C'$ on the adjacent sides of the triangle intersecting at a common point, $M$.",
    "Miquel's_theorem_2": "The Pivot Theorem for various triangles",
    "Miquel's_theorem_3": "Miquel and Steiner's Quadrilateral Theorem",
    "Miquel's_theorem_4": "Miquel's six circles theorem states that if five circles share four triple-points of intersection then the remaining four points of intersection lie on a sixth circle",
    "Miquel's_theorem_5": "The three-dimensional case: the four spheres intercepts other spheres on black circles.",
    "Divergence_theorem_1": "A region V bounded by the surface $S=\\partial V$ with the surface normal n",
    "Divergence_theorem_2": "A volume divided into two subvolumes. At right the two subvolumes are separated to show the flux out of the different surfaces.",
    "Divergence_theorem_3": "The volume can be divided into any number of subvolumes and the flux out of $V$ is equal to the sum of the flux out of each subvolume, because the flux through the green surfaces cancels out in the sum. In (b) the volumes are shown separated slightly, illustrating that each green partition is part of the boundary of two adjacent volumes",
    "Divergence_theorem_4": "The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, like any of the surfaces on the left. It can not directly be used to calculate the flux through surfaces with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)",
    "Divergence_theorem_5": "A region $V$ bounded by the surface $S = \\partial V$ with the surface normal $n$",
    "Tetrahedral-octahedral_honeycomb_4": "This diagram shows an exploded view of the cells surrounding each vertex.",
    "Tetrahedral-octahedral_honeycomb_1": "Vertex figure with nonplanar 3.3.3.3 vertex configuration for the triangular bipyramids",
    "Tetrahedral-octahedral_honeycomb_2": "Gyroelongated alternated cubic honeycomb",
    "Tetrahedral-octahedral_honeycomb_3": "Part of a truncated alternated cubic honeycomb - a space filling tesselation of cuboctahedra (green), truncated octahedra (red) and truncated tetrahedra (blue).",
    "Snub_(geometry)_3": "Transforming rhombicuboctahedron into a snub cube visualized.",
    "Snub_(geometry)_1": "Two chiral copies of the snub cube, as alternated (red or green) vertices of the truncated cuboctahedron.",
    "Snub_(geometry)_2": "Orthogonal projection of snub 24-cell",
    "Simson_line_1": "The Simson line $LN$ (red) of the triangle $ABC$ with respect to point $P$ on the circumcircle",
    "Simson_line_2": "The projections of $Ap$, $Bp$, $Cp$ onto $BC$, $CA$, $AB$ are three collinear points",
    "Complete_bipartite_graph_2": "A complete bipartite graph of $K_{4,7}$ showing that Turán's brick factory problem with 4 storage sites (yellow spots) and 7 kilns (blue spots) requires 18 crossings (red dots)",
    "Complete_bipartite_graph_1": "The star graphs $K_{1,3}$, $K_{1,4}$, $K_{1,5}$, and $K_{1,6}$.",
    "Angle_of_parallelism_2": "Angle of parallelism in hyperbolic geometry",
    "Angle_of_parallelism_1": "The angle of parallelism, $\\Phi$, formulated as: (a) The angle between the $x$-axis and the line running from $x$, the center of $Q$, to $y$, the $y$-intercept of $Q$, and (b) The angle from the tangent of $Q$ at $y$ to the $y$-axis. This diagram, with yellow ideal triangle, is similar to one found in a book by Smogorzhevsky.",
    "Box_plot_7": "Box plot of data from the Michelson experiment",
    "Box_plot_1": "Box-plot with whiskers from minimum to maximum",
    "Box_plot_2": "Box-plot with whiskers drawn within the 1.5 IQR value",
    "Box_plot_3": "This is a picture of a box plot representing data",
    "Box_plot_4": "Four box plots, with and without notches and variable width",
    "Box_plot_5": "Box-plot and a probability density function (pdf) of a standard Normal $\\mathcal{N}(0, 1\\sigma^2)$ population.",
    "Box_plot_6": "Box-plots displaying the skewness of the data set",
    "Gradient_1": "The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).",
    "Gradient_2": "Plot of the graph of $f(x, y) = -\\left(\\cos(x^2) + \\cos(y^2)\\right)^2$, also showing its gradient projected on the bottom plane.",
    "Addition_9": "$2 + (1 + 3) = (2 + 1) + 3$ with segmented rods",
    "Addition_10": "A number-line visualization of the unary addition $2+4=6$. A translation by $4$ is equivalent to four translations by $1$.",
    "Addition_1": "Adding $\\pi ^{2}/6$ and $e$ using Cauchy sequences of rationals.",
    "Addition_2": "One set has three shapes while the other set has two. The total of shapes is five, which is a consequence of the addition of the objects from the two sets: $3+2=5$.",
    "Addition_3": "A number-line visualization of the algebraic addition $2+4=6$. A jump that has a distance of $2$ followed by another that is as long as $4$, is the same as a translation by $6$.",
    "Addition_4": "$4 + 2 = 2 + 4$ with blocks",
    "Addition_5": "$5 + 0 = 5$ with bags of dots",
    "Addition_6": "An addition with carry",
    "Addition_7": "Adding $\\pi^{2}/6$ and $e$ using Cauchy sequences of rationals.",
    "Rose_(mathematics)_1": "Roses specified by the sinusoid $r = \\cos(k\\theta)$ for various rational-numbered values of the angular frequency $k = \\frac{n}{d}$. For proper mathematical analysis, $k$ must be expressed in irreducible form.",
    "Rose_(mathematics)_2": "The rose specified by $r = \\cos(7\\theta)$. Since $k = 7$ is an odd number, the rose has $k = 7$ petals. Line segments connecting successive peaks lie on the circle $r = 1$ and will form a heptagon. The rose is inscribed in the circle $r = 1$.",
    "Ceva's_theorem_1": "Ceva's theorem, case 2: the three lines are concurrent at a point $O$ inside $\\triangle ABC$.",
    "Sequence_1": "A tiling with squares whose sides are successive Fibonacci numbers in length.",
    "Sequence_2": "The plot of a convergent sequence ($a_n$) is shown in blue. From the graph we can see that the sequence is converging to the limit zero as $n$ increases.",
    "Sequence_3": "The plot of a Cauchy sequence ($X_n$), shown in blue, as $X_n$ versus $n$. In the graph the sequence appears to be converging to a limit as the distance between consecutive terms in the sequence gets smaller as $n$ increases. In the real numbers every Cauchy sequence converges to some limit.",
    "Klein_bottle_7": "Dissecting the Klein bottle results in two Möbius strips.",
    "Klein_bottle_1": "A two-dimensional representation of the Klein bottle immersed in three-dimensional space",
    "Klein_bottle_2": "A 6-colored Klein bottle, the only exception to the Heawood conjecture",
    "Klein_bottle_3": "The \"figure 8\" immersion of the Klein bottle.",
    "Klein_bottle_4": "Klein bagel cross section, showing a figure eight curve (the lemniscate of Gerono).",
    "Klein_bottle_5": "The pinched torus immersion of the Klein bottle.",
    "Klein_bottle_6": "Klein Bottle with slight transparency",
    "Tent_map_4": "Bifurcation diagram for the tent map. Higher density indicates increased probability of the $x$ variable acquiring that value for the given value of the $\\mu$ parameter.",
    "Tent_map_1": "Graph of tent map function",
    "Tent_map_2": "Time series of the Tent map for the parameter $m = 2.0$ which shows numerical error: the plot of the time series (plot of $x$ variable with respect to number of iterations) stops fluctuating and no values are observed after $n = 50$. Parameter $m = 2.0$, initial point is random.",
    "Tent_map_3": "Orbits of unit-height tent map",
    "Simplex_1": "The numbers of faces in the above table are the same as in Pascal's triangle, without the left diagonal.",
    "Simplex_2": "The total number of faces is always a power of two minus one. This figure (a projection of the tesseract) shows the centroids of the 15 faces of the tetrahedron.",
    "Simplex_3": "The standard 2-simplex in $R^3$",
    "Curvilinear_coordinates_1": "Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.",
    "Curvilinear_coordinates_2": "Coordinate surfaces, coordinate lines, and coordinate axes of spherical coordinates. Surfaces: $r$ - spheres, $\\theta$ - cones, $\\Phi$ - half-planes; Lines: $r$ - straight beams, $\\theta$ - vertical semicircles, $\\Phi$ - horizontal circles; Axes: $r$ - straight beams, $\\theta$ - tangents to vertical semicircles, $\\Phi$ - tangents to horizontal circles",
    "Curvilinear_coordinates_3": "Transformation of local covariant basis in the case of general curvilinear coordinates",
    "Curvilinear_coordinates_4": "A vector v (red) represented by • a vector basis (yellow, left: $e_1, e_2, e_3$), tangent vectors to coordinate curves (black) and • a covector basis or cobasis (blue, right: $e^1, e^2, e^3$), normal vectors to coordinate surfaces (grey) in general (not necessarily orthogonal) curvilinear coordinates ($q^1, q^2, q^3$). The basis and cobasis do not coincide unless the coordinate system is orthogonal",
    "Polar_coordinate_system_9": "One arm of an Archimedean spiral with equation $r(\\varphi) = \\frac{\\varphi}{2\\pi}$ for $0 < \\varphi < 6\\pi$",
    "Polar_coordinate_system_10": "A diagram illustrating the relationship between polar and Cartesian coordinates.",
    "Polar_coordinate_system_1": "A polar grid with several angles, increasing in counterclockwise orientation and labelled in degrees",
    "Polar_coordinate_system_2": "An illustration of a complex number $z$ plotted on the complex plane",
    "Polar_coordinate_system_3": "An illustration of a complex number plotted on the complex plane using Euler's formula",
    "Polar_coordinate_system_4": "Points in the polar coordinate system with pole $O$ and polar axis $L$. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or $(3, 60^\\circ)$. In blue, the point $(4, 210^\\circ)$.",
    "Polar_coordinate_system_5": "A circle with equation $r(\\varphi) = 1$",
    "Polar_coordinate_system_6": "A polar rose with equation $r(\\varphi) = 2 \\sin(4\\varphi)$",
    "Polar_coordinate_system_7": "The integration region $R$ is bounded by the curve $r(\\varphi)$ and the rays $\\varphi = a$ and $\\varphi = b$.",
    "Polar_coordinate_system_8": "The region $R$ is approximated by $n$ sectors (here, $n = 5$).",
    "Near_sets_5": "Examples of degree of nearness between two sets: (a) High degree of nearness, and (b) Low degree of nearness.",
    "Near_sets_1": "Descriptively, very near sets",
    "Near_sets_2": "Descriptively, minimally near sets",
    "Near_sets_3": "Example of a descriptive EF-proximity relation between sets $A,B$, and $C^{c}$",
    "Near_sets_4": "Example depicting $\\delta$-neighbourhoods",
    "Ptolemy's_theorem_8": "Side of the inscribed decagon",
    "Ptolemy's_theorem_1": "Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral. ${AC}\\cdot {BD}={AB}\\cdot {CD}+{BC}\\cdot {AD}$",
    "Ptolemy's_theorem_2": "The golden ratio follows from this application of Ptolemy's theorem",
    "Ptolemy's_theorem_3": "Constructions for a proof of Ptolemy's theorem",
    "Ptolemy's_theorem_4": "Proof of Ptolemy's theorem via circle inversion",
    "Ptolemy's_theorem_5": "Corollary 1: Pythagoras's theorem",
    "Ptolemy's_theorem_6": "Corollary 2: the law of cosines",
    "Ptolemy's_theorem_7": "This is not a cyclic quadrilateral. The equality never holds here, and is unequal in the direction indicated by Ptolemy's inequality.",
    "Nakagami_distribution_2": "Cumulative function for Nakagami distribution.",
    "Nakagami_distribution_1": "Probability distribution function for Nakagami distribution.",
    "Logistic_distribution_3": "Standard logistic CDF",
    "Logistic_distribution_1": "Fitted cumulative logistic distribution to October rainfalls using CumFreq, see also Distribution fitting",
    "Logistic_distribution_2": "Standard logistic PDF",
    "Dimension_(graph_theory)_5": "The wheel graph with one spoke removed is of dimension 2.",
    "Dimension_(graph_theory)_1": "The complete bipartite graph $K_{4,2}$ drawn in Euclidean 3-space.",
    "Dimension_(graph_theory)_2": "A star graph drawn in the plane with edges of unit length.",
    "Dimension_(graph_theory)_3": "The same wheel is of Euclidean dimension 3.",
    "Root_mean_square_1": "Sine, square, triangle, and sawtooth waveforms. In each, the centerline is at 0, the positive peak is at $y = A_1$ and the negative peak is at $y = -A_1$.",
    "Split-quaternion_2": "Hyperboloid of one sheet, source of hyperbolic units. (the vertical axis is called $x$ in the article)",
    "Split-quaternion_1": "Hyperboloid of two sheets, source of imaginary units",
    "Area_7": "Triangle area $A={\\tfrac {b\\cdot h}{2}}$",
    "Area_1": "Although there are 10 mm in 1 cm, there are 100 $mm^2$ in 1 $cm^2$.",
    "Area_2": "The area of this rectangle is $lw$.",
    "Area_3": "A parallelogram can be cut up and re-arranged to form a rectangle.",
    "Area_4": "A parallelogram split into two equal triangles",
    "Area_5": "A circle can be divided into sectors which rearrange to form an approximate parallelogram.",
    "Area_6": "Integration can be thought of as measuring the area under a curve, defined by $f(x)$, between two points (here $a$ and $b$).",
    "Trilinear_interpolation_1": "Trilinear interpolation as two bilinear interpolations followed by a linear interpolation.",
    "Trilinear_interpolation_2": "A geometric visualisation of trilinear interpolation. The product of the value at the desired point and the entire volume is equal to the sum of the products of the value at each corner and the partial volume diagonally opposite the corner.",
    "Jensen's_inequality_3": "This is a proof without words of Jensen's inequality for n variables. Without loss of generality, the sum of the positive weights is 1. It follows that the weighted point lies in the convex hull of the original points, which lies above the function itself by the definition of convexity.",
    "Jensen's_inequality_1": "Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph.",
    "Jensen's_inequality_2": "A graphical \"proof\" of Jensen's inequality for the probabilistic case. The dashed curve along the $X$ axis is the hypothetical distribution of $X$, while the dashed curve along the Y axis is the corresponding distribution of $Y$ values. Note that the convex mapping $Y(X)$ increasingly \"stretches\" the distribution for increasing values of $X$.",
    "Wiener_process_1": "A single realization of a one-dimensional Wiener process",
    "Wiener_process_2": "A single realization of a three-dimensional Wiener process",
    "Wiener_process_3": "Wiener processes with drift (blue) and without drift (red).",
    "Wiener_process_4": "2D Wiener processes with drift (blue) and without drift (red).",
    "Locus_(mathematics)_1": "Each curve in this example is a locus defined as the conchoid of the point $P$ and the line $l$. In this example, $P$ is 8 cm from $l$.",
    "Locus_(mathematics)_2": "Locus of point $C$",
    "Locus_(mathematics)_3": "The locus is a circle",
    "Penrose_tiling_28": "Matching rule for Penrose rhombs using circular arcs or edge modifications to enforce the tiling rules",
    "Penrose_tiling_29": "A P1 tiling using Penrose's original set of six prototiles",
    "Penrose_tiling_30": "Tie and Navette tiling (in red on a Penrose background)",
    "Penrose_tiling_1": "A Penrose tiling with rhombi exhibiting fivefold symmetry",
    "Penrose_tiling_2": "Part of a periodic tiling with two prototiles",
    "Penrose_tiling_3": "An aperiodic set of Wang dominoes.",
    "Penrose_tiling_4": "Robinson's six prototiles",
    "Penrose_tiling_5": "The pentagonal Penrose tiling (P1) drawn in black on a colored rhombus tiling (P3) with yellow edges.",
    "Penrose_tiling_6": "A non-Penrose tiling by pentagons and thin rhombs in the early 18th-century Pilgrimage Church of Saint John of Nepomuk at Zelená hora, Czech Republic",
    "Penrose_tiling_7": "Penrose P1 tiles with overlaid circular arcs and nodes to enforce the tiling rules",
    "Penrose_tiling_8": "Matching rule for Penrose rhombs using circular arcs or edge modifications to enforce the tiling rules",
    "Penrose_tiling_9": "Matching rule for Penrose rhombs using parabolic edges to enforce the tiling rules",
    "Penrose_tiling_10": "A Penrose Tiling of type P3 using Penrose Rhombuses with parabolic edges",
    "Penrose_tiling_11": "Pentagon with an inscribed thick rhomb (light), acute Robinson triangles (lightly shaded) and a small obtuse Robinson triangle (darker). Dotted lines give additional edges for inscribed kites and darts.",
    "Penrose_tiling_12": "A pentagon decomposed into six smaller pentagons (half a dodecahedral net) with gaps",
    "Penrose_tiling_13": "Robinson triangles and their decompositions",
    "Penrose_tiling_14": "8th deflation of the 'sun' vertex in a Penrose tiling of type P2",
    "Penrose_tiling_15": "Gummelt's decagon (left) with the decomposition into kites and darts indicated by dashed lines; the thicker darker lines bound an inscribed ace and thick rhomb; possible overlaps (right) are by one or two red aces.",
    "Penrose_tiling_16": "A variant tiling which is not a quasicrystal. It is not a Penrose tiling because it does not comply with the tile alignment rules.",
    "Penrose_tiling_17": "A Penrose tiling with rhombi exhibiting fivefold symmetry",
    "Penrose_tiling_18": "Part of a periodic tiling with two prototiles",
    "Penrose_tiling_19": "An aperiodic set of Wang dominoes.",
    "Penrose_tiling_20": "Robinson's six prototiles",
    "Penrose_tiling_21": "Penrose $P1$ tiles with overlaid circular arcs and nodes to enforce the tiling rules",
    "Penrose_tiling_22": "Kite and dart tiles used in Penrose tiling, with two sorts of matching rule",
    "Penrose_tiling_23": "The seven possible vertex figures in a Penrose tiling by kites and darts. In Conway's terminology these are, from left to right, Star\", \"Ace\" and \"Sun\" (top row) and \"King\", \"Jack\", \"Queen\" and \"Deuce\" (bottom row).\"",
    "Penrose_tiling_24": "A Penrose Tiling of type $P3$ using Penrose Rhombuses with parabolic edges",
    "Penrose_tiling_25": "Matching rule for Penrose rhombs using parabolic edges to enforce the tiling rules",
    "Penrose_tiling_26": "Tie and Navette tiling (in red on a Penrose background)",
    "Penrose_tiling_27": "8th deflation of the sun vertex in a Penrose tiling of type $P2$",
    "Connected_space_1": "The topologist's sine curve is connected, but it is not locally connected",
    "Connected_space_2": "Examples of unions and intersections of connected sets",
    "Connected_space_3": "Two connected sets whose difference is not connected",
    "J-invariant_1": "The usual choice of a fundamental domain (gray) for the modular group acting on the upper half plane.",
    "J-invariant_2": "Real part of the j-invariant as a function of the square of the nome on the unit disk",
    "J-invariant_3": "Phase of the j-invariant as a function of the square of the nome on the unit disk",
    "Ptolemy's_inequality_2": "A cycle graph in which the distances disobey Ptolemy's inequality",
    "Ptolemy's_inequality_1": "Four points and their six distances. The points are not co-circular, so Ptolemy's inequality is strict for these points.",
    "Law_of_cosines_1": "A triangle. The angles $\\alpha$ (or $A$), $\\beta$ (or $B$), and $\\gamma$ (or $C$) are respectively opposite the sides $a$, $b$, and $c$.",
    "Law_of_cosines_2": "Applications of the law of cosines: unknown side and unknown angle.",
    "Law_of_cosines_3": "Given triangle sides $b$ and $c$ and angle $\\gamma$, there are sometimes two solutions for $a$.",
    "Law_of_cosines_4": "Obtuse triangle $ABC$ with perpendicular $BH$",
    "Law_of_cosines_5": "Al-Kashi's version of the law of cosines (case where $\\gamma$ is obtuse), expressed with modern algebraic notation.",
    "Law_of_cosines_6": "Proof for the Law of cosines in acute and obtuse triangles using the Pythagorean theorem",
    "Law_of_cosines_7": "An acute triangle with perpendicular",
    "Law_of_cosines_8": "Coordinate geometry proof",
    "Law_of_cosines_9": "Proof of law of cosines using Ptolemy's theorem",
    "Law_of_cosines_10": "Proof of the law of cosines for acute angle $\\gamma$ by cutting and pasting.",
    "Law_of_cosines_11": "Proof of the law of cosines for obtuse angle $\\gamma$ by cutting and pasting.",
    "Law_of_cosines_12": "The triangle $ABC$ (pink), an auxiliary circle (light blue) and an auxiliary right triangle (yellow)",
    "Law_of_cosines_13": "The triangle $ABC$ (pink), an auxiliary circle (light blue) and two auxiliary right triangles (yellow)",
    "Law_of_cosines_14": "Proof of the law of cosines using the power of a point theorem.",
    "Law_of_cosines_15": "Spherical triangle solved by the law of cosines.",
    "Triangle_inequality_5": "Three examples of the triangle inequality for triangles with sides of lengths $x$, $y$, $z$. The top example shows a case where $z$ is much less than the sum $x+y$ of the other two sides, and the bottom example shows a case where the side $z$ is only slightly less than $x+y$.",
    "Triangle_inequality_1": "Euclid's construction for proof of the triangle inequality for plane geometry.",
    "Triangle_inequality_2": "The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations.",
    "Triangle_inequality_3": "Triangle inequality for norms of vectors.",
    "Great_circle_2": "The great circle $g$ (green) lies in a plane through the sphere's center $O$ (black). The perpendicular line $a$ (purple) through the center is called the axis of $g$, and its two intersections with the sphere, $P$ and $P'$ (red), are the poles of $g$. Any great circle s (blue) through the poles is secondary to $g$.",
    "Great_circle_1": "A great circle divides the sphere in two equal hemispheres.",
    "Real_projective_plane_1": "The fundamental polygon of the projective plane - $A$ is identified with $A$ and $B$ is identified with $B$, each with a twist",
    "Real_projective_plane_2": "The Möbius strip - because of the twist between the identified red $A$ sides of the square, the dotted line is a single edge",
    "Real_projective_plane_3": "The tetrahemihexahedron is a polyhedral representation of the real projective plane.",
    "Real_projective_plane_4": "Two views of a cross-capped disk.",
    "Real_projective_plane_5": "Two views of a cross-capped disk which has been sliced open.",
    "Real_projective_plane_6": "Two alternative views of a self-intersecting disk.",
    "Golden_rectangle_6": "Construction of half-golden rectangle (central right triangle) from polygons.",
    "Golden_rectangle_1": "Construction of a golden rectangle.",
    "Golden_rectangle_2": "Golden spiral and intersecting diagonals.",
    "Golden_rectangle_3": "A whirl of golden rectangles.",
    "Golden_rectangle_4": "Powers of $\\varphi$ within a golden rectangle.",
    "Golden_rectangle_5": "Diagonal segments of the golden rectangle measure nested pentagons. The ratio $AU:SV$ is $\\varphi^2$.",
    "Magnetic_flux_3": "Some examples of closed surfaces (left) and open surfaces (right). Left: Surface of a sphere, surface of a torus, surface of a cube. Right: Disk surface, square surface, surface of a hemisphere. (The surface is blue, the boundary is red.)",
    "Magnetic_flux_1": "Each point on a surface is associated with a direction, called the surface normal; the magnetic flux through a point is then the component of the magnetic field along this direction.",
    "Magnetic_flux_2": "The magnetic flux through a surface—when the magnetic field is variable—relies on splitting the surface into small surface elements, over which the magnetic field can be considered to be locally constant. The total flux is then a formal summation of these surface elements",
    "Rule_of_Sarrus_2": "Alternative vertical arrangement",
    "Rule_of_Sarrus_1": "Rule of Sarrus: The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals.",
    "Taylor_series_1": "As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows $\\sin x$ and its Taylor approximations by polynomials of degree $\\boxed{1}$, $\\boxed{3}$, $\\boxed{5}$, $\\boxed{7}$, $\\boxed{9}$, $\\boxed{11}$, and $\\boxed{13}$ at $x = 0$.",
    "Taylor_series_2": "The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.",
    "Taylor_series_3": "The Taylor polynomials for $\\ln(1 + x)$ only provide accurate approximations in the range $-1 < x \\leq 1$. For $x > 1$, Taylor polynomials of higher degree provide worse approximations.",
    "Taylor_series_4": "Second-order Taylor series approximation (in orange) of a function $f(x, y) = e^x \\ln(1 + y)$ around the origin.",
    "Field_(mathematics)_4": "The sum of three points $P$, $Q$, and $R$ on an elliptic curve E (red) is zero if there is a line (blue) passing through these points.",
    "Field_(mathematics)_5": "The geometric mean theorem asserts that $h^2 = pq$. Choosing $q = 1$ allows construction of the square root of a given constructible number $p$.",
    "Field_(mathematics)_1": "The regular heptagon cannot be constructed using only a straightedge and compass construction; this can be proven using the field of constructible numbers.",
    "Field_(mathematics)_2": "The multiplication of complex numbers can be visualized geometrically by rotations and scalings.",
    "Field_(mathematics)_3": "Each bounded real set has a least upper bound.",
    "Congruence_(geometry)_1": "This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle $ABC$ and triangle $A'B'C'$, triangle $ABC$ is congruent with triangle $A'B'C'$ if and only if: angle $CAB$ is congruent with angle $C'A'B'$, and angle $ABC$ is congruent with angle $A'B'C'$, and $BC$ is congruent with $B'C'$. Note hatch marks are used here to show angle and side equalities.",
    "Congruence_(geometry)_2": "The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same perimeter and area. (The ordering of the sides of the blue quadrilateral is \"mixed\" which results in two of the interior angles and one of the diagonals not being congruent.)",
    "Congruence_(geometry)_3": "The shape of a triangle is determined up to congruence by specifying two sides and the angle between them ($SAS$), two angles and the side between them ($ASA$) or two angles and a corresponding adjacent side ($AAS$). Specifying two sides and an adjacent angle ($SSA$), however, can yield two distinct possible triangles.",
    "Finite_difference_method_3": "The finite difference method relies on discretizing a function on a grid.",
    "Finite_difference_method_1": "The stencil for the most common explicit method for the heat equation.",
    "Finite_difference_method_2": "The implicit method stencil.",
    "Geometric_mean_theorem_5": "Area of grey square = area of grey rectangle: $h^2 = pq \\Leftrightarrow h = \\sqrt{pq}$",
    "Geometric_mean_theorem_1": "geometric mean theorem as a special case of the intersecting chords theorem: $|CD||DE|=|AD||DB|\\Leftrightarrow h^{2}=pq$",
    "Geometric_mean_theorem_2": "Construction of $\\sqrt{p}$ by setting $q = 1$.",
    "Geometric_mean_theorem_3": "AM-GM inequality",
    "Geometric_mean_theorem_4": "Shear mappings with their associated fixed lines (dotted), starting with the original square as preimage each parallelogram displays the image of a shear mapping of the figure left of it",
    "Morin_surface_2": "Morin surface seen from the side",
    "Morin_surface_1": "Morin surface seen from the top",
    "Proofs_of_quadratic_reciprocity_8": "The number of points with even $x$-coordinate inside $BCYX$ (marked by $O$'s) is equal modulo 2 to the number of such points in $CZY$ (marked by $X$'s)",
    "Proofs_of_quadratic_reciprocity_1": "Lattice point diagram",
    "Proofs_of_quadratic_reciprocity_2": "The number of points with even x-coordinate inside $BCYX$ (marked by $O$'s) is equal modulo 2 to the number of such points in $CZY$ (marked by $X$'s)",
    "Proofs_of_quadratic_reciprocity_3": "The number of points with even x-coordinate inside $CZY$ is equal to the number of points with odd $x$-coordinate inside $AXY$",
    "Proofs_of_quadratic_reciprocity_4": "Example showing lattice points inside $ABC$ with even $x$-coordinates, for $p = 11$ and $q = 7$",
    "Proofs_of_quadratic_reciprocity_5": "Example showing lattice points inside ABC with even $x$-coordinates, for $p = 11$ and $q = 7$",
    "Proofs_of_quadratic_reciprocity_6": "Lattice point diagram",
    "Proofs_of_quadratic_reciprocity_7": "The number of points with even $x$-coordinate inside $CZY$ is equal to the number of points with odd $x$-coordinate inside $AXY$",
    "Unbiased_estimation_of_standard_deviation_1": "Bias in standard deviation for autocorrelated data.",
    "Symmetry_(geometry)_7": "The triskelion has 3-fold rotational symmetry.",
    "Symmetry_(geometry)_8": "A continuous helix",
    "Symmetry_(geometry)_1": "A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.",
    "Symmetry_(geometry)_2": "In 2 dimensions, a point reflection is a 180 degree rotation.",
    "Symmetry_(geometry)_3": "A frieze pattern with translational symmetry",
    "Symmetry_(geometry)_4": "A frieze pattern with glide reflection symmetry",
    "Symmetry_(geometry)_5": "A pentagonal antiprism with marked edges shows rotoreflectional symmetry, with an order of 10.",
    "Symmetry_(geometry)_6": "A regular skew-apeirogon has a discrete (3-fold here) screw-axis symmetry, drawn in perspective.",
    "Euler's_formula_4": "Relationship between sine, cosine and exponential function",
    "Euler's_formula_1": "A plot of the first few terms of the Taylor series of $e^{it}$ for $t = 2$.",
    "Euler's_formula_2": "Euler's formula $e^{i\\varphi} = \\cos \\varphi + i \\sin \\varphi$ illustrated in the complex plane.",
    "Euler's_formula_3": "Euler's formula and identity combined in diagrammatic form",
    "Projective_plane_1": "The Moulton plane. Lines sloping down and to the right are bent where they cross the y-axis.",
    "Projective_plane_2": "The Fano plane. Points are shown as dots; lines are shown as lines or circles.",
    "Projective_plane_3": "(Non-empty) degenerate projective planes",
    "Projective_plane_4": "Graph of the projective plane of order 7, having 57 points, 57 lines, 8 points on each line and 8 lines passing through each point, where each point is denoted by a rounded rectangle and each line by a combination of letter and number. Only lines with letter A and H are drawn. In the Dobble or Spot It! game, two points are removed. In the SVG file, hover over a line to highlight it.",
    "Taxicab_geometry_1": "In taxicab geometry, the lengths of the red, blue, green, and yellow paths all equal 12, the taxicab distance between the opposite corners, and all four paths are shortest paths. Instead, in Euclidean geometry, the red, blue, and yellow paths still have length 12 but the green path is the unique shortest path, with length equal to the Euclidean distance between the opposite corners, $6\\sqrt{2} \\approx 8.49$.",
    "Taxicab_geometry_2": "Curves defined by monotone increasing or decreasing functions have the same taxicab arc length as long as they share the same endpoints.",
    "Taxicab_geometry_3": "Two taxicab right isoceles triangles. Three angles and two legs are congruent, but the triangles are not congruent. Therefore, ASASA is not a congruence theorem in taxicab geometry.",
    "Taxicab_geometry_4": "Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard",
    "Playfair's_axiom_1": "Antecedent of Playfair's axiom: a line and a point not on the line",
    "Playfair's_axiom_2": "Consequent of Playfair's axiom: a second line, parallel to the first, passing through the point",
    "Simplex_algorithm_1": "Polyhedron of simplex algorithm in 3D",
    "Edge_coloring_1": "A 3-edge-coloring of the Desargues graph",
    "Edge_coloring_2": "Geometric construction of a 7-edge-coloring of the complete graph $K_8$. Each of the seven color classes has one edge from the center to a polygon vertex, and three edges perpendicular to it.",
    "Edge_coloring_3": "This 3-regular planar graph has 16 vertices and 24 edges, but only 7 edges in any maximum matching. Therefore, it requires four colors in any edge coloring.",
    "Translational_symmetry_2": "For translational invariant functions it is $f(A)=f(A+t)$. The Lebesgue measure is an example for such a function.",
    "Translational_symmetry_1": "The less-than-relation on the real numbers is invariant under translation.",
    "One-_and_two-tailed_tests_1": "A two-tailed test applied to the normal distribution.",
    "One-_and_two-tailed_tests_2": "A one-tailed test, showing the p-value as the size of one tail.",
    "One-_and_two-tailed_tests_3": "Normal distribution, showing two tails",
    "Ladder_graph_1": "The ladder rung graphs $LR_1$, $LR_2$, $LR_3$, $LR_4$ and $LR_5$.",
    "Ladder_graph_2": "Two views of the Möbius ladder $M_{16}$.",
    "Lemniscate_elliptic_functions_1": "The lemniscate sine (red) and lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric sine $y = \\sin\\left(\\frac{\\pi x}{\\varpi}\\right)$ (pale dashed red).",
    "Lemniscate_elliptic_functions_2": "The lemniscate sine function and hyperbolic lemniscate sine functions are defined as inverses of elliptic integrals. The complete integrals are related to the lemniscate constant $\\varpi$.",
    "Lemniscate_elliptic_functions_3": "Curves $x^2 \\oplus y^2 = a$ for various values of $a$. Negative $a$ in green, positive $a$ in blue, $a = \\pm 1$ in red, $a = \\infty$ in black.",
    "Lemniscate_elliptic_functions_4": "The trigonometric sine and cosine analogously relate the arc length of an arc of a unit-diameter circle to the distance of one endpoint from the origin.",
    "Lemniscate_elliptic_functions_5": "A lemniscate divided into 15 sections of equal arclength (red curves). Because the prime factors of 15 (3 and 5) are both Fermat primes, this polygon (in black) is constructible using a straightedge and compass.",
    "Lemniscate_elliptic_functions_6": "The lemniscate sine relates the arc length to the x coordinate in the rectangular elastica.",
    "Lemniscate_elliptic_functions_7": "The lemniscate elliptic functions and an ellipse",
    "Lemniscate_elliptic_functions_8": "The hyperbolic lemniscate sine (red) and hyperbolic lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric tangent (pale dashed red).",
    "Lemniscate_elliptic_functions_9": "Superellipse with the relation $x^{4}+y^{4}=1$",
    "Exponential_function_4": "The exponential function $e^z$ plotted in the complex plane from $-2-2i$ to $2+2i$",
    "Exponential_function_1": "Exponential functions with bases $2$ and $\\frac{1}{2}$",
    "Exponential_function_2": "The derivative of the exponential function is equal to the value of the function. Since the derivative is the slope of the tangent, this implies that all green right triangles have a base length of 1.",
    "Exponential_function_3": "Exponential functions with bases $2$ and $\\frac{1}{2}$",
    "Anger_function_2": "Plot of the Anger function $J(v(z))$ with $n=2$ in the complex plane from $-2-2i$ to $2+2i$",
    "Anger_function_1": "Plot of the Weber function $E(v(z))$ with $n=2$ in the complex plane from $-2-2i$ to $2+2i$",
    "Spiric_section_1": "Spiric sections on a spindle torus",
    "Power_of_a_point_8": "Power of a point with respect to a sphere",
    "Power_of_a_point_1": "Orthogonal circle (green)",
    "Power_of_a_point_2": "Angle between two circles",
    "Power_of_a_point_3": "Similarity points of two circles: various cases",
    "Power_of_a_point_4": "Similarity points of two circles and their common power",
    "Power_of_a_point_5": "Common power of two circles: application",
    "Power_of_a_point_6": "Circles tangent to two circles",
    "Power_of_a_point_7": "Circles tangent to two circles",
    "Graph_partition_1": "The graph $G = (5,4)$ is analysed for spectral bisection. The linear combination of the smallest two eigenvectors leads to $[1 1 1 1 1]'$ having an eigen value $= 0$.",
    "Graph_partition_2": "Weighted graph $G$ may be partitioned to maximize $Q$ in (a) or to minimize the ratio-cut in (b). We see that (a) is a better balanced partition, thus motivating the importance of modularity in graph partitioning problems.",
    "Cube_1": "3D model of a cube",
    "Cube_2": "The Prince Rupert's cube",
    "Cube_3": "The graph of a cube",
    "Young_tableau_1": "Young diagram of shape $(5, 4, 1)$, French notation",
    "Young_tableau_2": "A standard Young tableau of shape $(5, 4, 1)$: the numbers 1-10 in the boxes increase in every row and every column.",
    "Young_tableau_3": "All standard Young tableaux with at most 5 boxes",
    "Young_tableau_4": "Skew tableau of shape $(5, 4, 2, 2) / (2, 1)$, English notation",
    "Planar_separator_theorem_1": "A planar separator for a grid graph",
    "Planar_separator_theorem_2": "A polyhedron formed by replacing each of the faces of an icosahedron by a mesh of 100 triangles, an example of the lower bound construction of Djidjev (1982)",
    "Point_in_polygon_1": "The number of intersections for a ray passing from the exterior of the polygon to any point: If odd, it shows that the point lies inside the polygon; if even, the point lies outside the polygon. This test also works in three dimensions.",
    "Point_in_polygon_2": "Visualization of Dan Sunday's winding number algorithm. A winding number of 0 means the point is outside the polygon; other values indicate the point is inside the polygon",
    "Harnack's_curve_theorem_1": "The elliptic curve (smooth degree 3) on the left is an M-curve, as it has the maximum (2) components, while the curve on the right has only 1 component.",
    "Cardinality_1": "A bijection, comparing a set of apples to a set of oranges, showing they have the same cardinality.",
    "Cardinality_2": "A one-to-one correspondence from $N$, the set of all non-negative integers, to the set $E$ of non-negative even numbers. Although $E$ is a proper subset of $N$, both sets have the same cardinality.",
    "Cardinality_3": "Algebraic numbers on the complex plane, colored by degree",
    "Algebraic_curve_3": "$x^2 + xy + y^2 = 1$",
    "Algebraic_curve_1": "The Tschirnhausen cubic is an algebraic curve of degree three.",
    "Algebraic_curve_2": "$x^3 = y^2$",
    "Bhargava_cube_1": "The Bhargava cube corresponding to the binary cubic form $px^{3}+3qx^{2}y+3rxy^{2}+sy^{3}$.",
    "Level_set_3": "Intersections of a co-ordinate function's level surfaces with a trefoil knot. Red curves are closest to the viewer, while yellow curves are farthest.",
    "Level_set_1": "Log-spaced level curve plot of Himmelblau's function",
    "Level_set_2": "Consider a function $f$ whose graph looks like a hill. The blue curves are the level sets; the red curves follow the direction of the gradient. The cautious hiker follows the blue paths; the bold hiker follows the red paths. Note that blue and red paths always cross at right angles.",
    "Interval_arithmetic_10": "Multiplication of positive intervals",
    "Interval_arithmetic_1": "Tolerance function (turquoise) and interval-valued approximation (red)",
    "Interval_arithmetic_2": "Values of a monotonic function",
    "Interval_arithmetic_3": "Mean value form",
    "Interval_arithmetic_4": "Outer bounds at different level of rounding",
    "Interval_arithmetic_5": "Approximate estimate of the value range",
    "Interval_arithmetic_6": "Treating each occurrence of a variable independently",
    "Interval_arithmetic_7": "Reduction of the search area in the interval Newton step in \"thick\" functions.",
    "Interval_arithmetic_8": "Rough estimate (turquoise) and improved estimates through \"mincing\" (red)",
    "Interval_arithmetic_9": "Approximation of the normal distribution by a sequence of intervals",
    "Conoid_2": "parabolic conoid: directrix is a parabola",
    "Conoid_1": "Right circular conoid: Directrix is a circle Axis is perpendicular to the directrix plane",
    "Laman_graph_3": "Henneberg construction of the Moser spindle",
    "Laman_graph_1": "The Moser spindle, a planar Laman graph drawn as a pointed pseudotriangulation",
    "Laman_graph_2": "The complete bipartite graph $K_{3,3}$, a non-planar Laman graph",
    "Hermite_polynomials_2": "Hermite functions: 0 (blue, solid), 2 (orange, dashed), 4 (green, dot-dashed), and 50 (red, solid)",
    "Hermite_polynomials_1": "Hermite functions: 0 (blue, solid), 1 (orange, dashed), 2 (green, dot-dashed), 3 (red, dotted), 4 (purple, solid), and 5 (brown, dashed)",
    "Inverse_trigonometric_functions_6": "Principal values of the $\\arcsec(x)$ and $\\text{arccsc}(x)$ functions graphed on the Cartesian plane.",
    "Inverse_trigonometric_functions_1": "The sine and cosine function in the unit circle.",
    "Inverse_trigonometric_functions_2": "For a circle of radius 1, arcsin and arccos are the lengths of actual arcs determined by the quantities in question.",
    "Inverse_trigonometric_functions_3": "The points labelled $1$, $\\sec(\\theta)$, $\\csc(\\theta)$ represent the length of the line segment from the origin to that point. $\\sin(\\theta)$, $\\tan(\\theta)$, and $1$ are the heights to the line starting from the x-axis, while $\\cos(\\theta)$, $1$, and $\\cot(\\theta)$ are lengths along the x-axis starting from the origin.",
    "Inverse_trigonometric_functions_4": "The usual principal values of the $\\arcsin(x)$ (red) and $\\arccos(x)$ (blue) functions graphed on the Cartesian plane.",
    "Inverse_trigonometric_functions_5": "The usual principal values of the $\\arctan(x)$ and $\\text{arccot}(x)$ functions graphed on the Cartesian plane.",
    "Euclidean_vector_7": "Vector in a cartesian coordinate system. $a = a_x + a_y + a_z$",
    "Euclidean_vector_1": "A vector $\\textstyle {\\stackrel {\\rightarrow }{a}}$ pointing from point $A$ to point $B$",
    "Euclidean_vector_2": "A vector in the Cartesian plane, showing the position of a point $A$ with coordinates $(2, 3)$.",
    "Euclidean_vector_3": "Illustration of tangential and normal components of a vector to a surface.",
    "Euclidean_vector_4": "The addition of two vectors $a$ and $b$",
    "Euclidean_vector_5": "The subtraction of two vectors $a$ and $b$",
    "Euclidean_vector_6": "Scalar multiplication of a vector by a factor of 3 stretches the vector out.",
    "Skewness_1": "Example distribution with positive skewness. These data are from experiments on wheat grass growth.",
    "Skewness_2": "Example of an asymmetric distribution with zero skewness. This figure serves as a counterexample that zero skewness does not imply symmetric distribution necessarily. (Skewness was calculated by Pearson's moment coefficient of skewness.)",
    "Skewness_3": "A general relationship of mean and median under differently skewed unimodal distribution.",
    "Boltzmann_distribution_2": "Boltzmann factor $\\tfrac{p_{i}}{p_{j}}$ (vertical axis) as a function of temperature $T$ for several energy differences $\\varepsilon_i - \\varepsilon_j$.",
    "Boltzmann_distribution_1": "Boltzmann's distribution is an exponential distribution.",
    "Solenoid_(mathematics)_2": "A solid torus wrapped twice around inside another solid torus in $R^3$",
    "Solenoid_(mathematics)_1": "The Smale-Williams solenoid.",
    "Bayes'_theorem_6": "Tree diagram illustrating the beetle example. $R$, $C$, $P$ and $\\overline{P}$ are the events rare, common, pattern, and no pattern. Percentages in parentheses are calculated. Three independent values are given, so it is possible to calculate the inverse tree.",
    "Bayes'_theorem_1": "Bayes' theorem applied to an event space generated by continuous random variables X and Y with known probability distributions. There exists an instance of Bayes' theorem for each point in the domain. In practice, these instances might be parametrized by writing the specified probability densities as a function of x and y.",
    "Bayes'_theorem_2": "A way to conceptualize event spaces generated by continuous random variables $X$ and $Y$",
    "Bayes'_theorem_3": "Visual proof of Bayes' theorem",
    "Bayes'_theorem_4": "A geometric visualization of Bayes' theorem using astronauts, from the online game Among Us, who may be suspicious (with eyebrows) and may be assassins (with daggers)",
    "Bayes'_theorem_5": "Illustration of frequentist interpretation with tree diagrams",
    "Cantellation_(geometry)_1": "A cantellated cube - Red faces are reduced. Edges are bevelled, forming new yellow square faces. Vertices are truncated, forming new blue triangle faces.",
    "Cantellation_(geometry)_2": "A cantellated cubic honeycomb - Purple cubes are cantellated. Edges are bevelled, forming new blue cubic cells. Vertices are truncated, forming new red rectified cube cells.",
    "Polytope_1": "A polygon is a 2-dimensional polytope. Polygons can be characterised according to various criteria. Some examples are: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (including both its boundary and its interior), and self-intersecting with varying densities of different regions.",
    "Poincare_half-plane_model_5": "Distance from the apex of a semicircle to another point on it is the inverse Gudermannian function of the central angle.",
    "Poincare_half-plane_model_1": "Stellated regular heptagonal tiling of the model",
    "Poincare_half-plane_model_2": "Distance between two points can alternately be computed using ratios of Euclidean distances to the ideal points at the ends of the hyperbolic line.",
    "Poincare_half-plane_model_3": "Parallel rays in Poincare half-plane model of hyperbolic geometry",
    "Poincare_half-plane_model_4": "The distance between two points in the half-plane model can be computed in terms of Euclidean distances in an isosceles trapezoid formed by the points and their reflection across the x-axis: a side length $s$, a diagonal $d$, and two heights $h_1$ and $h_2$. It is the logarithm $\\text{dist}(p_1, p_2) = \\log\\left(\\frac{(s + d)^2}{h_1h_2}\\right)$.",
    "Matroid_representation_1": "The Vámos matroid, not linear over any field",
    "Homology_(mathematics)_1": "Cycles on a hemispherical projective plane $P^{2}$",
    "Homology_(mathematics)_2": "Cycles on a Klein bottle $K^{2}$",
    "Homology_(mathematics)_3": "Cycles on a torus $T^{2}$",
    "Homology_(mathematics)_4": "Cycles on a 2-sphere $S^{2}$",
    "Homology_(mathematics)_5": "The torus $T=S^{1} \\times S^{1}$",
    "Homology_(mathematics)_6": "The solid disc or 2-ball $B^{2}$",
    "Homology_(mathematics)_7": "The 2-sphere $S^{2}$ is the outer shell, not the interior, of a ball",
    "Homology_(mathematics)_8": "The circle or 1-sphere $S^{1}$",
    "Power_graph_analysis_1": "Two different power graphs that represent the same graph.",
    "Cevian_2": "A triangle with a cevian of length $d$",
    "Cevian_1": "Three cevians passing through a common point",
    "Point_location_1": "A planar subdivision divided into slabs.",
    "Point_location_2": "A monotone planar subdivision with some monotone chains highlighted.",
    "Point_location_3": "A trapezoidal decomposition.",
    "Vicsek_fractal_4": "Vicsek fractal (5th iteration of cross form)",
    "Vicsek_fractal_1": "Self-affine fractal built from a $3 \\times 2$ grid",
    "Vicsek_fractal_2": "Four iterations of the saltire form of the fractal (top) and the cross form of the fractal (bottom).",
    "Vicsek_fractal_3": "Anticross-stitch curve, iterations 0-4",
    "Apollonian_gasket_8": "In the limiting case $(0,0,1,1)$, the two largest circles are replaced by parallel straight lines. This produces a family of Ford circles.",
    "Apollonian_gasket_1": "An example of an Apollonian gasket",
    "Apollonian_gasket_2": "Mutually tangent circles. Given three mutually tangent circles (black), there are in general two other circles mutually tangent to them (red).",
    "Apollonian_gasket_3": "Integral Apollonian circle packing defined by circle curvatures of $(-1, 2, 2, 3)$",
    "Apollonian_gasket_4": "Integral Apollonian circle packing defined by circle curvatures of $(-3, 5, 8, 8)$",
    "Apollonian_gasket_5": "Integral Apollonian circle packing defined by circle curvatures of $(-12, 25, 25, 28)$",
    "Apollonian_gasket_6": "Integral Apollonian circle packing defined by circle curvatures of $(-6, 10, 15, 19)$",
    "Apollonian_gasket_7": "Integral Apollonian circle packing defined by circle curvatures of $(-10, 18, 23, 27)$",
    "Kernel_principal_component_analysis_1": "Output after kernel PCA with $k({\\boldsymbol {x}},{\\boldsymbol {y}})=({\\boldsymbol {x}}^{\\mathrm {T} }{\\boldsymbol {y}}+1)^{2}$. The three groups are distinguishable using the first component only.",
    "Kernel_principal_component_analysis_2": "Output after kernel PCA, with a Gaussian kernel.",
    "Projections_onto_convex_sets_2": "Example on two circles",
    "Projections_onto_convex_sets_1": "Example of averaged projections variant",
    "Lattice_(order)_5": "Non-lattice poset: $c$ and $d$ have no common upper bound.",
    "Lattice_(order)_1": "Non-lattice poset: $a$ and $b$ have common lower bounds $0,d,g,h,$ and $i,$ but none of them is the greatest lower bound.",
    "Lattice_(order)_2": "Non-lattice poset: $b$ and $c$ have common upper bounds $d,e,$ and $f,$ but none of them is the least upper bound.",
    "Lattice_(order)_3": "Smallest non-modular (and hence non-distributive) lattice $N_5$. $b\\leq c$, but $b\\vee (a\\wedge c)=b$ and $(b\\vee a)\\wedge c=c$, so the modular law is violated.The labelled elements also violate the distributivity equation $c\\wedge (a\\vee b)=(c\\wedge a)\\vee (c\\wedge b),$ but satisfy its dual $c\\vee (a\\wedge b)=(c\\vee a)\\wedge (c\\vee b).$",
    "Lattice_(order)_4": "Smallest non-distributive (but modular) lattice $M_3$.",
    "Stereographic_projection_1": "Stereographic projection of the unit sphere from the north pole onto the plane $z = 0$, shown here in cross section",
    "Stereographic_projection_2": "Stereographic projection of the unit sphere from the north pole onto the plane $z = -1$, shown here in cross section",
    "Stereographic_projection_3": "Stereographic projection of a sphere from a point $Q$ onto the plane $E$, shown here in cross section",
    "Stereographic_projection_4": "A Cartesian grid on the plane appears distorted on the sphere. The grid lines are still perpendicular, but the areas of the grid squares shrink as they approach the north pole.",
    "Stereographic_projection_5": "A polar grid on the plane appears distorted on the sphere. The grid curves are still perpendicular, but the areas of the grid sectors shrink as they approach the north pole.",
    "Stereographic_projection_6": "The sphere, with various loxodromes shown in distinct colors",
    "Stereographic_projection_7": "Illustration for deriving the relations between a stereographic projection and a circle inversion of a plane.",
    "Stereographic_projection_8": "Wulff net or stereonet, used for making plots of the stereographic projection by hand",
    "Stereographic_projection_9": "The generation of a Wulff net (circular net within the red circle) by a stereographic projection with center $C$ and projection plane $\\pi$",
    "Stereographic_projection_10": "Stereographic projection of the unit circle onto a line through the center illustrating the change in the length element for the Weierstrass substitution.",
    "Sign_function_1": "Signum function $y=\\operatorname {sgn} x$",
    "Projective_variety_1": "An elliptic curve is a smooth projective curve of genus one.",
    "Dual_curve_1": "Curves, dual to each other; see below for properties.",
    "Fermat_curve_1": "The Fermat cubic surface $X^{3}+Y^{3}=Z^{3}$",
    "Gaussian_integral_1": "A graph of the function $f(x)=e^{-x^{2}}$ and the area between it and the $x$-axis, (i.e. the entire real line) which is equal to ${\\sqrt {\\pi }}$.",
    "Harnack's_inequality_1": "A harmonic function (green) over a disk (blue) is bounded from above by a function (red) that coincides with the harmonic function at the disk center and approaches infinity towards the disk boundary.",
    "Circular_segment_1": "A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area).",
    "X_+_Y_sorting_1": "Geometric visualization of the $X+Y$ sorting problem. The input sets $X$ and $Y$ are represented by the sets of vertical and horizontal black lines (respectively), and the goal of the problem is to sort the crossing points by the positions of the red diagonal lines through them.",
    "Petzval_field_curvature_1": "Field curvature: the image \"plane\" (the arc) deviates from a flat surface (the vertical line).",
    "Quintic_function_1": "Graph of a polynomial of degree 5, with 3 real zeros (roots) and 4 critical points",
    "Trigonometric_functions_1": "Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.",
    "Cantor_function_1": "The graph of the Cantor function on the unit interval",
    "Tangent_bundle_1": "Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).",
    "Neovius_surface_1": "Neovius' minimal surface in a unit cell.",
    "Null_vector_1": "A null cone where $q(x,y,z)=x^{2}+y^{2}-z^{2}.$",
    "Series_(mathematics)_1": "Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.",
    "Labelled_enumeration_theorem_1": "A set of cycles being re-labelled to form a permutation. (There are three slots and $G=S_{3}$.)",
    "Plane_of_rotation_1": "Two different reflections in two dimensions generating a rotation.",
    "Tautochrone_curve_1": "Schematic of a cycloidal pendulum",
    "Hendecagon_1": "Symmetries of a regular hendecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edge. Gyration orders are given in the center.",
    "Synchrotron_function_1": "First and second Synchrotron functions.",
    "Leaky_integrator_1": "A graph of a solution to a leaky integrator; the input changes at $T=5$.",
    "K-set_(geometry)_1": "A set of six points (red), its six 2-sets (the sets of points contained in the blue ovals), and lines separating each $k$-set from the remaining points (dashed black).",
    "Laguerre's_method_1": "Attraction zones of Laguerre's method for the polynomial $p(x)=x^{4}+2x^{3}+3x^{2}+4x+1.$",
    "Euler_spiral_1": "A double-end Euler spiral. The curve continues to converge to the points marked, as $t$ tends to positive or negative infinity.",
    "Inclined_plane_1": "The inclined plane's geometry is based on a right triangle. The horizontal length is sometimes called Run, the vertical change in height Rise.",
    "Maximum_modulus_principle_1": "A plot of the modulus of $\\cos(z)$ (in red) for $z$ in the unit disk centered at the origin (shown in blue). As predicted by the theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).",
    "Langevin_equation_1": "Phase portrait of a harmonic oscillator showing spreading due to the Langevin Equation.",
    "Procrustes_analysis_1": "Procrustes superimposition. The figure shows the three transformation steps of an ordinary Procrustes fit for two configurations of landmarks. (a) Scaling of both configurations to the same size; (b) Transposition to the same position of the center of gravity; (c) Rotation to the orientation that provides the minimum sum of squared distances between corresponding landmarks.",
    "Tucker's_lemma_1": "In this example, where $n=2$, the red 1-simplex has vertices which are labelled by the same number with opposite signs. Tucker's lemma states that for such a triangulation at least one such 1-simplex must exist.",
    "Perfect_number_1": "Illustration of the perfect number status of the number 6",
    "Annulus_(mathematics)_1": "Illustration of Mamikon's visual calculus method showing that the areas of two annuli with the same chord length are the same regardless of inner and outer radii.",
    "Commutative_property_1": "The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.",
    "Semisimple_Lie_algebra_1": "The simple Lie algebras are classified by the connected Dynkin diagrams.",
    "Confluent_hypergeometric_function_1": "Plot of the Kummer confluent hypergeometric function $1F1(a;b;z)$ with $a=1$ and $b=2$ and input $z^2$ with $1F1(1,2,z^2)$ in the complex plane from $-2-2i$ to $2+2i$ with colors.",
    "Faddeeva_function_1": "Faddeeva function on the complex plane",
    "Arrangement_of_hyperplanes_1": "A hyperplane arrangement in space",
    "Curved_space_1": "In a flat space, the sum of the squares of the side of a right-angled triangle is equal to the square of the hypotenuse. This relationship does not hold for curved spaces.",
    "216_(number)_1": "Visual proof that $3^3 + 4^3 + 5^3 = 6^3$",
    "Absorbing_Markov_chain_2": "A Markov Chain with 4 states for the String Generation problem.",
    "Absorbing_Markov_chain_3": "The cumulative probability of finishing a game of Snakes and Ladders by turn $N$",
    "Abstract_simplicial_complex_1": "Geometric realization of a 3-dimensional abstract simplicial complex",
    "Acnode_1": "An acnode at the origin.",
    "Acyclic_orientation_1": "The 14 different acyclic orientations of a 4-vertex cycle graph, assignments of a direction to each cycle edge that make the resulting directed graph acyclic. Two orientations are shown as adjacent when they differ in the direction of a single edge.",
    "Adjoint_functors_2": "The existence of the unit, a universal morphism, can prove the existence of an adjunction.",
    "Adjoint_functors_3": "Naturality of $\\Phi$",
    "Adjoint_functors_4": "String diagram for adjunction.",
    "Affine_connection_1": "An affine connection on the sphere rolls the affine tangent plane from one point to another. As it does so, the point of contact traces out a curve in the plane: the development.",
    "Affine_connection_2": "Parallel transport of a tangent vector along a curve in the sphere.",
    "Affine_connection_3": "Historically, people used the covariant derivative (or Levi-Civita connection given by the metric) to describe the variation rate of a vector along the direction of another vector. Here on the punctured 2-dimensional Euclidean space, the blue vector field $X$ sends the one-form $dr$ to $0.07$ everywhere. The red vector field $Y$ sends the one-form $rd\\theta$ to $0.5r$ everywhere. Endorsed by the metric $ds^2 = dr^2 + r^2 d\\theta^2$, the Levi-Civita connection $\\nabla_Y X$ is $0$ everywhere, indicating $X$ has no change along $Y$. In other words, $X$ parallel transports along each concentric circle. $\\nabla_X Y = Y/r$ everywhere, which sends $rd\\theta$ to $0.5$ everywhere, implying $Y$ has a constant changing rate in the radial direction.",
    "Affine_geometry_2": "Pappus' law: if the red lines are parallel and the blue lines are parallel, then the dotted black lines must be parallel.",
    "Affine_plane_(incidence_geometry)_1": "Affine plane of order 3 9 points, 12 lines",
    "Affine_space_1": "In $\\mathbb {R} ^{3},$ the upper plane (in blue) $P_{2}$ is not a vector subspace, since $\\mathbf {0} \\notin P_{2}$ and $\\mathbf {a} +\\mathbf {b} \\notin P_{2};$ it is an affine subspace. Its direction (the linear subspace associated with this affine subspace) is the lower (green) plane $P_{1},$, which is a vector subspace. Although $\\mathbf {a}$ and $\\mathbf {b}$ are in $P_{2},$ their difference is a displacement vector, which does not belong to $P_{2},$ but belongs to vector space $P_{1}.$",
    "Affine_variety_1": "A cubic plane curve given by $y^{2}=x^{2}(x+1)$",
    "Affine_variety_2": "A drawing of the real points of the curve $y^2 = x^3 - x^2 - 16x$.",
    "Airy_zeta_function_1": "The Airy functions $Ai$ and $Bi$",
    "Algebraic_connectivity_1": "An example graph, with 6 vertices, diameter 3, connectivity 1, and algebraic connectivity 0.722",
    "Algebraic_connectivity_2": "The truncated icosahedron or Buckminsterfullerene graph has a traditional connectivity of 3, and an algebraic connectivity of 0.243.",
    "Algebraic_connectivity_3": "Partitioning into two connected graphs",
    "Algebraic_geometry_1": "Parabola ($y = x^2$, red) and cubic ($y = x^3$, blue) in projective space",
    "Algebraic_geometry_2": "This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.",
    "Algebraic_geometry_3": "Sphere and slanted circle",
    "Algebraic_graph_theory_1": "A highly symmetrical graph, the Petersen graph, which is vertex-transitive, symmetric, distance-transitive, and distance-regular. It has diameter 2. Its automorphism group has 120 elements, and is in fact the symmetric group $S_{5}$.",
    "Algebraic_graph_theory_2": "A Cayley graph for the alternating group $A_4$, forming a truncated tetrahedron in three dimensions. All Cayley graphs are vertex-transitive, but some vertex-transitive graphs (like the Petersen graph) are not Cayley graphs.",
    "Algebraic_number_1": "The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.",
    "Almost_perfect_number_1": "Demonstration, with Cuisenaire rods, that the number 8 is almost perfect, and deficient.",
    "Altitude_(triangle)_1": "The altitude from $A$ (dashed line segment) intersects the extended base at $D$ (a point outside the triangle).",
    "Altitude_(triangle)_2": "The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle.",
    "Altitude_(triangle)_3": "In a right triangle, the altitude from each acute angle coincides with a leg and intersects the opposite side at (has its foot at) the right-angled vertex, which is the orthocenter.",
    "Altitude_(triangle)_4": "Comparison of the inverse Pythagorean theorem with the Pythagorean theorem",
    "Analytic_continuation_1": "Analytic continuation of natural logarithm (imaginary part)",
    "Analytic_continuation_2": "Analytic continuation from $U$ (centered at 1) to $V$ (centered at $\\frac{3+i}{2}$)",
    "Angle_2": "Angles $A$ and $B$ are a pair of vertical angles; angles $C$ and $D$ are a pair of vertical angles. Hatch marks are used here to show angle equality.",
    "Angle_3": "Angles $A$ and $B$ are adjacent.",
    "Angle_4": "The complementary angles $a$ and $b$ ($b$ is the complement of $a$, and $a$ is the complement of $b$.)",
    "Angle_5": "The angles a and b are supplementary angles.",
    "Angle_6": "The angles $a$ and $b$ are supplementary angles.",
    "Angle_7": "Angles $AOB$ and $COD$ are conjugate as they form a complete angle. Considering magnitudes, $45^\\circ + 315^\\circ = 360^\\circ$.",
    "Angle_8": "Internal and external angles",
    "Angle_bisector_theorem_1": "The theorem states: for any triangle, if $\\angle DAB$ and $\\angle DAC$ share a common bisector $AD$, then $|BD| : |CD| = |AB| : |AC|$.",
    "Angle_bisector_theorem_2": "A drawing to angle bisector theorem generalized proof",
    "Angle_bisector_theorem_3": "Exterior angle bisectors (dotted red): Points $D$, $E$, and $F$ are collinear, and the following ratio equations hold: $\\tfrac{|EB|}{|EC|} = \\tfrac{|AB|}{|AC|}$, $\\tfrac{|FB|}{|FA|} = \\tfrac{|CB|}{|CA|}$, and $\\tfrac{|DA|}{|DC|} = \\tfrac{|BA|}{|BC|}$.",
    "Angular_displacement_1": "Euler's rotation theorem. A great circle transforms to another great circle under rotations, leaving always a diameter of the sphere in its original position.",
    "Angular_distance_1": "Angular separation $\\theta$ between points $A$ and $B$ as seen from $O$",
    "Angular_distance_2": "Planar approximation of angular distance on sky",
    "Antiderivative_1": "The slope field of $F(x)={\\frac {x^{3}}{3}}-{\\frac {x^{2}}{2}}-x+C$, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant $C$.",
    "Antimatroid_1": "Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding path poset.",
    "Antimatroid_2": "A shelling sequence of a planar point set. The line segments show edges of the convex hulls after some of the points have been removed.",
    "Antisymmetric_relation_1": "Symmetric and antisymmetric relations",
    "Aperiodic_tiling_1": "The Penrose tiling is an example of an aperiodic tiling; every tiling it can produce lacks translational symmetry.",
    "Aperiodic_tiling_2": "An aperiodic tiling using a single shape and its reflection, discovered by David Smith",
    "Aperiodic_tiling_3": "A portion of tiling by the Robinson tiles",
    "Aperiodic_tiling_4": "The chair substitution tiling system.",
    "Aperiodic_tiling_5": "Some tilings obtained by the cut and project method. The cutting planes are all parallel to the one which defines Penrose tilings (the fourth tiling on the third line). These tilings are all in different local isomorphism classes, that is, they are locally distinguishable.",
    "Aperiodic_tiling_6": "The Trilobite and Cross tiles enforce the chair substitution structure they can only admit tilings in which the chair substitution can be discerned and so are aperiodic.",
    "Apollonian_circles_1": "Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points $C$, $D$, and every blue circle separates the two points.",
    "Apollonian_circles_2": "Apollonian circle, the angle bisectors in X yield ${\\frac {|CT_{i}|}{|DT_{i}|}}={\\frac {|CT_{o}|}{|DT_{o}|}}=r$, due $\\angle T_{i}XT_{o}={\\frac {180^{\\circ }}{2}}=90^{\\circ }$ and Thales's theorem $X$ is located on a half circle with diameter $CD$",
    "Apothem_1": "Apothem of a hexagon",
    "Arc_(projective_geometry)_1": "The red points are a 4-arc in Fano plane, the projective plane of order 2.",
    "Archimedean_property_1": "Illustration of the Archimedean property.",
    "Archimedes'_quadruplets_1": "Each of the Archimedes' quadruplets (green) have equal area to each other and to Archimedes' twin circles",
    "Arithmetic_mean_1": "Comparison of two log-normal distributions with equal median, but different skewness, resulting in various means and modes",
    "Arithmetic_mean_2": "Geometric proof without words that $\\max(a,b) > \\mathrm{RMS} > \\mathrm{AM} > \\mathrm{GM} > \\mathrm{HM} > \\min(a,b)$ for two distinct positive numbers $a$ and $b$.",
    "Arithmetic_mean_3": "Proof without words of the AM--GM inequality: $PR$ is the diameter of a circle centered at $O$; its radius $AO = \\frac{a + b}{2}$ is the arithmetic mean of $a$ and $b$. Using the geometric mean theorem, triangle $PGR$'s altitude $GQ = \\sqrt{ab}$ is the geometric mean. For any ratio $a : b$, $AO \\geq GQ$.",
    "Arithmetic_number_1": "Demonstration, with Cuisenaire rods, of the arithmetic nature of the number 6",
    "Arithmetical_hierarchy_1": "An illustration of how the levels of the hierarchy interact and where some basic set categories lie within it.",
    "Asymptote_2": "$(sec(t), cosec(t))$, or $x^2 + y^2 = (xy)^2$, with 2 horizontal and 2 vertical asymptotes",
    "Asymptote_3": "$x^2+2x+3$ is a parabolic asymptote to $\\frac{x^3+2x^2+3x+4}{x}.$",
    "Asymptote_4": "A curve intersecting an asymptote infinitely many times",
    "Asymptote_5": "$f(x)={\\tfrac {1}{x}}$ graphed on Cartesian coordinates. The x and y-axis are the asymptotes.",
    "Asymptote_6": "The arctangent function has two different asymptotes.",
    "Asymptote_7": "Black: the graph of $f(x)=(x^{2}+x+1)/(x+1)$. Red: the asymptote $y=x$. Green: difference between the graph and its asymptote for $x=1,2,3,4,5,6$.",
    "Asymptote_8": "A cubic curve, the folium of Descartes (solid) with a single real asymptote (dashed)",
    "Asymptote_9": "The graph of a function with a horizontal $(y = 0)$, vertical $(x = 0)$, and oblique asymptote (purple line, given by $y = 2 \\times x$)",
    "Automorphism_1": "An automorphism of the Klein four-group shown as a mapping between two Cayley graphs, a permutation in cycle notation, and a mapping between two Cayley tables.",
    "Axiom_of_power_set_1": "The elements of the power set of the set $\\{x, y, z\\}$ ordered with respect to inclusion.",
    "Axiom_schema_of_replacement_2": "Axiom schema of collection: the image $f[A]$ of the domain set $A$ under the definable class function $f$ falls inside a set $B$.",
    "Backward_Euler_method_1": "The pink region outside the disk shows the stability region of the backward Euler method.",
    "Banach's_matchbox_problem_1": "Distribution of probability of having $k$ matches remaining in the other pocket.",
    "Bankoff_circle_1": "A Bankoff circle with the center $C^{\\prime\\prime}_6$",
    "Barbier's_theorem_1": "These Reuleaux polygons have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters.",
    "Barnes_G-function_1": "The Barnes G-function along part of the real axis",
    "Barnes_G-function_2": "Plot of the Barnes $G$ (also known as the double gamma function $G(z)$) in the complex plane from $-2 - 2i$ to $2 + 2i$ with colors.",
    "Barycentric_subdivision_2": "Barycentric subdivision of a 2-simplex. The colored points added on the right are the barycenters of the simplexes on the left",
    "Barycentric_subdivision_3": "The disdyakis dodecahedron, the barycentric subdivision of a cube",
    "Basel_problem_1": "The sum of inverse squares of distances of $P1$ and $P2$ from $Q$ equals the inverse square distance from $P$ to $Q$.",
    "Basel_problem_2": "The inequality $\\frac{1}{2}r^{2}\\tan\\theta > \\frac{1}{2}r^{2}\\theta > \\frac{1}{2}r^{2}\\sin\\theta$ is shown pictorially for any $\\theta \\in (0, \\pi/2)$. The three terms are the areas of the triangle $OAC$, the circular sector $OAB$, and the triangle $OAB$, respectively. Taking reciprocals and squaring gives $\\cot^{2}\\theta < \\frac{1}{\\theta^{2}} < \\csc^{2}\\theta.$",
    "Basis_(linear_algebra)_1": "The same vector can be represented in two different bases (purple and red arrows).",
    "Basis_(linear_algebra)_2": "This picture illustrates the standard basis in $R^2$. The blue and orange vectors are the elements of the basis; the green vector can be given in terms of the basis vectors, and so is linearly dependent upon them.",
    "Belt_problem_1": "The belt problem",
    "Bernoulli's_inequality_1": "An illustration of Bernoulli's inequality, with the graphs of $y=(1+x)^{r}$ and $y=1+rx$ shown in red and blue respectively. Here, $r=3.$",
    "Bernstein_polynomial_1": "Bernstein basis polynomials for 4th degree curve blending",
    "Beta-binomial_distribution_1": "Probability mass function for the beta-binomial distribution",
    "Beta-binomial_distribution_2": "Cumulative probability distribution function for the beta-binomial distribution",
    "Beta_function_1": "Contour plot of the beta function",
    "Beta_skeleton_1": "The circle-based 1.1-skeleton (heavy dark edges) and 0.9-skeleton (light dashed blue edges) of a set of 100 random points in a square.",
    "Beta_skeleton_2": "The empty regions $R_{pq}$ defining the circle-based $\\beta$-skeleton. Left: $\\beta < 1$. Center: $\\beta = 1$. Right: $\\beta > 1$.",
    "Bethe_lattice_2": "A Bethe lattice with coordination number $z = 3$",
    "Betti_number_1": "For a torus, the first Betti number is $b_1 = 2$, which can be intuitively thought of as the number of circular \"holes\"",
    "Bicentric_polygon_1": "A regular pentagon",
    "Bicentric_polygon_2": "An equilateral triangle",
    "Bicentric_polygon_3": "A bicentric isosceles trapezoid",
    "Bicentric_quadrilateral_2": "A right kite",
    "Bicentric_quadrilateral_3": "A bicentric quadrilateral $ABCD$ and its contact quadrilateral $WXYZ$",
    "Bicentric_quadrilateral_4": "A bicentric quadrilateral $ABCD$ with the contact quadrilateral $WXYZ$. Animation see here",
    "Bicentric_quadrilateral_5": "A bicentric quadrilateral $ABCD$ with incenter $I$ and circumcenter $O$",
    "Bicorn_1": "A transformed bicorn with $a = 1$",
    "Bifolium_1": "Bifolium with $a = 1$",
    "Bijection_2": "A bijection from the natural numbers to the integers, which maps $2n$ to $-n$ and $2n - 1$ to $n$, for $n \\geq 0$.",
    "Bijection,_injection_and_surjection_1": "Injective composition: the second function need not be injective.",
    "Bijection,_injection_and_surjection_2": "Surjective composition: the first function need not be surjective.",
    "Bijection,_injection_and_surjection_3": "Bijective composition: the first function need not be surjective and the second function need not be injective.",
    "Bilunabirotunda_1": "Six bilunabirotundae around a cube",
    "Binary_tetrahedral_group_2": "Cayley graph of $SL(2,3)$",
    "Binomial_proportion_confidence_interval_2": "Wilson score intervals plotted on a logistic curve, revealing asymmetry and good performance for small $n$ and where $p$ is at or near 0 or 1.",
    "Binomial_proportion_confidence_interval_3": "The probability density function (PDF) for the Wilson score interval, plus PDFs at interval bounds. Tail areas are equal.",
    "Bipartite_graph_1": "A complete bipartite graph with $m=5$ and $n=3$",
    "Bipartite_graph_2": "The Heawood graph is bipartite.",
    "Bipartite_graph_3": "A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph.",
    "Birational_geometry_1": "The circle is birationally equivalent to the line. One birational map between them is stereographic projection, pictured here.",
    "Biregular_graph_1": "The graph of the rhombic dodecahedron is biregular.",
    "Birkhoff's_representation_theorem_1": "The distributive lattice of divisors of 120, and its representation as sets of prime powers.",
    "Birthday_problem_1": "The computed probability of at least two people sharing the same birthday versus the number of people",
    "Birthday_problem_2": "The probability that no two people share a birthday in a group of $n$ people. Note that the vertical scale is logarithmic (each step down is $10^{20}$ times less likely).",
    "Birthday_problem_3": "Graphs showing the approximate probabilities of at least two people sharing a birthday (red) and its complementary event (blue)",
    "Birthday_problem_4": "A graph showing the accuracy of the approximation $1 - e^{-n^2/730}$ (red)",
    "Birthday_problem_5": "Plot of the probability of at least one shared birthday between at least one man and one woman",
    "Birthday_problem_6": "Comparing $p(n) =$ probability of a birthday match with $q(n) =$ probability of matching your birthday",
    "Bisection_2": "Perpendicular bisector of a line segment",
    "Bisection_3": "Construction by straight edge and compass",
    "Bisection_4": "The interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram.",
    "Bisection_method_1": "A few steps of the bisection method applied over the starting range $[a_1;b_1]$. The bigger red dot is the root of the function.",
    "Bitangents_of_a_quartic_2": "The Trott curve and seven of its bitangents. The others are symmetric with respect to $90^\\circ$ rotations through the origin.",
    "Blancmange_curve_1": "A plot of the blancmange curve.",
    "Blowing_up_1": "Blowup of the affine plane.",
    "Boltzmann_machine_1": "A graphical representation of an example Boltzmann machine. Each undirected edge represents dependency. In this example there are 3 hidden units and 4 visible units. This is not a restricted Boltzmann machine.",
    "Bolza_surface_1": "The tiling of the Bolza surface by reflection domains is a quotient of the order-3 bisected octagonal tiling.",
    "Bolza_surface_2": "The fundamental domain of the Bolza surface in the Poincaré disk; opposite sides are identified.",
    "Bolza_surface_3": "The four generators of the symmetry group of the Bolza surface",
    "Boolean_algebra_1": "Venn diagrams for conjunction, disjunction, and complement",
    "Boundary_(topology)_2": "Boundary of hyperbolic components of Mandelbrot set",
    "Bounded_function_1": "A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not.",
    "Bounded_variation_2": "The function $f(x) = x \\sin\\left(\\frac{1}{x}\\right)$ is not of bounded variation on the interval $[0,2/\\pi]$.",
    "Bounded_variation_3": "The function $f(x) = x^2 \\sin\\left(\\frac{1}{x}\\right)$ is of bounded variation on the interval $[0,2/\\pi]$.",
    "Bounding_sphere_1": "Some instances of the smallest bounding circle, the case of the bounding sphere in 2 dimensions.",
    "Bour's_minimal_surface_1": "Bour's surface, leaving out the points with $r < 0.5$ to show the self-crossings more clearly.",
    "Boxcar_function_1": "A graphical representation of a boxcar function",
    "Brahmagupta_theorem_1": "${\\overline {BM}}\\perp {\\overline {AC}},{\\overline {EF}}\\perp {\\overline {BC}}$ $\\Rightarrow |{\\overline {AF}}|=|{\\overline {FD}}|$",
    "Breather_2": "Pseudospherical breather surface",
    "Bregman_divergence_2": "Bregman divergence interpreted as areas.",
    "Bring_radical_1": "Plot of the Bring radical for real argument",
    "Brocard_points_1": "The Brocard point of a triangle, constructed at the intersection point of three circles",
    "Brun's_theorem_1": "The convergence to Brun's constant.",
    "Bullet-nose_curve_1": "Bullet-nose curve with $a = 1$ and $b = 1$",
    "Bump_function_1": "The graph of the bump function $(x,y)\\in \\mathbb {R} ^{2}\\mapsto \\Psi (r),$ where $r=\\left(x^{2}+y^{2}\\right)^{1/2}$ and $\\Psi (r)=e^{-1/(1-r^{2})}\\cdot \\mathbf {1} _{\\{|r|<1\\}}.$",
    "Bump_function_2": "The 1d bump function $\\Psi (x).$",
    "Bump_function_3": "The non-analytic smooth function $f(x)$ considered in the article.",
    "Bump_function_4": "The smooth transition g from 0 to 1 defined here.",
    "Bump_function_5": "An illustration of the sets in the construction.",
    "Butterfly_theorem_2": "Butterfly theorem",
    "CORDIC_1": "An illustration of the CORDIC algorithm in progress",
    "Cage_(graph_theory)_1": "The Tutte $(3,8)$-cage.",
    "Cairo_pentagonal_tiling_2": "Geometry of pentagons for the dual snub square tiling",
    "Cairo_pentagonal_tiling_3": "Equilateral form of the Cairo tiling",
    "Cairo_pentagonal_tiling_4": "Type 2 Cairo tiling, with coloring showing reflected and non-reflected tiles",
    "Cairo_pentagonal_tiling_5": "In a type 4 Cairo tiling, the pentagons can be bilaterally symmetric even when the tiling is not",
    "Cairo_pentagonal_tiling_6": "Collinear form of Cairo tiling, with integer-coordinate pentagons, formed by flattening two perpendicular regular hexagonal tilings in perpendicular directions",
    "Calculus_1": "Archimedes used the method of exhaustion to calculate the area under a parabola in his work Quadrature of the Parabola.",
    "Calculus_2": "Tangent line at $(x_0, f(x_0))$. The derivative $f'(x)$ of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.",
    "Candido's_identity_2": "The (white) squares of side lengths $x^{2}$ and $y^{2}$ appear each twice and the colored areas equal the area of the white square of side length $(x+y)^{2}$, hence the area of the outer square equals twice the sum of the areas of the three (white) inner squares.",
    "Capsule_(geometry)_1": "A two-dimensional orthographic projection at the left with a three-dimensional one at the right depicting a capsule",
    "Caratheodory's_theorem_(convex_hull)_1": "An illustration of Carathéodory's theorem for a square in $R^2$",
    "Cartesian_oval_1": "Example of Cartesian ovals.",
    "Cartesian_product_1": "Cartesian product of the sets $\\{x,y,z\\}$ and $\\{1,2,3\\}$",
    "Casey's_theorem_1": "$t_{12}\\cdot t_{34}+t_{14}\\cdot t_{23}-t_{13}\\cdot t_{24}=0$",
    "Cassini_oval_2": "Cassini oval: $|PP_1| \\times |PP_2| = b^2$ for any location of $P$ on the curve.",
    "Cassini_oval_3": "Some Cassini ovals. ($b = 0.6a$, $0.8a$, $a$, $1.2a$, $1.4a$, $1.6a$)",
    "Cassini_oval_4": "Cassini ovals and their orthogonal trajectories (hyperbolas)",
    "Cassini_oval_5": "Cassini ovals as planar sections of a torus (the torus on the right is a spindle torus)",
    "Catalan's_minimal_surface_1": "Catalan's minimal surface.",
    "Categorical_distribution_1": "The possible probabilities for the categorical distribution with $k=3$ are the 2-simplex $p_{1}+p_{2}+p_{3}=1$, embedded in 3-space.",
    "Catenary_2": "Catenaries for different values of $a$",
    "Catenary_3": "Diagram of forces acting on a segment of a catenary from $c$ to $r$. The forces are the tension $T_0$ at $c$, the tension $T$ at $r$, and the weight of the chain $(0, -ws)$. Since the chain is at rest the sum of these forces must be zero.",
    "Catenary_4": "Three catenaries through the same two points, depending on the horizontal force $T_H$.",
    "Cathetus_1": "A right-angled triangle where $c_1$ and $c_2$ are the catheti and $h$ is the hypotenuse",
    "Cauchy's_theorem_(geometry)_1": "Convex regular icosahedron",
    "Cauchy_condensation_test_1": "Visualization of the above argument. Partial sums of the series $\\sum f(n)$, $\\sum 2^{n}f(2^{n})$, and $2\\sum f(n)$ are shown overlaid from left to right.",
    "Cayley_graph_1": "The Cayley graph of the free group on two generators $a$ and $b$",
    "Cayley_graph_2": "Cayley graph of the dihedral group $D_{4}$ on two generators $a$ and $b$",
    "Cayley_graph_3": "Cayley graph of $D_{4}$, on two generators which are both self-inverse",
    "Cayley_graph_4": "Part of a Cayley graph of the Heisenberg group. (The coloring is only for visual aid.)",
    "Cayley_transform_1": "Cayley transform of upper complex half-plane to unit disk",
    "Center_manifold_1": "Center (red) and unstable (green) manifolds of saddle-node equilibrium point of the system ${\\dot {x}}=x^{2},$ ${\\dot {y}}=y$.",
    "Centered_square_number_1": "Centered square numbers (in red) are in the center of odd rows of Floyd's triangle.",
    "Central_angle_1": "Angle $AOB$ is a central angle",
    "Central_angle_2": "Central angle. Convex. Is subtended by minor arc $L$",
    "Central_angle_3": "Central angle. Reflex. Is not subtended by $L$",
    "Central_binomial_coefficient_1": "Pascal's triangle, rows 0 through 7. The numbers in the central column are the central binomial coefficients.",
    "Chamfer_(geometry)_2": "Historical drawings of truncated tetrahedron and slightly chamfered tetrahedron.",
    "Chaotic_mixing_2": "Growth of advected contour",
    "Chaotic_mixing_3": "Advected contour fractal dimension",
    "Chebyshev_function_1": "The Chebyshev function $\\psi (x)$, with $x < 50$",
    "Chebyshev_nodes_1": "Chebyshev zeros (solid dots, red lines) and extrema (hollow squares, blue lines) are the projection of two sets of equispaced points on the unit circle onto the x-axis. $2n$ equispaced points on the circle project onto $n$ Chebyshev zeros or $n+1$ Chebyshev extrema. (Here $n = 5$.)",
    "Chebyshev_nodes_2": "The Chebyshev zeros (solid dots) are roots of a Chebyshev polynomial of the first kind (red). The Chebyshev extrema (hollow squares) are roots of a Chebyshev polynomial of the second kind (blue), and also the extrema (crosses) of a Chebyshev polynomial of the first kind.",
    "Chebyshev_nodes_3": "Chebyshev nodes of both kinds from $n=2$ to $n=50$.",
    "Chebyshev_nodes_4": "Chebyshev zeros (solid dots, red lines) and extrema (hollow squares, blue lines) are the projection of two sets of equispaced points on the unit circle onto the $x$-axis. $2n$ equispaced points on the circle project onto $n$ Chebyshev zeros or $n+1$ Chebyshev extrema. (Here $n = 5$.)",
    "Chebyshev_polynomials_1": "Plot of the first five $T_n$ Chebyshev polynomials (first kind)",
    "Chebyshev_polynomials_2": "Plot of the first five $U_n$ Chebyshev polynomials (second kind)",
    "Chebyshev_polynomials_3": "The first few Chebyshev polynomials of the first kind in the domain $-1 < x < 1$: The flat $T_0, T_1, T_2, T_3, T_4,$ and $T_5$.",
    "Chebyshev_polynomials_4": "The first few Chebyshev polynomials of the second kind in the domain $-1 < x < 1$: The flat $U_0, U_1, U_2, U_3, U_4$ and $U_5$. Although not visible in the image, $U_n(1) = n + 1$ and $U_n(-1) = (n + 1)(-1)^n$.",
    "Chebyshev_polynomials_5": "The non-smooth function (top) $y = -x^3 H(-x)$, where $H$ is the Heaviside step function, and (bottom) the 5th partial sum of its Chebyshev expansion. The 7th sum is indistinguishable from the original function at the resolution of the graph.",
    "Chernoff_bound_1": "Two-sided Chernoff bound for a chi-square random variable",
    "Chi-squared_distribution_2": "Plot of the cumulative chi-square distribution for values of $k = \\{1, 2, 3, 4, 6, 9\\}$.",
    "Chi-squared_distribution_3": "Plot of the chi-square distribution for values of $k = \\{1, 2, 3, 4, 6, 9\\}$.",
    "Chirp_spectrum_1": "Linear chirp with $T_B=25$ and $N=128$",
    "Chord_(geometry)_1": "Common lines and line segments on a circle, including a chord in blue",
    "Christensen_failure_criterion_1": "For plane stresses, $\\displaystyle \\sigma _{3}=0$ and $\\frac{T}{C}=0.3$ (brittle materials). Blue line is polynomial invariants failure criterion. Red lines are coordinated fracture criterion.",
    "Circle_1": "Arc, sector, and segment",
    "Circle_graph_1": "A circle with five chords and the corresponding circle graph.",
    "Circle_graph_2": "The chords forming the 220-vertex 5-chromatic triangle-free circle graph of Ageev (1996), drawn as an arrangement of lines in the hyperbolic plane.",
    "Circle_graph_3": "An interval system with five intervals and the corresponding overlap graph.",
    "Circle_group_1": "Multiplication on the circle group is equivalent to addition of angles.",
    "Circle_packing_theorem_1": "A circle packing for a five-vertex planar graph",
    "Circle_packing_theorem_2": "Circle packings can be used to approximate conformal mappings between specified domains. Each circle on the left corresponds to a circle on the right.",
    "Circle_packing_theorem_3": "A polyhedron and its midsphere. The circle packing theorem implies that every polyhedral graph can be represented as the graph of a polyhedron that has a midsphere.",
    "Circulant_graph_1": "The Paley graph of order 13, an example of a circulant graph.",
    "Circular-arc_graph_1": "A circular-arc graph (left) and a corresponding arc model (right).",
    "Circular_uniform_distribution_1": "Probability densities $P_{N}({\\bar {R}})$ for small values of $N$. Densities for $N>3$ are normalised to the maximum density, those for $N=1$ and $2$ are scaled to aid visibility.",
    "Circular_uniform_distribution_2": "A 10,000 point Monte Carlo simulation of the distribution of the sample mean of a circular uniform distribution for $N = 3$",
    "Circular_uniform_distribution_3": "A 10,000 point Monte Carlo simulation of the distribution of the sample mean of a circular uniform distribution for $N = 3$",
    "Circumcenter_of_mass_1": "Circumcenter of mass of a polygon.",
    "Cissoid_2": "Ellipse $r={\\frac {1}{2-\\cos \\theta }}$ in red, with its two cissoid branches in black and blue (origin)",
    "Cissoid_of_Diocles_1": "Cissoid of Diocles traced by points $M$ with ${\\overline {OM}}={\\overline {M_{1}M_{2}}}$",
    "Classical_modular_curve_1": "Knot at infinity of $X_0(11)$",
    "Clebsch_graph_1": "$K_{16}$ 3-coloured as three Clebsch graphs.",
    "Clebsch_graph_2": "The Clebsch graph is Hamiltonian.",
    "Clebsch_graph_3": "The achromatic number of the Clebsch graph is 8.",
    "Clebsch_graph_4": "The chromatic number of the Clebsch graph is 4.",
    "Clebsch_graph_5": "The chromatic index of the Clebsch graph is 5.",
    "Clebsch_graph_6": "Construction of the Clebsch graph from a hypercube graph.",
    "Clique-width_1": "Construction of a distance-hereditary graph of clique-width 3 by disjoint unions, relabelings, and label-joins. Vertex labels are shown as colors.",
    "Clique_(graph_theory)_1": "A graph with $23 \\times 1$-vertex cliques (the vertices), $42 \\times 2$-vertex cliques (the edges), $19 \\times 3$-vertex cliques (light and dark blue triangles), and $2 \\times 4$-vertex cliques (dark blue areas). The 11 light blue triangles form maximal cliques. The two dark blue 4-cliques are both maximum and maximal, and the clique number of the graph is 4.",
    "Clique_problem_1": "The graph shown has one maximum clique, the triangle $\\{1,2,5\\}$, and four more maximal cliques, the pairs $\\{2,3\\}$, $\\{3,4\\}$, $\\{4,5\\}$, and $\\{4,6\\}$.",
    "Clique_problem_2": "In this permutation graph, the maximum cliques correspond to the longest decreasing subsequences $(4,3,1)$ and $(4,3,2)$ of the defining permutation.",
    "Clique_problem_3": "A random graph with a planted clique",
    "Close-packing_of_equal_spheres_2": "FCC arrangement seen on 4-fold axis direction",
    "Close-packing_of_equal_spheres_3": "Cannonballs piled on a triangular (front) and rectangular (back) base, both face-centered cubic lattices.",
    "Close-packing_of_equal_spheres_4": "Collections of snowballs arranged in pyramid shape. The front pyramid is hexagonal close-packed and rear is face-centered cubic.",
    "Cluster_algebra_1": "Mutation between two triangulations of the heptagon",
    "Cobordism_1": "A cobordism $(W; M, N)$.",
    "Cobordism_2": "A cobordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).",
    "Cochleoid_1": "$r={\\frac {\\sin \\theta }{\\theta }},-20<\\theta <20$",
    "Cochleoid_2": "cochleoid (solid) and its polar inverse (dashed)",
    "Codomain_1": "A function $f$ from $X$ to $Y$. The blue oval $Y$ is the codomain of $f$. The yellow oval inside $Y$ is the image of $f$, and the red oval $X$ is the domain of $f$.",
    "Coefficient_of_determination_1": "Ordinary least squares regression of Okun's law. Since the regression line does not miss any of the points by very much, the $R^2$ of the regression is relatively high.",
    "Coefficient_of_determination_2": "$R^{2}=1-{\\frac {\\color {blue}{SS_{\\text{res}}}}{\\color {red}{SS_{\\text{tot}}}}}$ The better the linear regression (on the right) fits the data in comparison to the simple average (on the left graph), the closer the value of $R^2$ is to 1. The areas of the blue squares represent the squared residuals with respect to the linear regression. The areas of the red squares represent the squared residuals with respect to the average value.",
    "Coefficient_of_determination_3": "This is an example of residuals of regression models in smaller and larger spaces based on ordinary least square regression.",
    "Cograph_1": "The Turán graph $T(13,4)$ is a cograph",
    "Cograph_2": "A cotree and the corresponding cograph. Each edge $(u,v)$ in the cograph has a matching color to the least common ancestor of $u$ and $v$ in the cotree.",
    "Color_triangle_1": "A 1775 color triangle by Tobias Mayer.",
    "Combinatorial_explosion_1": "Using separate lines of communication, four organizations require six channels",
    "Combinatorial_explosion_2": "Using an intermediary, only one channel per organization is required",
    "Complement_(set_theory)_1": "The relative complement of A in B: $B\\cap A^{c}=B\\setminus A$",
    "Complete_graph_2": "The complete graph on 7 vertices (graphic illustrating language links between all languages).",
    "Complex_algebraic_variety_1": "The Riemann sphere is one of the simplest complex algebraic varieties.",
    "Complex_logarithm_1": "A single branch of the complex logarithm. The hue of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm.",
    "Complex_logarithm_2": "A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number $z$ goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a branch point of the function.",
    "Complex_logarithm_3": "Complex log mapping maps radii to horizontal lines and circles to vertical lines",
    "Complex_logarithm_4": "The real part of $\\log(z)$ is the natural logarithm of $|z|$. Its graph is thus obtained by rotating the graph of $\\ln(x)$ around the $z$-axis.",
    "Complex_logarithm_5": "The circles $\\operatorname{Re}(\\Log z) = \\text{constant}$ and the rays $\\operatorname{Im}(\\Log z) = \\text{constant}$ in the complex $z$-plane.",
    "Complex_plane_1": "The complex plane. The horizontal axis represents the real numbers $(\\mathbb {R})$, the vertical axis represents the imaginary numbers $(\\mathbb {I})$. The complex number $4+4i$ is shown at the point $(4,4).$",
    "Composition_(combinatorics)_1": "Bijection between 3 bit binary numbers and compositions of 4",
    "Composition_(combinatorics)_2": "The 32 compositions of $6$: $1+1+1+1+1+1$, $2+1+1+1+1$, $1+2+1+1+1$, $\\ldots$, $1+5$, $6$.",
    "Composition_(combinatorics)_3": "The 11 partitions of $6$: $1+1+1+1+1+1$, $2+1+1+1+1$, $3 + 1 + 1 + 1$, $\\ldots$, $3+3$, $6$.",
    "Composition_(combinatorics)_4": "The numbers of compositions of $n+1$ into $k+1$ ordered partitions form Pascal's triangle",
    "Composition_(combinatorics)_5": "Using the Fibonacci sequence to count the $\\{1, 2\\}$-restricted compositions of $n$, for example, the number of ways one can ascend a staircase of length $n$, taking one or two steps at a time",
    "Compound_prism_1": "A double-Amici prism, showing the apex angles ($\\alpha _{1}$ and $\\alpha _{2}$) of the three elements, and the angles of incidence $\\theta _{i}$ and refraction $\\theta '_{i}$ at each interface.",
    "Compound_prism_2": "A triplet prism, showing the apex angles ($\\alpha _{1}$, $\\alpha _{2}$, and $\\alpha _{3}$) of the three elements, and the angles of incidence $\\theta _{i}$ and refraction $\\theta '_{i}$ at each interface.",
    "Conchoid_of_de_Sluze_1": "The Conchoid of de Sluze for several values of a",
    "Concyclic_points_2": "Circumscribed circle, $C$, and circumcenter, $O$, of a cyclic polygon, $P$",
    "Concyclic_points_3": "Four concyclic points forming a cyclic quadrilateral, showing two equal angles",
    "Conditional_dependence_1": "A Bayesian network illustrating conditional dependence",
    "Cone_2": "A right circular cone and an oblique circular cone",
    "Cone_3": "A double cone, not infinitely extended",
    "Cone_(topology)_1": "Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.",
    "Configuration_(geometry)_2": "The Schläfli double six.",
    "Configuration_(geometry)_3": "Configurations $(4_36_2)$ (a complete quadrangle, at left) and $(6_24_3)$ (a complete quadrilateral, at right).",
    "Conformal_geometry_1": "A coordinate grid prior to a Möbius transformation",
    "Conformal_geometry_2": "The same grid after a Möbius transformation",
    "Congruent_number_1": "Right triangle with the area 6, a congruent number.",
    "Conic_constant_1": "An illustration of various conic constants",
    "Conical_surface_1": "An elliptic cone, a special case of a conical surface, shown truncated for simplicity",
    "Connected_sum_2": "Consider disjoint planar projections of each knot.",
    "Connected_sum_3": "Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots.",
    "Connected_sum_4": "Now join the two knots together by deleting these arcs from the knots and adding the arcs that form the other pair of sides of the rectangle.",
    "Connection_(principal_bundle)_1": "A principal bundle connection form $\\omega$ may be thought of as a projection operator on the tangent bundle $TP$ of the principal bundle $P$. The kernel of the connection form is given by the horizontal subspaces for the associated Ehresmann connection.",
    "Connection_(principal_bundle)_2": "A connection is equivalently specified by a choice of horizontal subspace $H_{p} \\subset T_{p}P$ for every tangent space to the principal bundle $P$.",
    "Connection_(principal_bundle)_3": "A principal bundle connection is required to be compatible with the right group action of $G$ on $P$. This can be visualized as the right multiplication $R_{g}$ taking the horizontal subspaces into each other. This equivariance of the horizontal subspaces $H\\subset TP$ interpreted in terms of the connection form $\\omega$ leads to its characteristic equivariance properties.",
    "Connection_(vector_bundle)_1": "A section of a bundle may be viewed as a generalized function from the base into the fibers of the vector bundle. This can be visualized by the graph of the section, as in the figure above.",
    "Connection_(vector_bundle)_2": "How to recover the covariant derivative of a connection from its parallel transport. The values $s(\\gamma (t))$ of a section $s\\in \\Gamma (E)$ are parallel transported along the path $\\gamma$ back to $\\gamma (0)=x$, and then the covariant derivative is taken in the fixed vector space, the fibre $E_{x}$ over  $x$.",
    "Conservative_vector_field_1": "Depiction of two possible paths to integrate. In green is the simplest possible path; blue shows a more convoluted curve",
    "Conservative_vector_field_2": "Line integral paths used to prove the following statement: if the line integral of a vector field is path-independent, then the vector field is a conservative vector field.",
    "Conservative_vector_field_3": "The above vector field $\\mathbf {v} =\\left(-{\\frac {y}{x^{2}+y^{2}}},{\\frac {x}{x^{2}+y^{2}}},0\\right)$ defined on $U=\\mathbb {R} ^{3}\\setminus \\{(0,0,z)\\mid z\\in \\mathbb {R} \\}$, i.e., $\\mathbb {R} ^{3}$ with removing all coordinates on the $z$-axis (so not a simply connected space), has zero curl in $U$ and is thus irrotational. However, it is not conservative and does not have path independence.",
    "Constant_(mathematics)_1": "Graph of $f(x)=5$.",
    "Constant_function_1": "An example of a constant function is $y(x) = 4$, because the value of $y(x)$ is 4 regardless of the input value $x$.",
    "Convergent_series_1": "If the blue series, $\\Sigma b_{n}$, can be proven to converge, then the smaller series, $\\Sigma a_{n}$ must converge. By contraposition, if the red series $\\Sigma a_{n}$ is proven to diverge, then $\\Sigma b_{n}$ must also diverge.",
    "Converse_(logic)_1": "Venn diagram of $P\\leftarrow Q$ The white area shows where the statement is false.",
    "Convex_hull_1": "The convex hull of the red set is the blue and red convex set.",
    "Convex_hull_2": "Convex hull of a bounded planar set: rubber band analogy",
    "Convex_hull_3": "3D convex hull of 120 point cloud",
    "Convex_hull_4": "The witch of Agnesi. The points on or above the red curve provide an example of a closed set whose convex hull is open (the open upper half-plane).",
    "Convex_hull_5": "Convex hull of points in the plane",
    "Convex_hull_6": "Convex hull (in blue and yellow) of a simple polygon (in blue)",
    "Convex_hull_7": "An oloid, the convex hull of two circles in 3d space",
    "Convex_hull_8": "Partition of seven points into three subsets with intersecting convex hulls, guaranteed to exist for any seven points in the plane by Tverberg's theorem",
    "Convex_metric_space_2": "A circle as a convex metric space.",
    "Convex_polygon_1": "An example of a convex polygon: a regular pentagon.",
    "Convex_polytope_1": "A 3-dimensional convex polytope",
    "Convex_polytope_2": "The face lattice of a square pyramid, drawn as a Hasse diagram; each face in the lattice is labeled by its vertex set.",
    "Convex_set_1": "Illustration of a convex set shaped like a deformed circle. The line segment joining points $x$ and $y$ lies completely within the set, illustrated in green. Since this is true for any potential locations of two points within the set, the set is convex.",
    "Convex_set_2": "Illustration of a non-convex set. The line segment joining points $x$ and $y$ partially extends outside of the set, illustrated in red, and the intersection of the set with the line occurs in two places, illustrated in black.",
    "Convex_set_3": "A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.",
    "Convex_set_4": "Blaschke-Santaló $(r, D, R)$ diagram for planar convex bodies. $\\mathbb {L}$ denotes the line segment, $\\mathbb {I} _{\\frac {\\pi }{3}}$ the equilateral triangle, $\\mathbb {RT}$ the Reuleaux triangle and $\\mathbb {B} _{2}$ the unit circle.",
    "Convex_set_5": "Minkowski addition of sets. The sum of the squares $Q_1=[0,1]^2$ and $Q_2=[1,2]^2$ is the square $Q_1+Q_2=[1,3]^2$.",
    "Coordinate_system_1": "The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance $r$, polar angle $\\theta$ (theta), and azimuthal angle $\\phi$ (phi). The symbol $\\rho$ (rho) is often used instead of $r$.",
    "Coordinate_system_2": "A small portion of the Cartesian coordinate system, showing the origin, axes, and the four quadrants, with illustrative points and grid.",
    "Coordinate_system_3": "The Cartesian coordinate system in three-dimensional space",
    "Coordinate_system_4": "Cylindrical coordinate system",
    "Coplanarity_1": "An example of coplanar points",
    "Coprime_integers_1": "The numbers 4 and 9 are coprime. Therefore, the diagonal of a $4 \\times 9$ lattice does not intersect any other lattice points",
    "Coprime_integers_2": "The numbers 4 and 9 are coprime. Therefore, the diagonal of a $4 \\times 9$ lattice does not intersect any other lattice points",
    "Coprime_integers_3": "The tree rooted at $(2, 1)$. The root $(2, 1)$ is marked red, its three children are shown in orange, third generation is yellow, and so on in the rainbow order.",
    "Coulomb_wave_function_1": "Irregular Coulomb wave function $G$ plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1",
    "Coulomb_wave_function_2": "image of complex plot of regular Coulomb wave function added",
    "Coulomb_wave_function_3": "Regular Coulomb wave function $F$ plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1",
    "Covariance_1": "The sign of the covariance of two random variables $X$ and $Y$",
    "Covariance_and_contravariance_of_vectors_1": "A vector, $v$, represented in terms of tangent basis $e_1$, $e_2$, $e_3$ to the coordinate curves (left), dual basis, covector basis, or reciprocal basis $e^1$, $e^2$, $e^3$ to coordinate surfaces (right), in 3-d general curvilinear coordinates $(q^1, q^2, q^3)$, a tuple of numbers to define a point in a position space. Note the basis and cobasis coincide only when the basis is orthonormal.",
    "Covariance_and_contravariance_of_vectors_2": "Aid for explaining the geometrical meaning of covariant and contravariant vector components.",
    "Covering_space_1": "Intuitively, a covering locally projects a \"stack of pancakes\" above an open neighborhood $U$ onto $U$",
    "Covering_space_2": "The Hawaiian earring. Only the ten largest circles are shown.",
    "Covering_space_3": "The space $Y = [0,1] \\times \\mathbb{R}$ is a covering space of $X = [0,1] \\times S^1$. The disjoint open sets $S_i$ are mapped homeomorphically onto $U$. The fiber of $x$ consists of the points $y_i$.",
    "Coxeter_group_1": "Coxeter diagrams for the affine Coxeter groups",
    "Coxeter_group_2": "Stiefel diagram for the $G_{2}$ root system",
    "Coxeter_group_3": "Coxeter graphs of the irreducible finite Coxeter groups",
    "Coxeter_notation_1": "A radical subgroup is similar to an alternation, but removes the rotational generators.",
    "Coxeter_notation_2": "Rank 2 example, $[6]$ trionic subgroups with 3 colors of mirror lines",
    "Coxeter_notation_3": "Example trionic subgroup on hexagonal symmetry $[6,3]$ maps onto a larger $[6,3]$ symmetry.",
    "Coxeter_notation_4": "Example trionic subgroups on octagonal symmetry $[8,3]$ maps onto larger $[4,8]$ symmetries.",
    "Coxeter_notation_5": "Trionic relations of $[3,3]$",
    "Coxeter_notation_6": "Trionic subgroup relations of $[3,3,4]$",
    "Coxeter_notation_7": "Hasse diagram subgroups of $[4,4]$, down to its commutator subgroup, index 8",
    "Cramer's_rule_1": "Geometric interpretation of Cramer's rule. The areas of the second and third shaded parallelograms are the same and the second is $x_{1}$ times the first. From this equality Cramer's rule follows.",
    "Crofton_formula_1": "The line defined by choices of $\\varphi ,p$ intersects the curve $\\gamma$ twice, therefore, $n_{\\gamma }(\\varphi ,p)=2$.",
    "Crofton_formula_2": "Application of the Crofton formula in a Monte-Carlo simulation.",
    "Cross_section_(geometry)_1": "Colored regions are cross-sections of the solid cone. Their boundaries (in black) are the named plane sections.",
    "Cross_section_(geometry)_2": "Cross-section of a solid cylinder",
    "Cross_section_(geometry)_3": "A graph of $z = x^2 + xy + y^2$. For the partial derivative at $(1, 1, 3)$ that leaves y constant, the corresponding tangent line is parallel to the $xz$-plane.",
    "Cross_section_(geometry)_4": "A slice of the graph showing the function in the $xz$-plane at $y= 1$ and the tangent at point .",
    "Crossed_ladders_problem_1": "Crossed ladders of lengths a and b. h is half the harmonic mean of A and B; equivalently, the reciprocals of A and B sum to the reciprocal of h (the optic equation). Given a, b, and h, find w.",
    "Crossed_ladders_problem_2": "Folding a rectangular sheet of paper into thirds using the crossed ladders problem",
    "Crossed_ladders_problem_3": "Crossed ladders problem, complex solution",
    "Crossed_ladders_problem_4": "Diagram is used to illustrate and to prove the crossed ladder problem (a geometric problem in recreational mathematics), which here has been extended to crossed ladders inside a triangle.",
    "Crossing_number_(graph_theory)_1": "A drawing of the Heawood graph with three crossings. This is the minimum number of crossings among all drawings of this graph, so the graph has crossing number $cr(G) = 3$.",
    "Crossing_number_(graph_theory)_2": "An optimal drawing of $K_{4,7}$ showing that Turán's brick factory problem with 4 storage sites (yellow spots) and 7 kilns (blue spots) requires 18 crossings (red dots)",
    "Crossing_number_(knot_theory)_1": "Trefoil knot without 3-fold symmetry with crossings labeled.",
    "Crossing_number_(knot_theory)_2": "A table of all prime knots with seven crossing numbers or fewer (not including mirror images).",
    "Crown_graph_1": "A biclique cover of the ten-vertex crown graph",
    "Crown_graph_2": "Crown graphs with six, eight, and ten vertices.",
    "Crunode_1": "A crunode at the origin of the curve defined by $y^{2}-x^{2}(x+1)=0.$",
    "Crystal_Ball_function_1": "Examples of the Crystal Ball function.",
    "Cubic_Hermite_spline_2": "Example with finite-difference tangents",
    "Cubic_Hermite_spline_3": "Cardinal spline example in 2D. The line represents the curve, and the squares represent the control points ${\\boldsymbol {p}}_{k}$. Notice that the curve does not reach the first and last points; these points do, however, affect the shape of the curve. The tension parameter used is 0.1",
    "Cubic_Hermite_spline_4": "Geometric interpretation of Catmull-Rom cubic interpolation of the black point with uniformly spaced abscissae.",
    "Cubic_graph_1": "The Petersen graph is a cubic graph.",
    "Cubic_graph_2": "The complete bipartite graph $K_{3,3}$ is an example of a bicubic graph",
    "Cubic_graph_3": "Representation of a planar embedding as a graph-encoded map",
    "Cubic_honeycomb_2": "Cantitruncated cubic tiling",
    "Cubic_honeycomb_3": "The alternated cantitruncated cubic honeycomb",
    "Cubic_honeycomb_4": "Truncated square prismatic honeycomb",
    "Cubic_honeycomb_5": "Snub square prismatic honeycomb",
    "Cubic_surface_1": "A smooth cubic surface (the Clebsch surface)",
    "Cubic_surface_2": "The Schläfli graph",
    "Cubic_surface_3": "Cayley's nodal cubic surface",
    "Cupola_(geometry)_1": "Plane \"hexagonal cupolae\" in the rhombitrihexagonal tiling",
    "Cupola_(geometry)_2": "A tetracontagonal cupola has: 40 isosceles triangles; 40 rectangles; A top regular tetracontagon; and a bottom regular octacontagon (hidden).",
    "Curve_1": "A parabola, one of the simplest curves, after (straight) lines",
    "Curve_2": "The curves created by slicing a cone (conic sections) were among the curves studied in ancient Greek mathematics.",
    "Curve_3": "Analytic geometry allowed curves, such as the Folium of Descartes, to be defined using equations instead of geometrical construction.",
    "Curve_of_constant_width_2": "A curve of constant width defined by an 8th-degree polynomial",
    "Curve_of_constant_width_3": "An irregular Reuleaux polygon",
    "Curve_of_constant_width_4": "Applying the crossed-lines method to an arrangement of four lines. The boundaries of the blue body of constant width are circular arcs from four nested pairs of circles (inner circles dark red and outer circles light red).",
    "Curve_of_constant_width_5": "Body of constant width (yellow) formed by intersecting disks (blue) centered on a semi-ellipse (black). The red circle shows a tangent circle to a supporting line, at a point of minimum curvature of the semi-ellipse. The eccentricity of the semi-ellipse in the figure is the maximum possible for this construction.",
    "Cut_(graph_theory)_2": "A maximum cut.",
    "Cycle_graph_1": "A directed cycle graph of length 8",
    "Cycle_graph_2": "Circle graph $C_5$",
    "Cycle_space_1": "The symmetric difference of two Eulerian subgraphs (red and green) is a Eulerian subgraph (blue).",
    "Cyclic_group_1": "The six 6th complex roots of unity form a cyclic group under multiplication. Here, $z$ is a generator, but $z^2$ is not, because its powers fail to produce the odd powers of $z$.",
    "Cyclic_permutation_1": "A cyclic permutation consisting of a single 8-cycle.",
    "Cyclic_permutation_2": "A permutation that is cyclic for the enlarged definition but not for the restricted one, with two fixed points (1-cycles) and a 6-cycle",
    "Cyclic_permutation_3": "Matrix of $\\pi$",
    "Cyclogon_1": "Cyclogon generated by a convex quadrilateral",
    "Cyclogon_2": "Cyclogon generated by a non-convex quadrilateral",
    "Cyclogon_3": "Cyclogon generated by a star-like quadrilateral",
    "Cyclohedron_2": "The 2-dimensional cyclohedron as the centrally symmetric triangulations of the regular hexagon",
    "Cycloid_2": "Demonstration of the properties of the involute of a cycloid",
    "Cycloid_3": "The length of the cycloid as consequence of the property of its involute",
    "Cycloid_4": "Schematic of a cycloidal pendulum.",
    "Cylinder_2": "A right and an oblique circular cylinder",
    "Cylinder_3": "A right circular cylinder with radius $r$ and height $h$",
    "Cylinder_4": "A solid elliptic right cylinder with the semi-axes $a$ and $b$ for the base ellipse and height $h$",
    "Cylinder_5": "A sphere has $\\frac{2}{3}$ the volume and surface area of its circumscribing cylinder including its bases",
    "Cylindrical_multipole_moments_1": "Definitions for cylindrical multipoles; looking down the $z'$ axis",
    "Dawson_function_2": "The Dawson function, $F(x) = D_+(x)$, around the origin.",
    "Dawson_function_3": "The Dawson function, $F(x) = D_-(x)$, around the origin.",
    "De_Bruijn_graph_1": "The three-dimensional binary De Bruijn graph",
    "De_Bruijn_graph_2": "The De Bruijn graph as a line graph.",
    "De_Bruijn_sequence_1": "A de Bruijn graph. Every four-digit sequence occurs exactly once if one traverses every edge exactly once and returns to one's starting point (a Eulerian cycle). Every three-digit sequence occurs exactly once if one visits every vertex exactly once (a Hamiltonian path).",
    "De_Bruijn_sequence_2": "Directed graphs of two $B(2,3)$ de Bruijn sequences and a $B(2,4)$ sequence. In $B(2,3)$, each vertex is visited once, whereas in $B(2,4)$, each edge is traversed once.",
    "De_Bruijn_sequence_3": "Standard permutation for the Inverse Burrows-Wheeler transformation that will construct a De Bruijn sequence",
    "De_Gua's_theorem_1": "Tetrahedron with a right-angle corner in O",
    "De_Rham_cohomology_1": "Vector field corresponding to a differential form on the punctured plane that is closed but not exact, showing that the de Rham cohomology of this space is non-trivial.",
    "De_Rham_curve_1": "Cesàro curve for $a = 0.3 + i 0.3$",
    "De_Rham_curve_2": "Cesàro curve for $a = 0.5 + i 0.5$. This is the Lévy C curve.",
    "De_Rham_curve_3": "Koch-Peano curve for $a = 0.6 + i 0.37$. This is close to, but not quite the Koch curve.",
    "De_Rham_curve_4": "Koch-Peano curve for $a = 0.6 + i 0.45$.",
    "De_Rham_curve_5": "Generic affine de Rham curve.",
    "De_Rham_curve_6": "Generic affine de Rham curve.",
    "De_Rham_curve_7": "Generic affine de Rham curve.",
    "De_Rham_curve_8": "Generic affine de Rham curve.",
    "Decomposition_method_(constraint_satisfaction)_1": "Choosing the node $b$ as the root, this is the tree similar to the ones created by the other decomposition methods",
    "Decomposition_method_(constraint_satisfaction)_2": "An hypertree decomposition of the same problem of the query decomposition above. $R(b,d,e,-)$ means that the last variable of $R$ is not a variable associated to the root. By grouping two variables in one constraint in the root, width decreases from three to two",
    "Decomposition_method_(constraint_satisfaction)_3": "The constraint passed from node $i$ to node $j$ summarizes the effects of the nodes on the \"side\" of i to the variables of $j$.",
    "Decomposition_method_(constraint_satisfaction)_4": "The biconnected decomposition: nodes of the tree are separating vertices and biconnected components",
    "Decomposition_method_(constraint_satisfaction)_5": "Its biconnected components (colored) and its separating vertices (black, only one in this case)",
    "Dedekind_cut_1": "Dedekind used his cut to construct the irrational, real numbers.",
    "Defective_coloring_1": "Example of defective colouring of a cycle on five vertices when $d = 0, 1, 2$",
    "Degeneracy_(mathematics)_1": "The three types of degenerate triangles, all of which contain zero area.",
    "Degenerate_conic_1": "The degenerate hyperbola $3x^{2}-2xy-y^{2}-6x+10y-9=0,$ which factors as $(x-y+1)(3x+y-9)=0,$ is the union of the red and blue loci.",
    "Degenerate_conic_2": "The degenerate parabola $9x^{2}+12xy+4y^{2}-54x-36y+72$ $=0,$ which factors as $(3x+2y-6)(3x+2y-12)=0,$ is the union of the red and blue loci.",
    "Degenerate_conic_3": "Pencils of circles: in the pencil of red circles, the only degenerate conic is the horizontal axis; the pencil of blue circles has three degenerate conics, the vertical axis and two circles of radius zero.",
    "Degree_(graph_theory)_1": "A graph with a loop having vertices labeled by degree",
    "Degree_of_a_continuous_mapping_1": "A degree two map of a sphere onto itself.",
    "Degree_of_curvature_1": "Diagram showing different parts of the curve used in the formula",
    "Dehn_twist_1": "General Dehn twist on a compact surface represented by a n-gon.",
    "Delta_set_1": "Three Delta-set structures on the circle, with the third also being a simplicial complex.",
    "Delta_set_2": "A diagram of Delta-maps.",
    "Delta_set_3": "Delta-set structures for the torus, the real projective plane, and the Klein bottle.",
    "Deltoidal_icositetrahedron_1": "3D model of a dyakis dodecahedron",
    "Demihypercube_2": "The rhombic disphenoid inside of a cuboid",
    "Dendrite_(mathematics)_1": "Dendrite Julia set",
    "Descartes'_theorem_1": "Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have.",
    "Descartes'_theorem_2": "Here, as all three circles are tangent to each other at the same point, Descartes' theorem does not apply.",
    "Dessin_d'enfant_1": "The dessin d'enfant arising from the rational function $f=-(x-1)^{3}(x-9)/64x$. Not to scale.",
    "Dessin_d'enfant_2": "Transforming a dessin d'enfant into a gluing pattern for halfspaces of a Riemann surface by including points at infinity.",
    "Dessin_d'enfant_3": "The triangulation of the hyperbolic plane with $(2,3,7)$ triangle group generated as the universal cover of the Klein quartic",
    "Dessin_d'enfant_4": "The dessin d'enfant corresponding to the sextic monomial $p(x)=x^{6}$.",
    "Dessin_d'enfant_5": "The Chebyshev polynomials and the corresponding dessins d'enfants, alternately-colored path graphs.",
    "Dessin_d'enfant_6": "Two conjugate dessins d'enfants",
    "Determinant_1": "The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.",
    "Determinant_2": "The volume of this parallelepiped is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.",
    "Difference_of_Gaussians_1": "Comparison between the difference of Gaussians and the Mexican hat wavelet",
    "Differentiable_function_2": "Differentiable functions can be locally approximated by linear functions.",
    "Differentiable_function_3": "The function $f:\\mathbb {R} \\to \\mathbb {R}$ with $f(x)=x^{2}\\sin \\left({\\tfrac {1}{x}}\\right)$ for $x\\neq 0$ and $f(0)=0$ is differentiable. However, this function is not continuously differentiable.",
    "Differentiable_function_4": "The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point $x = 0$, where it makes a sharp turn as it crosses the y-axis.",
    "Differentiable_manifold_1": "The transition map of two charts: $\\phi_{\\alpha\\beta}$ denotes $\\phi_{\\alpha} \\circ \\phi_{\\beta}^{-1}$ and $\\phi_{\\beta\\alpha}$ denotes $\\phi_{\\beta} \\circ \\phi_{\\alpha}^{-1}$.",
    "Differential_geometry_of_surfaces_1": "A local parametrization of Monge form for the upper hemisphere of the 2-sphere, obtained by projecting onto the $xy$-plane",
    "Differential_geometry_of_surfaces_2": "Coordinate changes between different local charts must be smooth",
    "Differential_geometry_of_surfaces_3": "A hyperboloid of two sheets",
    "Differential_geometry_of_surfaces_4": "Definition of second fundamental form",
    "Differential_geometry_of_surfaces_5": "A quadric ellipsoid",
    "Differential_geometry_of_surfaces_6": "The catenoid is a regular surface of revolution.",
    "Differential_of_a_function_1": "The differential of a function $f(x)$ at a point $x_{0}$.",
    "Dimension_1": "The first four spatial dimensions, represented in a two-dimensional picture. Two points can be connected to create a line segment. Two parallel line segments can be connected to form a square. Two parallel squares can be connected to form a cube. Two parallel cubes can be connected to form a tesseract.",
    "Dini's_surface_2": "Dini's surface with $0 \\leq u \\leq 4\\pi$ and $0.01 \\leq v \\leq 1$ and constants $a = 1.0$ and $b = 0.2$.",
    "Diophantus_II.VIII_2": "Diophantus II.VIII: Generalized solution in which the sides of triangle $OAB$ form a rational triple if line $CB$ has a rational gradient $t$.",
    "Dirac_delta_function_1": "Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.",
    "Dirac_delta_function_2": "A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of $T$",
    "Direct_proof_1": "Diagram of Pythagoras Theorem",
    "Director_circle_1": "An ellipse, its minimum bounding box, and its director circle.",
    "Dirichlet_beta_function_1": "The Dirichlet beta function",
    "Discrete_Poisson_equation_1": "Poisson convergence of various iterative methods with infinity norms of residuals against iteration count and computer time.",
    "Discrete_calculus_2": "The Riemann sum is measuring the total area of the bars, defined by $f$, between two points (here $a$ and $b$).",
    "Discrete_geometry_2": "Seven points are elements of seven lines in the Fano plane, an example of an incidence structure.",
    "Discrete_geometry_3": "A collection of circles and the corresponding unit disk graph",
    "Disjoint_sets_1": "Two disjoint sets",
    "Disjoint_sets_2": "A disjoint collection of sets",
    "Disjoint_union_(topology)_1": "Characteristic property of disjoint unions",
    "Disk_(mathematics)_1": "Disk with circumference $C$ diameter $D$ radius $R$ center or origin $O$",
    "Disk_(mathematics)_2": "The average distance to a location from points on a disc",
    "Disk_(mathematics)_3": "The average distance from a disk to an internal point",
    "Disk_(mathematics)_4": "The average distance from a disk to an external point",
    "Distance_correlation_1": "Several sets of $(x, y)$ points, with the distance correlation coefficient of $x$ and $y$ for each set. Compare to the graph on correlation",
    "Distance_from_a_point_to_a_line_1": "Diagram for geometric proof",
    "Distance_from_a_point_to_a_line_2": "Diagram for vector projection proof",
    "Distance_from_a_point_to_a_line_3": "Illustration of the vector formulation.",
    "Distance_geometry_1": "Problem of hyperbolic navigation",
    "Distributivity_(order_theory)_1": "Hasse diagram for the definition of distributivity for a meet-semilattice.",
    "Dividing_a_circle_into_areas_2": "Proof without words that summing up to the first 5 terms on each row is equivalent to summing up to the first 3 odd terms of the next row",
    "Dividing_a_circle_into_areas_3": "$r_{G}$ (purple) and other OEIS sequences in Bernoulli's triangle",
    "Dividing_a_circle_into_areas_4": "Maximum number of regions that an inscribed hexagon and its diagonals can divide a circle (top) compared to when the hexagon is regular (bottom)",
    "Divisor_summatory_function_2": "The summatory function, with leading terms removed, for $x<10^{7}$",
    "Divisor_summatory_function_3": "The summatory function, with leading terms removed, for $x<10^{7}$, graphed as a distribution or histogram. The vertical scale is not constant left to right.",
    "Dixon_elliptic_functions_1": "The Dixon elliptic functions \\$\\operatorname{cm}\\$, \\$\\operatorname{sm}\\$ applied to a real-valued argument \\$x\\$. Both functions are periodic with real period \\$\\pi\\_3 \\approx 5.29991625\\$.",
    "Dixon_elliptic_functions_2": "The function \\$t \\mapsto (\\operatorname{cm}, t, \\operatorname{sm}, t)\\$ parametrizes the cubic Fermat curve, with area of the sector equal to half the argument \\$t\\$.",
    "Dixon_elliptic_functions_3": "Elliptic curve $y^2 = 4x^3 - \\tfrac{1}{27}$ for the Weierstrass $\\wp$-function $z \\mapsto \\wp\\left(z;,0,,\\tfrac{1}{27}\\right)$ related to the Dixon elliptic functions.",
    "Domain_decomposition_methods_1": "Domain decomposition methods",
    "Dominance_order_1": "The dominance order on standard Young tableaux for the partition $6=4+2$",
    "Dominance_order_2": "Example of dominance ordering of partitions of $n$. Here, $n = 6$, nodes are partitions of 6, edges indicate that the upper node dominates the lower node. While this particular partial ordering is graded, this is not true for the dominance ordering on partitions of any number $n > 6$.",
    "Double_exponential_function_1": "A double exponential function (red curve) compared to a single exponential function (blue curve).",
    "Doubling_space_1": "In the Euclidean plane, seven disks of radius $\\frac{r}{2}$ can cover any disk of radius $r$, so the plane is a doubling space with doubling constant 7 and doubling dimension $\\log_2 7$.",
    "Dragon_curve_2": "Tiling of the plane by dragon curves",
    "Dragon_curve_3": "Twindragon curve constructed from two Heighway dragons",
    "Dragon_curve_4": "Lévy C curve.",
    "Droz-Farny_line_theorem_1": "The line through $A_{0},B_{0},C_{0}$ is Droz-Farny line",
    "Droz-Farny_line_theorem_2": "Dao's first generalization",
    "Droz-Farny_line_theorem_3": "Dao's second generalization",
    "Du_Val_singularity_1": "Du Val singularies are classified by the simply laced Dynkin diagrams, a form of ADE classification.",
    "Dual_cone_and_polar_cone_1": "A set $C$ and its polar cone $C^{\\circ}$. The dual cone and the polar cone are symmetric to each other with respect to the origin.",
    "Dual_cone_and_polar_cone_2": "The polar of the closed convex cone $C$ is the closed convex cone $C^{\\circ}$, and vice versa.",
    "Dual_cone_and_polar_cone_3": "A set $C$ and its dual cone $C^{*}$.",
    "Dual_polyhedron_2": "The dual of a Platonic solid can be constructed by connecting the face centers. In general this creates only a topological dual.Images from Kepler's Harmonices Mundi (1619)",
    "Dual_polyhedron_3": "Canonical dual compound of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common midsphere.",
    "Dual_polyhedron_4": "The square tiling, $\\{4,4\\}$, is self-dual, as shown by these red and blue tilings",
    "Dual_polyhedron_5": "The Infinite-order apeirogonal tiling, $\\{\\infty, \\infty\\}$ in red, and its dual position in blue",
    "Duality_(mathematics)_1": "A set $C$ (blue) and its dual cone $C^*$ (red).",
    "Duality_(projective_geometry)_1": "Pole and polar with respect to circle $C$. $P$ and $Q$ are inverse points, $p$ is the polar of $P$, $P$ is the pole of $p$.",
    "Duality_(projective_geometry)_2": "Three pairs of dual points and lines: one red pair, one yellow pair, and one blue pair.",
    "Duality_(projective_geometry)_3": "Diagonal triangle $P$, $Q$, $R$ of quadrangle $A, B, J, K$ on conic. Polars of diagonal points are colored the same as the points.",
    "Duffing_equation_1": "A Poincaré section of the forced Duffing equation suggesting chaotic behaviour ($\\alpha =1$, $\\beta =5$, $\\delta =0.02$, $\\gamma =8$, and $\\omega =0.5$).",
    "Duffing_equation_2": "Jumps in the frequency response. The parameters are: $\\alpha =\\gamma =1,$, $\\beta =0.04$, and $\\delta =0.1$.",
    "Dynamic_rectangle_1": "A root-phi rectangle divides into a pair of Kepler triangles (right triangles with edge lengths in geometric progression).",
    "Dynamic_rectangle_2": "Caskey's 1922 illustration of the property that a root-N rectangle divides into N reciprocal rectangles of the same proportions.",
    "Dynamical_system_2": "Linear vector fields and a few trajectories.",
    "E8_(mathematics)_2": "Hasse diagram of $E_8$ root poset with edge labels identifying added simple root position",
    "E8_(mathematics)_3": "An incomplete simple subgroup tree of $E_8$",
    "E8_(mathematics)_4": "Embeddings of the maximal subgroups of $E_8$ up to dimension 248 with associated projection matrix.",
    "Ear_decomposition_1": "An example of an ear decomposition of a graph containing 3 ears.",
    "Edge-graceful_labeling_1": "An edge-graceful labeling of $C_5$",
    "Edge-transitive_graph_1": "The Gray graph is edge-transitive and regular, but not vertex-transitive.",
    "Edwards_curve_1": "Edwards curves of equation $x^2 + y^2 = 1 + d \\cdot x^2 y^2$ over the real numbers for $d = -300$ (red), $d = -\\sqrt{8}$ (yellow), and $d = 0.9$ (blue).",
    "Edwards_curve_2": "Sum of two points on the Edwards curve with $d = -30$",
    "Edwards_curve_3": "Doubling a point on the Edwards curve with $d=-30$",
    "Ehresmann_connection_1": "An Ehresmann connection is a choice of horizontal subspace $H_{p}\\subset T_{p}P$ for every $p\\in P$, where $P$ is some fiber bundle, typically a principal bundle.",
    "Ehresmann_connection_2": "A principal bundle connection form $\\omega$ may be thought of as a projection operator on the tangent bundle $TP$ of the principal bundle $P$. The kernel of the connection form is given by the horizontal subspaces for the associated Ehresmann connection.",
    "Eigenvalues_and_eigenvectors_2": "In this shear mapping the red arrow changes direction, but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it does not change direction, and since its length is unchanged, its eigenvalue is 1.",
    "Eigenvalues_and_eigenvectors_3": "Matrix $A$ acts by stretching the vector $x$, not changing its direction, so $x$ is an eigenvector of $A$.",
    "Eisenstein_integer_1": "Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form $3n+2$. All others have an absolute value equal to 3 or square root of a natural prime of the form $3n+1$.",
    "Eisenstein_triple_2": "Triangle with ${120}^\\circ$ angle and integer sides",
    "Elkies_trinomial_curves_1": "Elkies trinomial curve $C_{168}$",
    "Elliptic_coordinate_system_1": "Elliptic coordinate system",
    "Elliptic_curve_point_multiplication_1": "Elliptic curve point operations: Addition (shown in facet 1), doubling (facets 2 and 4) and negation (facet 3).",
    "Elliptic_function_1": "The fundamental domain of an elliptic function as the unit cell of its period lattice.",
    "Enumerative_geometry_1": "Circles of Apollonius",
    "Epigraph_(mathematics)_2": "A function (in black) is convex if and only if the region above its graph (in green) is a convex set. This region is the function's epigraph.",
    "Epispiral_1": "An epispiral with equation $r(\\theta) = 2\\sec(2\\theta)$.",
    "Equal_incircles_theorem_1": "If the blue circles are equal, the green circles are also equal.",
    "Equal_incircles_theorem_2": "Geometry for the proof of equal incircles theorem.",
    "Equidiagonal_quadrilateral_2": "An equidiagonal kite that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux triangle",
    "Equilateral_pentagon_2": "Convex equilateral pentagon dissected into 3 triangles, which helps to calculate the value of angle $\\delta$ as a function of $\\alpha$ and $\\beta$.",
    "Equilateral_pentagon_3": "Cairo pentagonal tiling by equilateral pentagons with two non-adjacent right angles",
    "Equilateral_pentagon_4": "Concave pentagonal tile as union of equilateral triangle and 30 degree rhombus",
    "Equilateral_polygon_1": "Four Reinhardt pentadecagons",
    "Equilateral_polygon_2": "Convex equilateral pentagon",
    "Equilateral_polygon_3": "Concave equilateral pentagon",
    "Equilateral_triangle_2": "The regular octahedron is a deltahedron, as well as a member of the family of antiprisms.",
    "Equilateral_triangle_3": "Visual proof of Viviani's theorem",
    "Equipollence_(geometry)_1": "If the segments $AB$ and $CD$ are equipollent, then $AC$ and $BD$ are also equipollent",
    "Equitable_coloring_1": "An equitable coloring of the star $K_{1,5}$.",
    "Equivariant_map_1": "The centroid of a triangle (where the three red segments meet) is equivariant under affine transformations: the centroid of a transformed triangle is the same point as the transformation of the centroid of the triangle.",
    "Errera_graph_1": "Tangled Kempe chains in the Errera graph.",
    "Errera_graph_2": "Errera graph in polyhedron",
    "Errera_graph_3": "The Errera graph is planar.",
    "Errera_graph_4": "The chromatic index of the Errera graph is 6.",
    "Errera_graph_5": "The chromatic index of the Errera graph is 4.",
    "Error_function_2": "Plot of the complementary error function $\\mathrm{erfc}(z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors",
    "Error_function_3": "Plot of the imaginary error function $\\mathrm{erfi}(z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors",
    "Error_function_4": "A graph of the error function between -3 and 3.",
    "Error_function_5": "Inverse error function",
    "Error_function_6": "the normal cumulative distribution function plotted in the complex plane",
    "Escaping_set_1": "Escaping set of $(\\exp(z)-1)/2$.",
    "Euclid's_orchard_1": "One corner of Euclid's orchard, blue trees visible from the origin",
    "Euclid's_orchard_2": "Perspective view of Euclid's orchard from the origin. Red trees denote rows two off the main diagonal.",
    "Euclid's_orchard_3": "Plan view of one corner of Euclid's orchard, in which trees are labelled with the x co-ordinate of their projection on the plane $x + y = 1$.",
    "Euclid's_orchard_4": "Plan view of one corner of Euclid's orchard, in which trees are labelled with the $x$ co-ordinate of their projection on the plane $x + y = 1$.",
    "Euclid's_orchard_5": "One corner of Euclid's orchard, blue trees visible from the origin",
    "Euclid's_orchard_6": "Perspective view of Euclid's orchard from the origin. Red trees denote rows two off the main diagonal.",
    "Euclidean_distance_1": "Using the Pythagorean theorem to compute two-dimensional Euclidean distance",
    "Euclidean_distance_2": "Deriving the $n$-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem",
    "Euclidean_minimum_spanning_tree_1": "Euclidean minimum spanning tree of 25 random points in the plane",
    "Euclidean_plane_isometry_1": "Isometries as mirrors",
    "Euclidean_plane_isometry_2": "Translation addition with mirrors",
    "Euclidean_plane_isometry_3": "Dihedral group of regular pentagon symmetries",
    "Euclidean_space_1": "A point in three-dimensional Euclidean space can be located by three coordinates.",
    "Euclidean_space_2": "Origin-free illustration of the Euclidean plane",
    "Euclidean_space_3": "Positive and negative angles on the oriented plane",
    "Euclidean_space_4": "3-dimensional skew coordinates",
    "Euler's_rotation_theorem_2": "Blue great circle on sphere transforms into red great circle when rotated about diameter through $O$.",
    "Euler's_rotation_theorem_3": "Arcs connecting preimage $\\alpha$ and image $a$ of $A$ with bisector $AO$ of the angle at $A$.",
    "Euler's_rotation_theorem_4": "$O$ goes to $O'$, but $O'$ must coincide with $O$.",
    "Euler's_rotation_theorem_5": "Euler's original drawing where $ABC$ is the blue circle and $ACc$ is the red circle",
    "Euler's_theorem_in_geometry_1": "Euler's theorem: $d=|IO|={\\sqrt {R(R-2r)}}$",
    "Euler_characteristic_1": "Vertex, edge and face of a cube",
    "Euler_characteristic_2": "First steps of the proof in the case of a cube",
    "Euler_characteristic_3": "Comparison of Euler characteristics of hypercubes and simplices of dimensions 1 to 4",
    "Euler_line_1": "Euler's line, with the center of the nine-point circle Medians (intersect at the centroid) Altitudes (intersect at the orthocenter) Perpendicular lines from the side midpoints (intersect at the circumcenter)",
    "Euler_method_2": "Solution of $y'=-2.3y$ computed with the Euler method with step size $h=1$ (blue squares) and $h=0.7$ (red circles). The black curve shows the exact solution.",
    "Euler_method_3": "The pink disk shows the stability region for the Euler method.",
    "Euler_method_4": "Illustration of numerical integration for the equation $y'=y$, $y(0)=1$. Blue is the Euler method; green, the midpoint method; red, the exact solution, $y=e^t$. The step size is $h=1.0\".",
    "Euler_tour_technique_1": "Euler tour of a tree, with edges labeled to show the order in which they are traversed by the tour",
    "Eulerian_number_1": "A plot of the Eulerian numbers with the second argument fixed to 5.",
    "Eulerian_path_2": "Every vertex of this graph has an even degree. Therefore, this is an Eulerian graph. Following the edges in alphabetical order gives an Eulerian circuit/cycle.",
    "Eutactic_star_2": "A eutactic star in the plane $(n = 2, s = 4)$",
    "Even_and_odd_functions_1": "The sine function and all of its Taylor polynomials are odd functions.",
    "Even_and_odd_functions_2": "The cosine function and all of its Taylor polynomials are even functions.",
    "Even_and_odd_functions_3": "$f(x)=x^{2}$ is an example of an even function.",
    "Even_and_odd_functions_4": "$f(x)=x^{3}$ is an example of an odd function.",
    "Even_and_odd_functions_5": "$f(x)=x^{3}+1$ is neither even nor odd.",
    "Ex-tangential_quadrilateral_1": "Ex-tangential quadrilateral $ABCD$ Extended sides of $ABCD$ Excircle of $ABCD$",
    "Exact_coloring_2": "Exact coloring of the complete graph $K_6$",
    "Exact_sequence_1": "Illustration of an exact sequence of groups $G_{i}$ using Euler diagrams.",
    "Exact_sequence_2": "Shows how any exact sequence is weaved together from several overlapping short exact sequences.",
    "Examples_of_vector_spaces_1": "Planar analytic geometry uses the coordinate space $R^2$. Depicted: description of a line as the solution set in ${\\vec {x}}$ of the vector equation ${\\vec {x}}\\cdot {\\vec {n}}=d$.",
    "Exceptional_isomorphism_1": "The compound of five tetrahedra expresses the exceptional isomorphism between the chiral icosahedral group and the alternating group on five letters.",
    "Expected_value_1": "An illustration of the convergence of sequence averages of rolls of a dice to the expected value of 3.5 as the number of rolls (trials) grows",
    "Expected_value_2": "Expected value $\\mu$ and median $m$",
    "Expected_value_3": "The mass of a probability distribution is balanced at the expected value, here a $\\text{Beta}(\\alpha, \\beta)$ distribution with expected value $\\frac{\\alpha}{\\alpha + \\beta}$.",
    "Exponential_growth_1": "The graph illustrates how exponential growth (green) eventually surpasses both linear (red) and cubic (blue) growth. Linear growth Cubic growth Exponential growth",
    "Exponential_growth_2": "exponential growth: ${\\begin{aligned}a&=3\\\\b&=2\\\\r&=5\\end{aligned}}$",
    "Exponential_growth_3": "exponential decay: ${\\begin{aligned}a&=24\\\\b&={\\frac {1}{2}}\\\\r&=5\\end{aligned}}$",
    "Exponential_growth_4": "The J-shaped exponential growth (left, blue) and the S-shaped logistic growth (right, red).",
    "Exponential_growth_5": "Graphs comparing doubling times and half-lives of exponential growths (bold lines) and decay (faint lines), and their $\\frac{70}{t}$ and $\\frac{72}{t}$ approximations.",
    "Exponential_object_1": "Universal property of the exponential object",
    "Exponential_type_1": "The graph of the function in gray is $e^{-\\pi z^{2}}$, the Gaussian restricted to the real axis. The Gaussian does not have exponential type, but the functions in red and blue are one sided approximations that have exponential type $2\\pi$.",
    "Exterior_algebra_2": "The cross product (blue vector) in relation to the exterior product (light blue parallelogram). The length of the cross product is to the length of the parallel unit vector (red) as the size of the exterior product is to the size of the reference parallelogram (light red).",
    "Exterior_algebra_3": "Basis Decomposition of a 2-vector",
    "Exterior_algebra_4": "Universal property of the exterior algebra",
    "Exterior_algebra_5": "Geometric interpretation for the exterior product of n 1-forms $(\\varepsilon, \\eta, \\omega)$ to obtain an n-form (mesh of coordinate surfaces, here planes), for $n = 1, 2, 3$. The circulations show orientation.",
    "Exterior_angle_theorem_2": "The diagram for the remote exterior angle theorem.",
    "Extrapolation_1": "Example illustration of the extrapolation problem, consisting of assigning a meaningful value at the blue box, at $x=7$, given the red data points.",
    "Extremal_graph_theory_1": "The Petersen graph has chromatic number 3.",
    "Extremal_graph_theory_2": "The edges between parts in a regular partition behave in a \"random-like\" fashion.",
    "Extreme_value_theorem_1": "A continuous function $f(x)$ on the closed interval $[a,b]$ showing the absolute max (red) and the absolute min (blue).",
    "F-distribution_1": "The pdf of the F distribution.",
    "F-distribution_2": "The cdf of the F distribution.",
    "Fabius_function_1": "Graph of the Fabius function on the interval $[0,1]$.",
    "Fabius_function_2": "Extension of the function to the nonnegative real numbers.",
    "Factorization_1": "Visual proof of the differences between two squares and two cubes",
    "Factorization_2": "Visualisation of binomial expansion up to the 4th power",
    "Factorization_3": "The polynomial $x^2 + cx + d$, where $a + b = c$ and $ab = d$, can be factorized into $(x + a)(x + b)$.",
    "Fano_plane_2": "Bipartite Heawood graph. Points are represented by vertices of one color and lines by vertices of the other color.",
    "Fano_plane_3": "A collineation of the Fano plane corresponding to the 3-bit Gray code permutation",
    "Fano_plane_4": "Bijection between the Fano plane as field with eight elements minus the origin and the projective line over the field with seven elements. Symmetries are made explicit.",
    "Fano_plane_5": "Duality in the Fano plane: Each point corresponds to a line and vice versa.",
    "Fano_plane_6": "A nimber numbering of the Fano plane",
    "Farey_sequence_1": "Farey diagram to $F_9$ represented with circular arcs.",
    "Farey_sequence_2": "Plotting $F_6$ numerators vs denominators",
    "Farey_sequence_3": "Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of $1 + \\frac{96}{2} - 1 = 48$, according to Pick's theorem.",
    "Farey_sequence_4": "Farey diagram with straight lines in a square, order $F_9$.",
    "Farey_sequence_5": "Pattern made by the denominators of the Farey sequence, order = $F_{25}$.",
    "Feasible_region_1": "A problem with five linear constraints (in blue, including the non-negativity constraints). In the absence of integer constraints the feasible set is the entire region bounded by blue, but with integer constraints it is the set of red dots.",
    "Feasible_region_2": "A closed feasible region of a linear programming problem with three variables is a convex polyhedron.",
    "Feasible_region_3": "A bounded feasible set (top) and an unbounded feasible set (bottom). The set at the bottom continues forever towards the right.",
    "Fekete_polynomial_1": "Roots of the Fekete polynomial for $p = 43$",
    "Fence_(mathematics)_1": "The Hasse diagram of a six-element fence.",
    "Fermat's_right_triangle_theorem_1": "Two right triangles with the two legs of the top one equal to the leg and hypotenuse of the bottom one. For these lengths, $a^{2}$, $b^{2}$, and $c^{2}$ form an arithmetic progression separated by a gap of $d^{2}$. It is not possible for all four lengths $a$, $b$, $c$, and $d$ to be integers.",
    "Fermat's_right_triangle_theorem_2": "The elliptic curve $y^2 = x(x + 1)(x - 1)$. The three rational points $(-1,0)$, $(0,0)$, and $(1,0)$ are the crossings of this curve with the $x$-axis.",
    "Fermat's_spiral_1": "Fermat's spiral, both branches",
    "Fermat's_spiral_2": "A Fermat's spiral divides the plane into two connected and congruent regions (diagram: black and white)",
    "Fermat's_spiral_3": "Fermat's spiral: $a>0$, one branch $r=+a\\sqrt{\\varphi}$",
    "Fermat_point_2": "Geometry of the Fermat point",
    "Fermat_point_3": "The two isogonic centers are the intersections of three vesicae piscis whose paired vertices are the vertices of the triangle",
    "Fermat_point_4": "Construction of the first isogonic center, $X(13)$. When no angle of the triangle exceeds $120^\\circ$, this point is the Fermat point.",
    "Feuerbach_point_1": "Feuerbach's theorem: the nine-point circle is tangent to the incircle and excircles of a triangle. The incircle tangency is the Feuerbach point.",
    "Fiber_bundle_1": "The Möbius strip is a nontrivial bundle over the circle.",
    "Fiber_bundle_2": "The Klein bottle immersed in three-dimensional space.",
    "Fibered_knot_1": "Figure-eight knot is fibered.",
    "Fibered_knot_2": "The stevedore knot is not fibered",
    "Figure-eight_knot_(mathematics)_2": "Simple squared depiction of figure-eight configuration.",
    "Figure-eight_knot_(mathematics)_3": "Symmetric depiction generated by parametric equations.",
    "Figure-eight_knot_(mathematics)_4": "Mathematical surface Illustrating Figure-eight knot",
    "Final_stellation_of_the_icosahedron_1": "3D model of the final stellation of the icosahedron",
    "Final_stellation_of_the_icosahedron_2": "Stellation diagram of the icosahedron with numbered cells. The complete icosahedron is formed from all the cells in the stellation, but only the outermost regions, labelled \"13\" in the diagram, are visible.",
    "Final_stellation_of_the_icosahedron_3": "A polyhedral model can be constructed by 12 sets of faces, each folded into a group of five pyramids.",
    "Final_stellation_of_the_icosahedron_4": "Complete stellation of the icosahedron, orthogonal projection, centered on inner vertex",
    "Final_stellation_of_the_icosahedron_5": "Complete icosahedron, orthogonal projection, centered on edge",
    "Finite_geometry_2": "$PG(3,2)$ but not all the lines are drawn",
    "Finite_geometry_3": "Square model of Fano 3-space",
    "Finite_geometry_4": "Finite affine plane of order 3, containing 9 points and 12 lines.",
    "Finite_geometry_5": "Duality in the Fano plane: Each point corresponds to a line and vice versa.",
    "Finite_group_1": "A Cayley graph of the symmetric group $S_4$",
    "Fish_curve_1": "The fish curve with scale parameter $a = 1$",
    "Fisher's_method_1": "Under Fisher's method, two small p-values $P_1$ and $P_2$ combine to form a smaller p-value. The darkest boundary defines the region where the meta-analysis p-value is below 0.05. For example, if both p-values are around 0.10, or if one is around 0.04 and one is around 0.25, the meta-analysis p-value is around 0.05.",
    "Fisher_transformation_1": "A graph of the transformation (in orange). The untransformed sample correlation coefficient is plotted on the horizontal axis, and the transformed coefficient is plotted on the vertical axis. The identity function (gray) is also shown for comparison.",
    "Fisher_transformation_2": "Fisher Transformation with $\\rho = 0.9$ and $N = 30$. Illustrated is the exact probability density function of $r$ (in black), together with the probability density functions of the usual Fisher transformation (blue) and that obtained by including extra terms that depend on $N$ (red). The latter approximation is visually indistinguishable from the exact answer (its maximum error is 0.3\\%, compared to 3.4\\% of basic Fisher).",
    "Five-point_stencil_1": "An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively).",
    "Fixed-point_iteration_1": "The fixed-point iteration $x_{n+1} = \\sin x_n$ with initial value $x_0 = 2$ converges to 0. This example does not satisfy the assumptions of the Banach fixed-point theorem and so its speed of convergence is very slow.",
    "Fixed-point_iteration_2": "The fixed point iteration $x_{n+1} = \\cos x_n$ with initial value $x_1 = -1$.",
    "Flat_module_1": "Factor property of a flat module",
    "Flat_module_2": "Module properties in commutative algebra",
    "Flattening_2": "A sphere of radius a compressed to an oblate ellipsoid of revolution.",
    "Flow_graph_(mathematics)_1": "An example of a signal-flow graph",
    "Flow_graph_(mathematics)_2": "Flow graph for three simultaneous equations. The edges incident on each node are colored differently just for emphasis.",
    "Focus_(geometry)_1": "Point $F$ is a focus point for the red ellipse, green parabola and blue hyperbola.",
    "Focus_(geometry)_2": "The foci of an ellipse (purple crosses) are at intersects of the major axis (red) and a circle (cyan) of radius equal to the semi-major axis (blue), centred on an end of the minor axis (grey)",
    "Foliation_2": "2-dimensional section of Reeb foliation",
    "Foliation_3": "3-dimensional model of Reeb foliation",
    "Folium_of_Descartes_1": "The folium of Descartes (green) with asymptote (blue) when $a=1$",
    "Folium_of_Descartes_2": "Folium of Descartes in polar coordinates",
    "Folium_of_Descartes_3": "Trisectrix of Maclaurin",
    "Folkman_graph_1": "Construction of the Folkman graph from the complete graph $K_{5}$. The green vertices subdivide each edge of $K_{5}$, and the red pairs of vertices are the result of doubling the five vertices of $K_{5}$.",
    "Folkman_graph_2": "The Folkman graph with its vertices arranged in a Hamiltonian cycle. The edges that are not used in this cycle form the second Hamiltonian cycle of a Hamiltonian decomposition.",
    "Four_color_theorem_2": "A four-colored map of the states of the United States (ignoring lakes and oceans)",
    "Four_color_theorem_3": "A map with four regions, and the corresponding planar graph with four vertices.",
    "Four_color_theorem_4": "A graph containing a Kempe chain consisting of alternating blue and red vertices",
    "Four_color_theorem_5": "Proof without words that a map of US states needs at least four colors.",
    "Four_color_theorem_6": "By joining the single arrows together and the double arrows together, one obtains a torus with seven mutually touching regions; therefore seven colors are necessary.",
    "Four_color_theorem_7": "This construction shows the torus divided into the maximum of seven regions, each one of which touches every other.",
    "Four_color_theorem_8": "Proof without words that the number of colours needed is unbounded in three or more dimensions",
    "Fourier_series_1": "The top graph shows a non-periodic function $s(x)$ in blue defined only over the red interval from $0$ to $P$. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function $s(x)$ is not.",
    "Fourier_series_2": "Plot of the sawtooth wave, a periodic continuation of the linear function $s(x)=x/\\pi$ on the interval $(-\\pi ,\\pi ]$",
    "Fourier_series_3": "The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums converge to (become more and more like) the square wave.",
    "Fourier_series_4": "The blue curve is the cross-correlation of a square wave and a cosine template, as the phase lag of the template varies over one cycle. The amplitude and phase at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding rectangular coordinates can be determined by evaluating the correlation at just two samples separated by $90^\\circ$.",
    "Fractional_Fourier_transform_2": "Fractional Fourier transform in digital signal processing",
    "Fractional_Fourier_transform_3": "Time/frequency distribution of fractional Fourier transform",
    "Fractional_coloring_1": "5:2-coloring of Dodecahedral graph. A 4:2-coloring of this graph does not exist.",
    "Fractional_coloring_2": "Above:A 3:1-coloring of the cycle on 5 vertices, and the corresponding 6:2-coloring.Below: A 5:2 coloring of the same graph.",
    "Free_Boolean_algebra_1": "The Hasse diagram of the free Boolean algebra on two generators, p and q. Take p (left circle) to be \"John is tall\" and q (right circle)to be \"Mary is rich\". The atoms are the four elements in the row just above FALSE.",
    "Free_abelian_group_2": "The rational function $z^{4}/(z^{4}-1)$ has a zero of order four at 0 (the black point at the center of the plot), and simple poles at the four complex numbers $\\pm 1$ and $\\pm i$ (the white points at the ends of the four petals). It can be represented (up to a scalar) by the divisor $4e_{0}-e_{1}-e_{-1}-e_{i}-e_{-i}$ where  $e_{z}$ is the basis element for a complex number $z$ in a free abelian group over the complex numbers.",
    "Freshman's_dream_1": "An illustration of the Freshman's dream in two dimensions. Each side of the square is $X+Y$ in length. The area of the square is the sum of the area of the yellow region $(=X^2)$, the area of the green region $(=Y^2)$, and the area of the two white regions $(=2\\times X\\times Y)$.",
    "Frobenius_method_1": "Some solutions of a differential equation having a regular singular point with indicial roots $r={\\frac {1}{2}}$ and $-1$.",
    "Fuhrmann_circle_2": "Fuhrmann circle",
    "Fuhrmann_triangle_1": "Fuhrmann triangle (red): $\\triangle M_{c}^{\\prime }M_{b}^{\\prime }M_{a}^{\\prime }$ mid arc points: $M_{a}, M_{b}, M_{c}$",
    "Fuhrmann_triangle_2": "Fuhrmann triangle (red): $\\triangle M_{c}^{\\prime }M_{b}^{\\prime }M_{a}^{\\prime} \\sim \\triangle M_{a}M_{b}M_{c}$.",
    "Function_composition_1": "Concrete example for the composition of two functions.",
    "Function_composition_2": "Compositions of two real functions, the absolute value and a cubic function, in different orders, show a non-commutativity of composition.",
    "Function_of_a_real_variable_1": "The graph of a real function",
    "Function_of_a_real_variable_2": "Limit of a real function of a real variable.",
    "Function_of_a_real_variable_3": "Space curve in 3d. The position vector $r$ is parametrized by a scalar $t$. At $r = a$ the red line is the tangent to the curve, and the blue plane is normal to the curve.",
    "Function_of_a_real_variable_4": "Kinematic quantities of a classical particle: mass $m$, position $r$, velocity $v$, acceleration $a$.",
    "Fundamental_group_2": "A loop on a 2-sphere (the surface of a ball) being contracted to a point",
    "Fundamental_group_3": "Elements of the homotopy group of the circle",
    "Fundamental_group_4": "The fundamental group of the figure eight is the free group on two generators $a$ and $b$.",
    "Fundamental_group_5": "A trefoil knot.",
    "Fundamental_group_6": "The map $\\mathbb {R} \\times [0,1]\\to S^{1}\\times [0,1]$ is a covering: the preimage of $U$ (highlighted in gray) is a disjoint union of copies of $U$. Moreover, it is a universal covering since $\\mathbb {R} \\times [0,1]$ is contractible and therefore simply connected.",
    "Fundamental_theorem_of_Galois_theory_1": "Lattice of subfields (left) for $K/\\mathbb {Q}$ and inverted Lattice of subgroups of ${\\text{Gal}}(K/\\mathbb {Q} )$",
    "Fundamental_theorem_of_Galois_theory_2": "Lattice of subfields (left) for $K/\\mathbb {Q}$ and inverted Lattice of subgroups of ${\\text{Gal}}(K/\\mathbb {Q} )$",
    "G2_(mathematics)_1": "The 12 vector root system of $G_2$ in 2 dimensions.",
    "G2_(mathematics)_2": "The $A2$ Coxeter plane projection of the 12 vertices of the cuboctahedron contain the same 2D vector arrangement.",
    "G2_(mathematics)_3": "Graph of $G2$ as a subgroup of $F4$ and $E8$ projected into the Coxeter plane",
    "G2_(mathematics)_4": "Embeddings of the maximal subgroups of $G_2$ up to dimension 77 with associated projection matrix.",
    "Galois_theory_1": "On the left, the lattice diagram of the field obtained from Q by adjoining the positive square roots of 2 and 3, together with its subfields; on the right, the corresponding lattice diagram of their Galois groups.",
    "Gamma_distribution_2": "Cumulative distribution plots of gamma distributions",
    "Gamma_distribution_3": "Bounds and asymptotic approximations to the median of the gamma distribution. The cyan-colored region indicates the large gap between published lower and upper bounds before 2021.",
    "Gamma_distribution_4": "Two gamma distribution median asymptotes which were proved in 2023 to be bounds (upper solid red and lower dashed red), of the form $\\nu(\\alpha) \\approx 2^{-1/\\alpha}(A + \\alpha)$, and an interpolation between them that makes an approximation (dotted red) that is exact at $\\alpha = 1$ and has maximum relative error of about $0.6\\%$.",
    "Gamma_distribution_5": "Illustration of the gamma PDF for parameter values over $\\alpha$ and $x$ with $\\theta$ set to 1, 2, 3, 4, 5, and 6. One can see each $\\theta$ layer by itself here as well as by $\\alpha$ and $x$.",
    "Gamma_function_2": "$\\Gamma (x+1)$ interpolates the factorial function to non-integer values.",
    "Gamma_function_3": "The gamma function, $\\Gamma(z)$ in blue, plotted along with $\\Gamma(z) + \\sin(\\pi z)$ in green. Notice the intersection at positive integers. Both are valid extensions of the factorials to a meromorphic function on the complex plane.",
    "Gamma_function_4": "Absolute value (vertical) and argument (color) of the gamma function on the complex plane",
    "Gamma_function_5": "Colors showing the argument of the gamma function in the complex plane from $-2 - 2i$ to $6 + 2i$",
    "Gamma_function_6": "Representation of the gamma function in the complex plane. Each point $z$ is colored according to the argument of $\\Gamma(z)$. The contour plot of the modulus $|\\Gamma(z)|$ is also displayed.",
    "Gamma_function_7": "The analytic function $\\log\\Gamma(z)$",
    "Gamma_function_8": "Logarithmic gamma function in the complex plane from $-2 - 2i$ to $2 + 2i$ with colors",
    "Gamma_function_9": "Comparison of the gamma function (blue line) with the factorial (blue dots) and Stirling's approximation (red line)",
    "Gauss's_lemma_(Riemannian_geometry)_1": "The exponential map as a radial isometry",
    "Gauss_iterated_map_2": "Bifurcation diagram of the Gauss map with $\\alpha =4.90$ and $\\beta$ in the range -1 to +1. This graph resembles a mouse.",
    "Gauss_iterated_map_3": "Bifurcation diagram of the Gauss map with $\\alpha =6.20$ and $\\beta$ in the range -1 to +1.",
    "Gauss_map_1": "The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere. In this example, the curvature of a 2D-surface is mapped onto a 1D unit circle.",
    "Gauss_map_2": "A surface with a parabolic line and its Gauss map. A ridge passes through the parabolic line giving rise to a cusp on the Gauss map.",
    "Gaussian_curvature_1": "From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere).",
    "Gaussian_curvature_2": "Some points on the torus have positive, some have negative, and some have zero Gaussian curvature.",
    "Gaussian_curvature_3": "Saddle surface with normal planes in directions of principal curvatures",
    "Gaussian_curvature_4": "The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle.",
    "Gaussian_curvature_5": "Two surfaces which both have constant positive Gaussian curvature but with either an open boundary or singular points.",
    "Gaussian_filter_1": "Shape of the impulse response of a typical Gaussian filter",
    "Gaussian_integer_1": "Gaussian integers as lattice points in the complex plane",
    "Gaussian_integer_2": "Visualization of maximal distance to some Gaussian integer",
    "Gaussian_integer_3": "All 13 residue classes with their minimal residues (blue dots) in the square $Q_{00}$ (light green background) for the modulus $z_0 = 3 + 2i$. One residue class with $z = 2 - 4i \\equiv -i \\pmod{z_0}$ is highlighted with yellow/orange dots.",
    "Gaussian_integer_4": "The distribution of the small Gaussian primes in the complex plane",
    "General_linear_group_1": "Cayley table of $GL(2, 2)$, which is isomorphic to $S_3$.",
    "General_topology_1": "The Topologist's sine curve, a useful example in point-set topology. It is connected but not path-connected.",
    "General_topology_2": "This subspace of $R^2$ is path-connected, because a path can be drawn between any two points in the space.",
    "General_topology_3": "Three iterations of a Peano curve construction, whose limit is a space-filling curve. The Peano curve is studied in continuum theory, a branch of general topology.",
    "Generalised_circle_1": "A circle in the complex plane specified as an implicit equation in terms of center and radius",
    "Generalized_hypergeometric_function_1": "Plot of the generalized hypergeometric function $pFq(a, b, z)$ with $a=(2,4,6,8)$ and $b=(2,3,5,7,11)$ in the complex plane from $-2-2i$ to $2+2i$",
    "Generalized_mean_1": "Plot of several generalized means $M_{p}(1,x)$.",
    "Generalized_polygon_1": "The split Cayley hexagon of order 2",
    "Generalized_quadrangle_1": "Line graph of generalized quadrangle $GQ(2,4)$",
    "Geographic_coordinate_conversion_1": "The length PQ, called the prime vertical radius, is $N(\\phi )$. The length IQ is equal to $\\,e^{2}N(\\phi )$. $R=(X,\\,Y,\\,Z)$.",
    "Geometric_group_theory_1": "The Cayley graph of a free group with two generators. This is a hyperbolic group whose Gromov boundary is a Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs.",
    "Geometric_integrator_1": "Simple pendulum: trajectories",
    "Geometric_topology_1": "The Whitney trick requires $2+1$ dimensions, hence surgery theory requires 5 dimensions.",
    "Giacinto_Morera_1": "A curve $C$ in a domain $D$, as required by the statement of Morera's theorem.",
    "Gibbs_sampling_1": "A pictorial description of the Gibbs sampling algorithm",
    "Gibbs_sampling_2": "Schematic description of the information equality associated with the Gibbs sampler at the i-th step within a cycle",
    "Girsanov_theorem_1": "Visualisation of the Girsanov theorem. The left side shows a Wiener process with negative drift under a canonical measure $P$; on the right side each path of the process is colored according to its likelihood under the martingale measure $Q$. The density transformation from $P$ to $Q$ is given by the Girsanov theorem.",
    "Goldbach's_conjecture_1": "Sums of two primes at the intersections of three lines",
    "Goldbach's_conjecture_2": "Number of ways to write an even number n as the sum of two primes (sequence A002375 in the OEIS)",
    "Golden_angle_2": "The angle between successive florets in some flowers is the golden angle.",
    "Golden_ratio_2": "If $\\varphi$ were rational, then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the positive integers have a lower bound, so $\\varphi$ cannot be rational.",
    "Golden_ratio_3": "The golden ratio $\\varphi$ and its negative reciprocal $-\\varphi^{-1}$ are the two roots of the quadratic polynomial $x^2 - x - 1$. The golden ratio's negative $-\\varphi$ and reciprocal $\\varphi^{-1}$ are the two roots of the quadratic polynomial $x^2 + x - 1$.",
    "Golden_ratio_4": "Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers",
    "Golden_ratio_5": "A spiral formed by drawing quarter-arcs inside squares whose side lengths correspond to the Lucas numbers.",
    "Golden_ratio_6": "Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.",
    "Golden_ratio_7": "A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.",
    "Golden_ratio_8": "A golden triangle $ABC$ can be subdivided by an angle bisector into a smaller golden triangle $CXB$ and a golden gnomon $XAC$.",
    "Golden_ratio_9": "The kite and dart tiles of the Penrose tiling. The colored arcs divide each edge in the golden ratio; when two tiles share an edge, their arcs must match.",
    "Golden_ratio_10": "Geometric progression of areas of squares on the sides of a Kepler triangle",
    "Golden_ratio_11": "An isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length",
    "Golden_ratio_12": "To construct a golden rectangle with only a straightedge and compass in four simple steps: Draw a square. Draw a line from the midpoint of one side of the square to an opposite corner. Use that line as the radius to draw an arc that defines the height of the rectangle. Complete the golden rectangle.",
    "Golden_ratio_13": "The golden spiral (red) and its approximation by quarter-circles (green), with overlaps shown in yellow",
    "Golden_ratio_14": "A logarithmic spiral whose radius grows by the golden ratio per $108^\\circ$ of turn, surrounding nested golden isosceles triangles. This is a different spiral from the golden spiral, which grows by the golden ratio per $90^\\circ$ of turn.",
    "Golden_ratio_15": "Three golden rectangles touch all of the 12 vertices of a regular icosahedron.",
    "Golygon_1": "The smallest golygon has 8 sides. It is the only solution with fewer than 16 sides. It contains two concave corners, and fits on an $8 \\times 10$ grid. It is also a spirolateral, $8_{90^{\\circ}} ^{1,5}$.",
    "Golygon_2": "32-sided golygon. Spirolateral $32_{90^{\\circ}}^{1,3,5,7,11,12,14,17,19,21,23,26,29,31}$",
    "Golygon_3": "16-sided golygon. Spirolateral $16_{90^{\\circ}}^{1,3,6,8,11}$.",
    "Gosper_curve_2": "The line from the red to the green point shows a single step of the Gosper curve construction",
    "Gradient_descent_1": "Illustration of gradient descent on a series of level sets",
    "Gradient_descent_2": "The steepest descent algorithm applied to the Wiener filter",
    "Graph_amalgamation_1": "An amalgamation of the complete graph on five vertices.",
    "Graph_isomorphism_1": "The exception to Whitney's theorem: these two graphs are not isomorphic but have isomorphic line graphs.",
    "Graph_of_a_function_1": "Graph of the function $f(x)={\\frac {x^{3}+3x^{2}-6x-8}{4}}.$",
    "Graph_of_a_function_2": "Graph of the function $f(x) = x^4 - 4^x$ over the interval $[-2, +3]$. Also shown are the two real roots and the local minimum that are in the interval.",
    "Graph_of_a_function_3": "Graph of the function $f(x, y) = \\sin(x^2) \\cdot \\cos(y^2)$.",
    "Graph_of_a_function_4": "Plot of the graph of $f(x, y) = -\\left(\\cos(x^2) + \\cos(y^2)\\right)^2$, also showing its gradient projected on the bottom plane.",
    "Graph_theory_1": "An undirected graph with three vertices and three edges.",
    "Graph_toughness_1": "In this graph, removing the four red vertices would produce four connected components (depicted in four different colours). However, there is no set of k vertices whose removal leaves more than k components. Therefore, its toughness is exactly 1.",
    "Graphic_matroid_2": "Two different graphs (red) that are duals of the same planar graph (pale blue). Despite being non-isomorphic as graphs, they have isomorphic graphic matroids.",
    "Great_retrosnub_icosidodecahedron_1": "3D model of a great retrosnub icosidodecahedron",
    "Great_retrosnub_icosidodecahedron_2": "Great retrosnub icosidodecahedron",
    "Green's_theorem_1": "If $D$ is a simple type of region with its boundary consisting of the curves $C_1$, $C_2$, $C_3$, $C_4$, half of Green's theorem can be demonstrated.",
    "Grinberg's_theorem_1": "A graph that can be proven non-Hamiltonian using Grinberg's theorem",
    "Group_extension_2": "Commutative diagram of equivalence of group extensions",
    "Group_homomorphism_1": "Depiction of a group homomorphism (h) from $G$ (left) to $H$ (right). The oval inside $H$ is the image of $h$. $N$ is the kernel of $h$ and $aN$ is a coset of $N$.",
    "Group_of_rational_points_on_the_unit_circle_1": "The Pythagorean triple $(4,3,5)$ is associated to the rational point $\\left(\\frac{4}{5},\\frac{3}{5}\\right)$ on the unit circle.",
    "Group_representation_1": "A representation of a group \"acts\" on an object. A simple example is how the symmetries of a regular polygon, consisting of reflections and rotations, transform the polygon.",
    "Gyrobifastigium_1": "3D model of a gyrobifastigium",
    "Gyrobifastigium_2": "The Schmitt–Conway–Danzer biprism",
    "Gyrobifastigium_3": "The gyrobifastigium honeycomb",
    "Haar_wavelet_1": "The Haar wavelet",
    "Hadamard_product_(matrices)_2": "The penetrating face product of matrices",
    "Hadamard_space_1": "In a Hadamard space, a triangle is hyperbolic; that is, the middle one in the picture. In fact, any complete metric space where a triangle is hyperbolic is a Hadamard space.",
    "Hadwiger_conjecture_(combinatorial_geometry)_1": "A triangle can be covered by three smaller copies of itself; a square requires four smaller copies",
    "Halin_graph_1": "A Halin graph",
    "Hamming_graph_1": "$H(3,3)$ drawn as a unit distance graph",
    "Handle_decomposition_1": "A 3-ball with three 1-handles attached.",
    "Hann_function_2": "Hann function (left), and its frequency response (right)",
    "Happy_ending_problem_1": "The happy ending problem: every set of five points in general position contains the vertices of a convex quadrilateral",
    "Happy_ending_problem_2": "A set of eight points in general position with no convex pentagon",
    "Happy_ending_problem_3": "A set of sixteen points in general position with no convex hexagon",
    "Harmonic_function_1": "A harmonic function defined on an annulus.",
    "Harmonic_measure_1": "Harmonic measure is the exit distribution of Brownian motion",
    "Harmonic_measure_2": "Harmonic Measure on Simply Connected Planar Domains",
    "Harmonic_series_(mathematics)_2": "A wave and its harmonics, with wavelengths $1,{\\tfrac {1}{2}},{\\tfrac {1}{3}},\\dots$",
    "Harmonic_series_(mathematics)_3": "Rectangles with area given by the harmonic series, and the hyperbola $y=1/x$ through the upper left corners of these rectangles",
    "Harmonic_series_(mathematics)_4": "The digamma function on the complex numbers",
    "Harmonic_series_(mathematics)_5": "Solution to the jeep problem for $n=3$, showing the amount of fuel in each depot and in the jeep at each step",
    "Harmonic_series_(mathematics)_6": "The block-stacking problem: blocks aligned according to the harmonic series can overhang the edge of a table by the harmonic numbers",
    "Harmonic_series_(mathematics)_7": "Graph of number of items versus the expected number of trials needed to collect all items",
    "Harmonic_series_(mathematics)_8": "The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).",
    "Harris_affine_region_detector_1": "Overlap region of two elliptical regions.",
    "Hasse_diagram_2": "A class diagram depicting multiple inheritance",
    "Hasse_diagram_3": "A Hasse diagram of the factors of 60 ordered by the is-a-divisor-of relation",
    "Hausdorff_distance_1": "Components of the calculation of the Hausdorff distance between the green curve $X$ and the blue curve $Y$.",
    "Hausdorff_space_1": "The points $x$ and $y$, separated by their respective neighbourhoods $U$ and $V$.",
    "Haversine_formula_1": "A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown.",
    "Hawaiian_earring_1": "The Hawaiian earring.",
    "Hedetniemi's_conjecture_1": "Example of Hedetniemi's conjecture: the tensor product of C5 and C3 (on the left) produces a graph that contains a cycle with length 15 (on the right) so: the resulting graph requires 3 colors.",
    "Hedgehog_space_1": "A hedgehog space with a large but finite number of spines",
    "Heisenberg_group_1": "A portion of the Cayley graph of the discrete Heisenberg group, with generators $x$, $y$, $z$ as in the text (the coloring is only for visual aid)",
    "Helly's_theorem_1": "Helly's theorem for the Euclidean plane: if a family of convex sets has a nonempty intersection for every triple of sets, then the whole family has a nonempty intersection.",
    "Hemi-icosahedron_1": "The complete graph $K_6$ represents the 6 vertices and 15 edges of the hemi-icosahedron",
    "Hemi-icosahedron_2": "Image of Hemi-icosahedron",
    "Hemicontinuity_1": "This set-valued function is upper hemicontinuous everywhere, but not lower hemicontinuous at $x$: for a sequence of points $\\left(x_{m}\\right)$ that converges to $x,$ we have a $y$ ( $y\\in f(x)$) such that no sequence of $\\left(y_{m}\\right)$ converges to $y$ where each $y_{m}$ is in  $f\\left(x_{m}\\right).$",
    "Hemicontinuity_2": "This set-valued function is lower hemicontinuous everywhere, but not upper hemicontinuous at $x,$ because the graph (set) is not closed.",
    "Heptadecagon_2": "Construction of the regular heptadecagon according to \"sent by T. P. Stowell, credited to Leybourn's Math. Repository, 1818\"",
    "Heptadecagon_3": "Construction of the regular heptadecagon according to H. W. Richmond",
    "Heptadecagon_4": "Heptadecagon in principle according to H.W. Richmond, a variation of the design regarding to point $N$",
    "Heptadecagon_5": "Symmetries of a regular heptadecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.",
    "Heptagon_1": "Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.",
    "Heptagon_2": "The densest double lattice packing of the Euclidean plane by regular heptagons, conjectured to have the lowest maximum packing density of any convex set",
    "Heptagon_3": "A neusis construction of the interior angle in a regular heptagon.",
    "Heronian_mean_1": "A square frustum, with volume equal to the height times the Heronian mean of the square areas",
    "Herschel_graph_2": "The medial graph of the Herschel graph is a 4-regular planar graph with no Hamiltonian decomposition. The shaded regions correspond to the vertices of the underlying Herschel graph.",
    "Hesse_normal_form_1": "Distance from the origin $O$ to the line $E$ calculated with the Hesse normal form. Normal vector in red, line in green, point $O$ shown in blue.",
    "Hessian_form_of_an_elliptic_curve_1": "A Hessian curve of equation $x^{3}+y^{3}+1=0.3xy$",
    "Hexagon_1": "$R = \\text{Circumradius}; r = \\text{Inradius}; t = \\text{side length}$",
    "Hexagon_2": "Equilateral triangles on the sides of an arbitrary hexagon",
    "Hexagonal_number_1": "Proof without words that a hexagonal number (middle column) can be rearranged as rectangular and odd-sided triangular numbers",
    "Hexagonal_number_2": "The first four hexagonal numbers.",
    "High-dimensional_statistics_1": "Illustration of the linear model in high-dimensions: a data set consists of a response vector $Y\\in \\mathbb {R} ^{n}$ and a design matrix $X\\in \\mathbb {R} ^{n\\times p}$ with $p\\gg n$. Our goal is to estimate the unknown vector $\\beta =(\\beta _{1},\\dots ,\\beta _{p})\\in \\mathbb {R} ^{p}$ of regression coefficients where $\\beta$ is often assumed to be sparse, in the sense that the cardinality of the set $S:=\\{j:\\beta _{j}\\neq 0\\}$ is small by comparison with $p$.",
    "Hilbert's_third_problem_1": "Two polyhedra of equal volume, cut into two pieces which can be reassembled into either polyhedron",
    "Hill_climbing_1": "A surface with only one maximum. Hill-climbing techniques are well-suited for optimizing over such surfaces, and will converge to the global maximum.",
    "Hill_climbing_2": "A surface with two local maxima. (Only one of them is the global maximum.) If a hill-climber begins in a poor location, it may converge to the lower maximum.",
    "Hille_equation_1": "Parameters in the Hille equation.",
    "Hippopede_1": "Hippopedes with $a = 1$, $b = 0.1$, $0.2$, $0.5$, $1.0$, $1.5$, and $2.0$.",
    "Hippopede_2": "Hippopedes with $b = 1$, $a = 0.1$, $0.2$, $0.5$, $1.0$, $1.5$, and $2.0$.",
    "Hodge_conjecture_1": "Topological features of a space $X$, such as a hole (labelled by $A$) are usually detected using singular (co)homology, where the presence of a non-zero class $[\\alpha ]\\in H_{\\text{sing}}^{k}(X)$ indicates the space $X$ has a (dimension $k$) hole. Such a class is represented by a (co)chain of simplices, depicted by the red polygon built out of 1-simplices (line segments) on the left. This class detects the hole  $A$ by looping around it. In this case, there is in fact a polynomial equation whose zero set, depicted in green on the right, also detects the hole by looping around it. The Hodge conjecture generalises this statement to higher dimensions.",
    "Homoclinic_connection_1": "Homoclinic and heteroclinic connections and intersections",
    "Homogeneous_space_1": "A torus. The standard torus is homogeneous under its diffeomorphism and homeomorphism groups, and the flat torus is homogeneous under its diffeomorphism, homeomorphism, and isometry groups.",
    "Homotopy_groups_of_spheres_1": "The Hopf fibration is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere.",
    "Homotopy_groups_of_spheres_2": "A homotopy from a circle around a sphere down to a single point",
    "Homotopy_groups_of_spheres_3": "Homotopy of two circle maps keeping base point fixed",
    "Homotopy_groups_of_spheres_4": "Addition of two circle maps keeping base point fixed",
    "Hopf_bifurcation_1": "Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis.",
    "Hopf_bifurcation_2": "Dynamics of the Hopf bifurcation near $\\lambda =0$. Possible trajectories in red, stable structures in dark blue and unstable structures in dashed light blue. Supercritical Hopf bifurcation: 1a) stable fixed point 1b) unstable fixed point, stable limit cycle 1c) phase space dynamics. Subcritical Hopf bifurcation: 2a) stable fixed point, unstable limit cycle 2b) unstable fixed point 2c) phase space dynamics. $\\omega$ determines the angular dynamics and therefore the direction of winding for the trajectories.",
    "Horizon_2": "Geometrical distance to the horizon, Pythagorean theorem",
    "Hosford_yield_criterion_1": "The plane stress, anisotropic, Hosford yield surface for four values of $n$ and $R=2.0$",
    "Hosford_yield_criterion_2": "The plane stress, isotropic, Hosford yield surface for three values of $n$",
    "Huber_loss_2": "Comparison of Huber loss with other loss functions used for robust regression.",
    "Hurwitz_zeta_function_1": "Hurwitz zeta function corresponding to $a = \\frac{1}{3}$.",
    "Hurwitz_zeta_function_2": "Hurwitz zeta function corresponding to $a = \\frac{24}{25}$.",
    "Hurwitz_zeta_function_3": "Hurwitz zeta function as a function of $a$ with $s = 3 + 4i$.",
    "Hyperbolic_angle_1": "The unit hyperbola has a sector with an area half of the hyperbolic angle",
    "Hyperbolic_angle_2": "The curve represents $xy = 1$. A hyperbolic angle has magnitude equal to the area of the corresponding hyperbolic sector, which is in standard position if $a = 1$.",
    "Hyperbolic_coordinates_1": "Hyperbolic coordinates plotted on the Euclidean plane: all points on the same blue ray share the same coordinate value $u$, and all points on the same red hyperbola share the same coordinate value $v$.",
    "Hyperbolic_coordinates_2": "Right triangles with legs proportional to $\\sinh$ and $\\cosh$",
    "Hyperbolic_growth_1": "The reciprocal function, exhibiting hyperbolic growth.",
    "Hyperbolic_motion_1": "For some hyperbolic motions in the half-plane see the Ultraparallel theorem.",
    "Hyperbolic_orthogonality_1": "A represents an event connected by light to the origin. The hyperbolically-orthogonal blue axes have coordinates that measure light speed as the same ratio as the rectangular coordinates of the rest frame.",
    "Hyperbolic_spiral_1": "Hyperbolic spiral (blue) as image of an Archimedean spiral (green) by inversion through a circle (red)",
    "Hyperbolic_spiral_2": "Hyperbolic spiral as central projection of a helix",
    "Hyperbolic_spiral_3": "Definition of sector (light blue) and pitch angle $\\alpha$.",
    "Hyperbolic_spiral_4": "Hyperbolic spiral: branch for $\\varphi > 0$.",
    "Hyperbolic_spiral_5": "Hyperbolic spiral: both branches",
    "Hyperbolic_trajectory_2": "The blue path in this image is an example of a hyperbolic trajectory.",
    "Hyperboloid_1": "hyperboloid of one sheet: generation by a rotating hyperbola (top) and line (bottom: red or blue)",
    "Hyperboloid_2": "hyperboloid of one sheet: plane sections",
    "Hyperboloid_3": "hyperboloid of two sheets: generation by rotating a hyperbola",
    "Hyperboloid_4": "hyperboloid of two sheets: plane sections",
    "Hyperboloid_model_1": "Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid.",
    "Hypercube_graph_2": "Construction of $Q_3$ by connecting pairs of corresponding vertices in two copies of $Q_2$",
    "Hypercube_graph_3": "A Hamiltonian cycle on a tesseract with vertices labelled with a 4-bit cyclic Gray code",
    "Hypercube_graph_4": "Maximum lengths of snakes ($L_s$) and coils ($L_c$) in the snakes-in-the-box problem for dimensions $n$ from 1 to 4",
    "Hyperelliptic_curve_1": "The graph of the hyperelliptic curve $C:y^{2}=f(x)$ where $f(x)=x^{5}-2x^{4}-7x^{3}+8x^{2}+12x=x(x+1)(x-3)(x+2)(x-2).$",
    "Hyperexponential_distribution_1": "Diagram showing queueing system equivalent of a hyperexponential distribution",
    "Hypergeometric_function_1": "Plot of the hypergeometric function $2F_1(a,b;c;z)$ with $a=2$ and $b=3$ and $c=4$ in the complex plane from $-2-2i$ to $2+2i$ with colors.",
    "Hyperplane_1": "Two intersecting planes: Two-dimensional planes are the hyperplanes in three-dimensional space.",
    "Hypograph_(mathematics)_1": "Hypograph of a function",
    "Hypotenuse_1": "A right-angled triangle and its hypotenuse",
    "Hypotenuse_2": "A right triangle with the hypotenuse c",
    "I-bundle_2": "An annulus is an orientable I-bundle. This example is embedded in 3-space with an even number of twists",
    "I-bundle_3": "This image represents the twisted I-bundle over the 2-torus, which is also fibered as a Möbius strip times the circle. So, this space is also a circle bundle",
    "I-spline_1": "An I-spline family of order three with four interior knots.",
    "Icosagon_1": "Symmetries of a regular icosagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.",
    "Icosagon_2": "A regular icosagon",
    "Icosian_calculus_1": "Stereographic projection of dodecahedron used for Hamilton's icosian game",
    "Icosian_calculus_2": "Geometrical illustration of operation iota in icosian calculus",
    "Icosidodecahedron_1": "The graph of an icosidodecahedron",
    "Icositetragon_2": "A regular icositetragon",
    "Ideal_triangle_1": "Three ideal triangles in the Poincaré disk model creating an ideal pentagon",
    "Ideal_triangle_2": "Two ideal triangles in the Poincaré half-plane model",
    "Identity_(mathematics)_1": "Visual proof of the Pythagorean identity: for any angle $\\theta$, the point $(x,y)=(\\cos \\theta ,\\sin \\theta )$ lies on the unit circle, which satisfies the equation $x^{2}+y^{2}=1$. Thus, $\\cos ^{2}\\theta +\\sin ^{2}\\theta =1$.",
    "Iitaka_dimension_1": "The m-pluricanonical map of complex manifolds $M$ to $W $induces a fiber space structure.",
    "Iitaka_dimension_2": "The m-pluricanonical map is birational invariant. $P_m(M)=P_m(W)$",
    "Iitaka_dimension_3": "The existence of the birational map $\\psi : W_{m_1} \\rightarrow W_{m_2}$ in the projective space.",
    "Illustration_of_the_central_limit_theorem_1": "A probability density function.",
    "Illustration_of_the_central_limit_theorem_2": "Density of a sum of two variables.",
    "Illustration_of_the_central_limit_theorem_3": "Density of a sum of three variables.",
    "Illustration_of_the_central_limit_theorem_4": "Density of a sum of four variables",
    "Illustration_of_the_central_limit_theorem_5": "Sum of all permutations of length 1 selected from the set of integers 1, 2, 3",
    "Illustration_of_the_central_limit_theorem_6": "Sum of all permutations of length 2 selected from the set of integers 1, 2, 3",
    "Illustration_of_the_central_limit_theorem_7": "Sum of all permutations of length 3 selected from the set of integers 1, 2, 3",
    "Image_(mathematics)_2": "$f$ is a function from domain $X$ to codomain $Y$. The image of element $x$ is element $y$. The preimage of element  $y$ is the set $\\{x,x'\\}$. The preimage of element $y'$ is $\\varnothing$.",
    "Image_(mathematics)_3": "$f$ is a function from domain $X$ to codomain $Y$. The image of all elements in subset $A$ is subset $B$. The preimage of $B$ is subset $C$",
    "Image_(mathematics)_4": "$f$ is a function from domain $X$ to codomain $Y.$ The yellow oval inside $Y$ is the image of $f$. The preimage of $Y$ is the entire domain $X$",
    "Imaginary_hyperelliptic_curve_1": "Example of adding two elements of the Jacobian",
    "Imaginary_hyperelliptic_curve_2": "Example of a hyperelliptic curve",
    "Imaginary_number_1": "90-degree rotations in the complex plane",
    "Imaginary_unit_1": "The imaginary unit $i$ in the complex plane: Real numbers are conventionally drawn on the horizontal axis, and imaginary numbers on the vertical axis.",
    "Imaginary_unit_2": "The two square roots of $i$ in the complex plane",
    "Imaginary_unit_3": "The three cube roots of $i$ in the complex plane",
    "Immersion_(mathematics)_1": "An injectively immersed submanifold that is not an embedding.",
    "Immersion_(mathematics)_2": "The Möbius strip does not immerse in codimension 0 because its tangent bundle is non-trivial.",
    "Immersion_(mathematics)_3": "The quadrifolium, the 4-petaled rose.",
    "Improper_integral_1": "An improper Riemann integral of the first kind, where the region in the plane implied by the integral is infinite in extent horizontally. The area of such a region, which the integral represents, may be finite (as here) or infinite.",
    "Improper_integral_2": "An improper Riemann integral of the second kind, where the implied region is infinite vertically. The region may have either finite (as here) or infinite area.",
    "Improper_integral_3": "The improper integral $\\int _{0}^{\\infty }{\\frac {dx}{(x+1){\\sqrt {x}}}}=\\pi$ has unbounded intervals for both domain and range.",
    "Improper_integral_4": "The improper integral $\\int _{-1}^{1}{\\frac {dx}{\\sqrt[{3}]{x^{2}}}}=6$ converges, since both left and right limits exist, though the integrand is unbounded near an interior point.",
    "Incenter_1": "The point of intersection of angle bisectors of the 3 angles of triangle ABC is the incenter (denoted by I). The incircle (whose center is I) touches each side of the triangle.",
    "Incidence_coloring_2": "Incidence coloring of a Petersen graph",
    "Incidence_structure_1": "Examples of incidence structures: Example 1: points and lines of the Euclidean plane (top) Example 2: points and circles (middle), Example 3: finite incidence structure defined by an incidence matrix (bottom)",
    "Incidence_structure_2": "Heawood graph with labeling",
    "Incidence_structure_3": "Seven points are elements of seven lines in the Fano plane",
    "Incidence_structure_4": "Levi graph of the Möbius–Kantor configuration",
    "Inclusion_map_1": "$A$ is a subset of $B,$ and $B$ is a superset of $A$.",
    "Incomplete_gamma_function_2": "The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple).",
    "Incomplete_gamma_function_3": "The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple).",
    "Incomplete_gamma_function_4": "Plot of the regularized incomplete gamma function $Q(2,z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors",
    "Indefinite_orthogonal_group_1": "Squeeze mappings, here $r = \\frac{3}{2}$, are the basic hyperbolic symmetries.",
    "Independence_(probability_theory)_1": "Pairwise independent, but not mutually independent, events",
    "Independence_(probability_theory)_2": "Mutually independent events",
    "Indicator_function_1": "A three-dimensional plot of an indicator function, shown over a square two-dimensional domain (set $X$): the \"raised\" portion overlays those two-dimensional points which are members of the \"indicated\" subset ($A$).",
    "Indifference_graph_1": "An indifference graph, formed from a set of points on the real line by connecting pairs of points whose distance is at most one",
    "Indifference_graph_2": "Forbidden induced subgraphs for the indifference graphs: the claw, sun, and net (top, left–right) and cycles of length four or more (bottom)",
    "Inequality_(mathematics)_1": "If $x < y$, then $x + a < y + a$.",
    "Inequality_(mathematics)_2": "If $x < y$ and $a > 0$, then $ax < ay$.",
    "Inequality_(mathematics)_3": "If $x < y$ and $a < 0$, then $ax > ay$.",
    "Inequality_(mathematics)_4": "The graph of $y = \\ln x$",
    "Infimum_and_supremum_1": "A set $P$ of real numbers (hollow and filled circles), a subset $S$ of $P$ (filled circles), and the infimum of $S.$ Note that for totally ordered finite sets, the infimum and the minimum are equal.",
    "Infimum_and_supremum_2": "A set $A$ of real numbers (blue circles), a set of upper bounds of $A$ (red diamond and circles), and the smallest such upper bound, that is, the supremum of $A$ (red diamond).",
    "Infimum_and_supremum_3": "supremum = least upper bound",
    "Inscribed_square_problem_1": "The black dashed curve goes through all corners of several blue squares.",
    "Integer-valued_function_1": "The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles.",
    "Integer_lattice_1": "Approximations of regular pentagrams with vertices on a square lattice with coordinates indicated",
    "Integer_lattice_2": "Rational approximants of irrational values can be mapped to points lying close to lines having gradients corresponding to the values",
    "Integral_of_the_secant_function_1": "A graph of the secant function (red) and its antiderivative (blue)",
    "Integral_of_the_secant_function_2": "The Gudermannian function relates the area of a circular sector to the area of a hyperbolic sector, via a common stereographic projection. If twice the area of the blue hyperbolic sector is $\\psi$, then twice the area of the red circular sector is $\\phi = \\operatorname{gd} \\psi$. Twice the area of the purple triangle is the stereographic projection $s = \\tan \\tfrac{1}{2} \\phi = \\tanh \\tfrac{1}{2} \\psi$. The blue point has coordinates $(\\cosh \\psi, \\sinh \\psi)$. The red point has coordinates $(\\cos \\phi, \\sin \\phi)$. The purple point has coordinates $(0, s)$.",
    "Integrally_convex_set_1": "Non-integrally convex set",
    "Integrally_convex_set_2": "Integrally convex set",
    "Intercept_theorem_2": "${\\tfrac {|SA|}{|SB|}}={\\tfrac {|AC|}{|BD|}}$ does not necessarily imply $AC$ is parallel to $BD$.",
    "Intercept_theorem_3": "intercept theorem with a pair of intersecting lines",
    "Intercept_theorem_4": "intercept theorem with more than two lines",
    "Intercept_theorem_5": "Arranging two similar triangles, so that the intercept theorem can be applied",
    "Intercept_theorem_6": "${\\tfrac {|SA|}{|SB|}}={\\tfrac {|SC|}{|SD|}}$ does not necessarily imply $AC$ is parallel to $BD$.",
    "Interior_(topology)_1": "The point $x$ is an interior point of $S$. The point $y$ is on the boundary of $S$.",
    "Interpolation_1": "An interpolation of a finite set of points on an epitrochoid. The points in red are connected by blue interpolated spline curves deduced only from the red points. The interpolated curves have polynomial formulas much simpler than that of the original epitrochoid curve.",
    "Interpolation_2": "Plot of the data points as given in the table",
    "Interpolation_3": "Piecewise constant interpolation, or nearest-neighbor interpolation",
    "Interpolation_4": "Plot of the data with linear interpolation superimposed",
    "Interpolation_5": "Plot of the data with polynomial interpolation applied",
    "Interpolation_6": "Plot of the data with spline interpolation applied",
    "Intersection_(set_theory)_1": "Intersection of three sets: $~A\\cap B\\cap C$",
    "Intersection_(set_theory)_2": "Intersections of the unaccented modern Greek, Latin, and Cyrillic scripts, considering only the shapes of the letters and ignoring their pronunciation",
    "Intersection_(set_theory)_3": "Example of an intersection with sets",
    "Intersection_(set_theory)_4": "Conjunctions of the arguments in parenthesesThe conjunction of no argument is the tautology (compare: empty product); accordingly the intersection of no set is the universe.",
    "Intersection_curve_2": "Intersection curve: principle of the marching algorithm",
    "Intersection_curve_3": "Intersection curve between polyhedrons: three houses",
    "Intersection_curve_4": "Intersection of polyhedrons: two tori",
    "Interval_exchange_transformation_1": "Graph of interval exchange transformation (in black) with $\\lambda =(1/15,2/15,3/15,4/15,5/15)$ and $\\pi =(3,5,2,4,1)$. In blue, the orbit generated starting from $1/2$.",
    "Interval_exchange_transformation_2": "Dyadic odometer $T$",
    "Interval_exchange_transformation_3": "Dyadic odometer iterated twice; that is $T^{2}.$",
    "Interval_exchange_transformation_4": "Dyadic odometer thrice iterated; that is $T^{3}.$",
    "Interval_exchange_transformation_5": "Dyadic odometer iterated four times; that is $T^{4}.$",
    "Inverse_Gaussian_distribution_1": "Example of stopped random walks with $\\alpha =1,\\nu =0.1,\\sigma =0.2$. The upper figure shows the histogram of waiting times, along with the prediction according to inverse gaussian distribution. The lower figure shows the trajectories.",
    "Inverse_Gaussian_distribution_2": "Cumulative Distribution Function of the Inverse Gaussian distribution for several combinations of the parameters $\\mu$ and $\\lambda$.",
    "Inverse_Gaussian_distribution_3": "Inverse Gaussian Probability Density Function for several combinations of the parameters $\\mu$ and $\\lambda$.",
    "Inverse_distribution_1": "Graph of the density of the inverse of the standard normal distribution",
    "Inverse_transform_sampling_1": "Inverse transform sampling for normal distribution",
    "Inverse_transform_sampling_2": "Graph of the inversion technique from $x$ to $F(x)$. On the bottom right we see the regular function and in the top left its inversion.",
    "Inverse_transform_sampling_3": "Schematic of the inverse transform sampling. The inverse function of $y=F_{X}(x)$ can be defined by $F_{X}^{-1}(y)=\\mathrm {inf} \\{x|F_{X}(x)\\geq y\\}$.",
    "Involute_1": "Two involutes (red) of a parabola",
    "Involute_2": "Involute: properties. The angles depicted are 90 degrees.",
    "Involute_3": "Involutes of a semicubic parabola (blue). Only the red curve is a parabola. Notice how the involutes and tangents make up an orthogonal coordinate system. This is a general fact.",
    "Involute_4": "Involutes of a cycloid (blue): Only the red curve is another cycloid",
    "Irrational_number_1": "The number ${\\sqrt {2}}$ is irrational.",
    "Irrational_number_2": "An Euler diagram showing the set of real numbers ($\\mathbb {R}$), which include the rationals $(\\mathbb {Q}})$, which include the integers $(\\mathbb {Z})$, which include the natural numbers $(\\mathbb {N})$. The real numbers also include the irrationals $(\\mathbb {R}\\ \\mathbb {Q})$.",
    "Isodynamic_point_2": "Three circles, each making angles of $\\pi/3$ with the circumcircle and each other, meet at the first isodynamic point.",
    "Isodynamic_point_3": "Construction of the isodynamic point from reflected copies of the given triangle and inwards-pointing equilateral triangles.",
    "Isolated_point_1": "\"0\" is an isolated point of $A=\\{0\\}\\cup [1,2]$",
    "Isometry_1": "A composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like $R_1$ or $R_2$ on the image. Translation $T$ is a direct isometry: a rigid motion.",
    "Isoperimetric_inequality_1": "If a region is not convex, a \"dent\" in its boundary can be \"flipped\" to increase the area of the region while keeping the perimeter unchanged.",
    "Isoperimetric_inequality_2": "An elongated shape can be made more round while keeping its perimeter fixed and increasing its area.",
    "Isosceles_triangle_1": "Isosceles triangle showing its circumcenter (blue), centroid (red), incenter (green), and symmetry axis (purple)",
    "Isosceles_triangle_2": "Partition of a cyclic pentagon into isosceles triangles by radii of its circumcircle",
    "Jacobi_coordinates_1": "A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are $r_1$, $r_2$, $r_3$ and the center of mass $R$.",
    "Jacobi_elliptic_functions_2": "Plot of the Jacobi ellipse $(x^2 + y^2/b^2 = 1$, $b$ real) and the twelve Jacobi elliptic functions $pq(u,m)$ for particular values of angle $\\varphi$ and parameter $b$. The solid curve is the ellipse, with $m = 1 - 1/b^2$ and $u = F(\\varphi, m)$ where $F(\\cdot,\\cdot)$ is the elliptic integral of the first kind (with parameter $m = k^2$). The dotted curve is the unit circle. Tangent lines from the circle and ellipse at $x = \\operatorname{cd}$ crossing the $x$-axis at $\\operatorname{dc}$ are shown in light grey.",
    "Jacobi_elliptic_functions_3": "The region $F_{1}$ in the complex plane. It is bounded by two semicircles from below, by a ray from the left and by a ray from the right.",
    "Jacobi_elliptic_functions_4": "Plot of the degenerate Jacobi curve $(x^2 + y^2/b^2 = 1$, $b = \\infty)$ and the twelve Jacobi elliptic functions $pq(u, 1)$ for a particular value of angle $\\varphi$. The solid curve is the degenerate ellipse $(x^2 = 1)$ with $m = 1$ and $u = F(\\varphi, 1)$ where $F(\\cdot,\\cdot)$ is the elliptic integral of the first kind. The dotted curve is the unit circle. Since these are the Jacobi functions for $m = 0$ (circular trigonometric functions) but with imaginary arguments, they correspond to the six hyperbolic trigonometric functions.",
    "Jacobi_elliptic_functions_5": "Plot of the Jacobi hyperbola $(x^2 + y^2/b^2 = 1$, $b$ imaginary) and the twelve Jacobi elliptic functions $pq(u, m)$ for particular values of angle $\\varphi$ and parameter $b$. The solid curve is the hyperbola, with $m = 1 - 1/b^2$ and $u = F(\\varphi, m)$ where $F(\\cdot,\\cdot)$ is the elliptic integral of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, $\\sigma = \\sin(\\varphi)\\cos(\\varphi)$.",
    "Jacobi_polynomials_1": "Plot of the Jacobi polynomial function $P_{n}^{(\\alpha ,\\beta )}$ with $n=10$ and $\\alpha =2$ and $\\beta =2$ in the complex plane from $-2-2i$ to $2+2i$ with colors",
    "Jacobian_curve_1": "A Jacobi quartic of equation $y^{2}=x^{4}-1.9x^{2}+1$",
    "Jacobian_matrix_and_determinant_1": "A nonlinear map $f\\colon \\mathbb {R} ^{2}\\to \\mathbb {R} ^{2}$ sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.",
    "Join_(topology)_1": "Geometric join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid, a disphenoid, in gray.",
    "Jordan's_inequality_2": "unit circle with angle x and a second circle with radius $|EG|=\\sin(x)$ around E. ${\\begin{aligned}&|DE|\\leq |{\\widehat {DC}}|\\leq |{\\widehat {DG}}|\\\\\\Leftrightarrow &\\sin(x)\\leq x\\leq {\\tfrac {\\pi }{2}}\\sin(x)\\\\\\Rightarrow &{\\tfrac {2}{\\pi }}x\\leq \\sin(x)\\leq x\\end{aligned}}$",
    "Jordan's_lemma_1": "The path $C$ is the concatenation of the paths $C_1$ and $C_2$.",
    "Joukowsky_transform_1": "Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below.",
    "Julia_set_1": "A Julia set",
    "K-edge-connected_graph_1": "A 2-edge-connected graph",
    "K-vertex-connected_graph_1": "A graph with connectivity 4 and not 5: The graph remains connected whichever 3 vertices are removed but removing all but 2 opposing vertices (i.e. removing 4) leaves a disconnected graph.",
    "Kakeya_set_1": "\"Sprouting the Perron tree\": a method for constructing a Kakeya set of small measure. Shown here are two possible ways of dividing a triangle and overlapping the pieces to get a smaller set, the first with two triangles, and the second with eight. The method can be used to construct an arbitrarily small set by cutting up the original triangle to  $2^{n}$ pieces.",
    "Kan_fibration_1": "The striped blue simplex in the domain has to exist in order for this map to be a Kan fibration",
    "Kan_fibration_2": "Lifting diagram for a Kan fibration",
    "Kappa_curve_1": "The kappa curve has two vertical asymptotes",
    "Kautz_graph_1": "Example of Kautz graph on 3 characters with string length 2 (on the left) and 3 (on the right); the edges on the left correspond to the vertices on the right.",
    "Kelly_criterion_1": "Example of the optimal Kelly betting fraction, versus expected return of other fractional bets.",
    "Kelly_criterion_2": "3D figure representing the optimal Kelly bet size (vertical axis) as a function of win probability and amount gained with win.",
    "Kelly_criterion_3": "The figure plots the amount gained with a win on the $x$-axis against the fraction of portfolio to bet on the $y$-axis. This figure assumes $p = 0.5$ (that the probability of both a win and a loss is 50\\%). If the amount gained with a win is $1$, then the Kelly betting amount is $0$, which makes sense in a fair bet with no expected gain.",
    "Kelly_criterion_4": "The figure plots the amount gained with a win on the $x$-axis against the fraction of portfolio to bet on the $y$-axis. This figure assumes $p = 0.6$ (that the probability of a win is 60%).",
    "Kelvin_functions_1": "$\\mathrm {ber} (x)$ for $x$ between 0 and 20.",
    "Kelvin_functions_2": "$\\mathrm {ber} (x)/e^{x/{\\sqrt {2}}}$ for $x$ between 0 and 50.",
    "Kelvin_functions_3": "$\\mathrm {bei} (x)$ for x between 0 and 20.",
    "Kelvin_functions_4": "$\\mathrm {bei} (x)/e^{x/{\\sqrt {2}}}$ for $x$ between 0 and 50.",
    "Kelvin_functions_5": "$\\mathrm {ker} (x)$ for $x$ between 0 and 14.",
    "Kelvin_functions_6": "$\\mathrm {ker} (x)e^{x/{\\sqrt {2}}}$ for $x$ between 0 and 50.",
    "Kelvin_functions_7": "$\\mathrm {kei} (x)$ for $x$ between 0 and 14.",
    "Kelvin_functions_8": "$\\mathrm {kei} (x)e^{x/{\\sqrt {2}}}$ for $x$ between 0 and 50.",
    "Kelvin_functions_9": "Plot of the Kelvin function $\\mathrm {kei} (z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors created with Mathematica 13.1 function ComplexPlot3D",
    "Kelvin_functions_10": "Plot of the Kelvin function $\\mathrm {ker} (z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors created with Mathematica 13.1 function ComplexPlot3D",
    "Kelvin_functions_11": "Plot of the Kelvin function $\\mathrm {ber} (z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors created with Mathematica 13.1 function ComplexPlot3D",
    "Kempe_chain_1": "A graph containing a Kempe chain consisting of alternating blue and red vertices",
    "Kempner_function_1": "Graph of the Kempner function",
    "Kendall_rank_correlation_coefficient_1": "All points in the gray area are concordant and all points in the white area are discordant with respect to point $(X_{1},Y_{1})$. With $n=30$ points, there are a total of ${\\binom {30}{2}}=435$ possible point pairs. In this example there are 395 concordant point pairs and 40 discordant point pairs, leading to a Kendall rank correlation coefficient of 0.816.",
    "Kepler_triangle_1": "A Kepler triangle is a right triangle formed by three squares with areas in geometric progression according to the golden ratio.",
    "Kepler_triangle_2": "A right triangle formed by an edge midpoint, base center point, and apex of a square pyramid. Some pyramidologists have theorized that the triangle formed in this way for the Great Pyramid of Giza was intended as a Kepler triangle.",
    "Kepler_triangle_3": "When an isosceles triangle is formed from two Kepler triangles, reflected across their long sides, it has the maximum possible inradius among all isosceles triangles having legs of a given size.",
    "Kepler_triangle_4": "In Coxeter's loxodromic sequence of tangent circles, the centers of each three consecutive circles form an angle from the Kepler triangle.",
    "Killing_vector_field_1": "Killing field on the upper-half plane model, on a semi-circular selection of points. This Killing vector field generates the special conformal transformation. The colour indicates the magnitude of the vector field at that point.",
    "Kinetic_tournament_1": "Kinetic tournament overview",
    "Kissing_number_2": "A highly symmetrical realization of the kissing number 12 in three dimensions is by aligning the centers of outer spheres with vertices of a regular icosahedron. This leaves slightly more than 0.1 of the radius between two nearby spheres.",
    "Kissing_number_3": "Kissing number in one dimension is 2",
    "Kite_(geometry)_2": "Part of an aperiodic tiling with prototiles made from eight kites",
    "Kite_(geometry)_3": "A kite, showing its pairs of equal-length sides and its inscribed circle.",
    "Kite_(geometry)_4": "A fractal rosette of Penrose kites, quadrilaterals with angles $72^\\circ$, $72^\\circ$, $72^\\circ$, $144^\\circ$. These rosettes are not Penrose tilings, and can be formed for any kite for which one apex angle and the side angle sum to $180^\\circ(1 - \\frac{1}{n})$, for some $n$; here $n = 5$.",
    "Kite_(geometry)_5": "A kite and its dual isosceles trapezoid",
    "Klee's_measure_problem_1": "A set of rectangular ranges ('trellis') whose area has to be measured.",
    "Klein_graphs_2": "Klein quartic tiled with 56 triangles (dual of the Klein map)",
    "Klein_quartic_1": "The Klein quartic with the two dual Klein graphs (14-gon edges marked with the same number are equal.) The Klein quartic is a quotient of the heptagonal tiling (compare the 3-regular graph in green) and its dual triangular tiling (compare the 7-regular graph in violet).",
    "Klein_quartic_2": "A pants decomposition of the Klein quartic. The figure on the left shows the boundary geodesics in the $(2,3,7)$ tessellation of the fundamental domain. In the figure to the right, the pants have each been coloured differently to make it clear which part of the fundamental domain belongs to which pair of pants.",
    "Klein_quartic_3": "The fundamental domain of the Klein quartic. The surface is obtained by associating sides with equal numbers.",
    "Klein_quartic_4": "The tiling of the quartic by reflection domains is a quotient of the 3-7 kisrhombille.",
    "Kleinian_group_2": "Limit set of a quasi-Fuchsian group",
    "Kneser_graph_1": "Kneser graph $O_4 = K(7, 3)$",
    "Knight's_tour_1": "The knight's tour as solved by the Turk, a chess-playing machine hoax. This particular solution is closed (circular), and can thus be completed from any point on the board.",
    "Knight's_tour_2": "Knight's graph showing all possible paths for a knight's tour on a standard $8 \\times 8$ chessboard. The numbers on each node indicate the number of possible moves that can be made from that position.",
    "Knot_(mathematics)_2": "An overhand knot becomes a trefoil knot by joining the ends.",
    "Knot_(mathematics)_3": "The triangle is associated with the trefoil knot.",
    "Knot_(mathematics)_4": "A knot can be untied if the loop is broken.",
    "Knot_(mathematics)_5": "A wild knot",
    "Knot_(mathematics)_6": "A knot whose complement has a non-trivial JSJ decomposition",
    "Knot_(mathematics)_7": "A table of all prime knots with up to seven crossings represented as knot diagrams with their medial graph",
    "Knot_(mathematics)_8": "The signed planar graph associated with a knot diagram.",
    "Kobon_triangle_problem_1": "Five lines forming a pentagram, with one more horizontal line below them, form seven triangles: five in the pentagram, and two more formed by pairs of rays emanating from the corners of the pentagram. If the lower horizontal line is moved to the line at infinity of the projective plane, all five pairs of rays emanating from the pentagram would form triangles with it.",
    "Koch_snowflake_1": "The first four iterations of the Koch snowflake",
    "Koch_snowflake_2": "A fractal rough surface built from multiple Koch curve iterations",
    "Koch_snowflake_3": "Tessellation by two sizes of Koch snowflake",
    "Koch_snowflake_4": "Graph of the Koch's function",
    "Kosnita's_theorem_1": "$X(54)$ is the Kosnita point of the triangle $ABC$",
    "Kostant_partition_function_2": "Values of the Kostant partition function for the root system $B_{2}$. The root system is given the Euclidean coordinates $\\alpha _{1}=(1,0),\\alpha _{2}=(-1,1)$.",
    "Kriging_1": "Example of one-dimensional data interpolation by kriging, with credible intervals. Squares indicate the location of the data. The kriging interpolation, shown in red, runs along the means of the normally distributed credible intervals shown in gray. The dashed curve shows a spline that is smooth, but departs significantly from the expected values given by those means.",
    "Kummer_surface_1": "3D model of a Kummer surface",
    "Kurtosis_2": "pdf for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); and 0 (black)",
    "Kurtosis_3": "Probability density functions for selected distributions with mean 0, variance 1 and different excess kurtosis",
    "Kurtosis_4": "Logarithms of probability density functions for selected distributions with mean 0, variance 1 and different excess kurtosis",
    "Kurtosis_5": "Log-pdf for the Pearson type VII distribution with excess kurtosis of infinity (red); $2$ (blue); $1$, $1/2$, $1/4$, $1/8$, and $1/16$ (gray); and $0$ (black).",
    "L-estimator_1": "Simple L-estimators can be visually estimated from a box plot, and include interquartile range, midhinge, range, mid-range, and trimean.",
    "LU_decomposition_2": "Illustration of operation of Banachiewicz LU algorithm to obtain 3-rd row and column of respectively matrices $U$ and $L$. Involved matrices are named above squares marking their content. Matrix products and subtractions are applied only to elements in the thick frame boxes. Green filled thin frame boxes indicate values already known, from previous stages. Blue boxes indicate places in $U$ and $L$ matrices for storing of results.",
    "Laguerre_plane_2": "minimal model of a Laguerre plane (only 4 of 8 cycles are shown)",
    "Laguerre_plane_3": "Theorem of Miquel (circles drawn instead of parabolas)",
    "Laguerre_plane_4": "classical Laguerre plane: 2d/3d-model",
    "Laguerre_plane_5": "Laguerre-plane: axioms",
    "Lah_number_1": "Illustration of the unsigned Lah numbers for $n$ and $k$ between 1 and 4",
    "Lambda_cube_1": "The lambda cube. Direction of each arrow is direction of inclusion.",
    "Lambert's_problem_2": "Hyperbola with the points $P_{1}$ and $P_{2}$ as foci passing through $F_{1}$",
    "Lambert's_problem_3": "$F_{1}$ is the centre of attraction, $P_{1}$ is the point corresponding to vector ${\\bar {r}}_{1}$, and $P_{2}$ is the point corresponding to vector ${\\bar {r}}_{2}$",
    "Lanczos_resampling_1": "Lanczos windows for $a = 1, 2, 3.$",
    "Laplace's_method_1": "$f(x)={\\tfrac {\\sin(x)}{x}}$ has a global maximum at $x=0$. $e^{Mf(x)}$ is shown on top for $M=0.5$ and at the bottom for $M=3$ (both in blue). As $M$ grows, the approximation of this function by a Gaussian function (shown in red) improves. This observation underlies Laplace's method.",
    "Laplace's_method_2": "The figure of $e^{M[f(sy+x_{0})-f(x_{0})]}$ with $M$ equals 1, 2 and 3, and the red line is the curve of function $e^{-\\pi y^{2}}$ .",
    "Laplace_distribution_1": "Fitted Laplace distribution to maximum one-day rainfalls",
    "Laplace_distribution_2": "Probability density plots of Laplace distributions",
    "Laplace_distribution_3": "Cumulative distribution plots of Laplace distributions",
    "Laplace_transform_1": "$\\Re (e^{-st})$ for various complex frequencies in the s-domain $(s=\\sigma +i\\omega ),$ which can be expressed as $e^{-\\sigma t}\\cos(\\omega t).$ The $\\sigma =0$ axis contains pure cosines. Positive $\\sigma$ contains damped cosines. Negative $\\sigma$ contains exponentially growing cosines.",
    "Largest_empty_rectangle_1": "Maximum Empty Rectangles (in green) with different bounding objects (with black outline) . The light green rectangle would be suboptimal (non-maximal) solution. A-C are axis oriented - parallel to axes of the light blue \"floor\" and also examples of. E shows a maximal empty rectangle with arbitrary orientation",
    "Lattice_(discrete_subgroup)_1": "A portion of the discrete Heisenberg group, a discrete subgroup of the continuous Heisenberg Lie group. (The coloring and edges are only for visual aid.)",
    "Lattice_(group)_2": "A rectangular lattice.",
    "Lattice_(group)_3": "An oblique lattice.",
    "Lattice_(group)_4": "An equilateral triangle lattice.",
    "Lattice_of_subgroups_1": "Hasse diagram of the lattice of subgroups of the dihedral group $Dih_4$, with the subgroups represented by their cycle graphs",
    "Lattice_reduction_1": "Lattice reduction in two dimensions: the black vectors are the given basis for the lattice (represented by blue dots), the red vectors are the reduced basis",
    "Laurent_series_1": "A Laurent series is defined with respect to a particular point $c$ and a path of integration $\\gamma$. The path of integration must lie in an annulus, indicated here by the red color, inside which $f(z)$ is holomorphic (analytic).",
    "Laurent_series_2": "3-dimensional plot of the absolute value of the complex gamma function",
    "Laurent_series_3": "$e^{-1/x^2}$ and its Laurent approximations. As the negative degree of the Laurent series rises, it approaches the correct function.",
    "Laves_graph_1": "The Laves graph",
    "Law_of_cotangents_1": "A triangle, showing the incircle and the partitioning of the sides. The angle bisectors meet at the incenter, which is the center of the incircle.",
    "Law_of_sines_1": "Deriving the ratio of the sine law equal to the circumscribing diameter. Note that triangle ADB passes through the center of the circumscribing circle with diameter d.",
    "Law_of_sines_2": "Ambiguous triangle with acute and obtuse solutions. Redrawn from source with Greek letters for angle labels. Capital letters retained as vertex labels.",
    "Law_of_sines_3": "Spherical trigonometry: the vectors used in the proof of the cosine rule.",
    "Law_of_sines_4": "Diagram for the geometric proof of spherical law of sine",
    "Least_common_multiple_1": "A Venn diagram showing the least common multiples of all subsets of {2, 3, 4, 5, 7}.",
    "Least_squares_2": "Conic fitting a set of points using least-squares approximation",
    "Least_squares_3": "The residuals are plotted against corresponding $x$ values. The random fluctuations about $r_{i}=0$ indicate a linear model is appropriate.",
    "Least_squares_4": "The residuals are plotted against the corresponding $x$ values. The parabolic shape of the fluctuations about $r_{i}=0$ indicates a parabolic model is appropriate.",
    "Least_squares_5": "\"Fanning Out\" Effect of Heteroscedasticity",
    "Lebesgue_covering_dimension_1": "Refinement of the cover of a circle",
    "Lebesgue_covering_dimension_2": "Refinement of the cover of a square",
    "Lebesgue_measure_1": "Translation invariance: The Lebesgue measure of $A$ and $A+t$ are the same.",
    "Legendre_polynomials_1": "Diagram for the multipole expansion of electric potential.",
    "Legendre_polynomials_2": "Plot of the six first Legendre polynomials.",
    "Lemniscate_of_Bernoulli_2": "Sinusoidal spirals ($r^n = -\\frac{1}{n} \\cos(n\\theta),\\ \\theta = \\frac{\\pi}{2}$) in polar coordinates and their equivalents in rectangular coordinates: $n = -2$: Equilateral hyperbola, $n = -1$: Line, $n = -\\frac{1}{2}$: Parabola, $n = \\frac{1}{2}$: Cardioid, $n = 1$: Circle, $n = 2$: Lemniscate of Bernoulli.",
    "Lemniscate_of_Bernoulli_3": "The lemniscate sine and cosine relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.",
    "Lemniscate_of_Bernoulli_4": "relation between angles at Bernoulli's lemniscate",
    "Lemniscate_of_Bernoulli_5": "The inversion of hyperbola yields a lemniscate",
    "Lemniscate_of_Gerono_1": "The lemniscate of Gerono",
    "Lemniscate_of_Gerono_2": "Dual to the lemniscate of Gerono",
    "Lemoine_hexagon_1": "The Lemoine hexagon, shown with self-intersecting connectivity, circumscribed by the first Lemoine circle",
    "Leonhard_Euler_1": "A geometric interpretation of Euler's formula",
    "Lester's_theorem_1": "The Fermat points $X_{13},X_{14}$, the center $X_{5}$ of the nine-point circle (light blue), and the circumcenter $X_{3}$ of the green triangle lie on the Lester circle (black).",
    "Levi's_lemma_1": "The $uw = x$ and $v = wy$ case of Levi's lemma",
    "Lie_sphere_geometry_1": "A Dupin cyclide.",
    "Lift_(mathematics)_1": "The morphism $h$ is a lift of $f$ (commutative diagram)",
    "Likelihood_function_1": "The likelihood function ($\\textstyle p_{\\text{H}}^{2}$) for the probability of a coin landing heads-up (without prior knowledge of the coin's fairness), given that we have observed $HH$.",
    "Likelihood_function_2": "The likelihood function ($\\textstyle p_{\\text{H}}^{2}(1-p_{\\text{H}})$) for the probability of a coin landing heads-up (without prior knowledge of the coin's fairness), given that we have observed $HHT$.",
    "Limit_(category_theory)_1": "A universal cone",
    "Limit_(category_theory)_2": "A universal co-cone",
    "Limit_cycle_1": "Stable limit cycle (shown in bold) and two other trajectories spiraling into it",
    "Limit_cycle_2": "Stable limit cycle (shown in bold) for the Van der Pol oscillator",
    "Limit_cycle_3": "Examples of limit cycles branching from fixed points near Hopf bifurcation. Trajectories in red, stable structures in dark blue, unstable structures in light blue. The parameter choice determines the occurrence and stability of limit cycles.",
    "Lindley's_paradox_1": "Distribution of $p$ under the null hypothesis, and the posterior distribution of $p$",
    "Line_(geometry)_2": "A line on polar coordinates without passing though the origin, with the general parametric equation written above",
    "Line_(geometry)_3": "A ray with a terminus at $A$, with two points $B$ and $C$ on the right",
    "Line_(geometry)_4": "Drawing of a line segment \"AB\" on the line \"a\"",
    "Line_(geometry)_5": "A number line, with variable $x$ on the left and $y$ on the right. Therefore, $x$ is smaller than $y$.",
    "Line_(geometry)_6": "Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.",
    "Line_(geometry)_7": "A red line near the origin on the two-dimensional Cartesian coordinate system",
    "Line_(geometry)_8": "Distance from the origin $O$ to the line $E$ calculated with the Hesse normal form. Normal vector in red, line in green, point $O$ shown in blue.",
    "Line_segment_1": "The geometric definition of a closed line segment: the intersection of all points at or to the right of $A$ with all points at or to the left of $B$",
    "Linear_approximation_1": "Tangent line at $(a, f(a))$",
    "Linear_discriminant_analysis_1": "Linear discriminant analysis on a two dimensional space with two classes. The Bayes boundary is calculated based on the true data generation parameters, the estimated boundary on the realised data points.",
    "Linear_discriminant_analysis_2": "Fisher's Linear Discriminant visualised as an axis",
    "Linear_discriminant_analysis_3": "Visualisation for one-versus-all LDA axes for 4 classes in 3d",
    "Linear_discriminant_analysis_4": "Projections along linear discriminant axes for 4 classes",
    "Linear_equation_1": "Two graphs of linear equations in two variables",
    "Linear_equation_2": "Vertical line of equation $x = a$",
    "Linear_equation_3": "Horizontal line of equation $y = b$",
    "Linear_flow_on_the_torus_1": "Irrational rotation on a 2-torus",
    "Linear_independence_1": "Linearly independent vectors in $\\mathbb {R} ^{3}$",
    "Linear_independence_2": "Linearly dependent vectors in a plane in $\\mathbb {R} ^{3}.$",
    "Linear_inequality_1": "Graph of linear inequality: $x + 3y < 9$",
    "Linear_interpolation_2": "In this geometric visualisation, the value at the green circle multiplied by the horizontal distance between the red and blue circles is equal to the sum of the value at the red circle multiplied by the horizontal distance between the green and blue circles, and the value at the blue circle multiplied by the horizontal distance between the green and red circles.",
    "Linear_interpolation_3": "Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines).",
    "Linear_interpolation_4": "Example of bilinear interpolation on the unit square with the $z$ values 0, 1, 1, and 0.5 as indicated. Interpolated values in between are represented by colour.",
    "Linear_interpolation_5": "A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom)",
    "Linear_interpolation_6": "Comparison of linear and bilinear interpolation some 1- and 2-dimensional interpolations. Black and red/yellow/green/blue dots correspond to the interpolated point and neighbouring samples, respectively. Their heights above the ground correspond to their values.",
    "Linear_regression_1": "Visualization of heteroscedasticity in a scatter plot against 100 random fitted values using Matlab",
    "Linear_regression_2": "To check for violations of the assumptions of linearity, constant variance, and independence of errors within a linear regression model, the residuals are typically plotted against the predicted values (or each of the individual predictors). An apparently random scatter of points about the horizontal midline at 0 is ideal, but cannot rule out certain kinds of violations such as autocorrelation in the errors or their correlation with one or more covariates.",
    "Linear_regression_3": "The data sets in the Anscombe's quartet are designed to have approximately the same linear regression line (as well as nearly identical means, standard deviations, and correlations) but are graphically very different. This illustrates the pitfalls of relying solely on a fitted model to understand the relationship between variables.",
    "Linear_regression_4": "Example of simple linear regression, which has one independent variable",
    "Linear_regression_5": "In linear regression, the observations (red) are assumed to be the result of random deviations (green) from an underlying relationship (blue) between a dependent variable $(y)$ and an independent variable $(x)$.",
    "Linear_separability_1": "The existence of a line separating the two types of points means that the data is linearly separable",
    "Linear_separability_2": "$H_1$ does not separate the sets. $H_2$ does, but only with a small margin. $H_3$ separates them with the maximum margin.",
    "Linear_span_1": "The cross-hatched plane is the linear span of $u$ and $v$ in both $R^2$ and $R^3$, here shown in perspective.",
    "Linear_subspace_2": "In $R^3$, the intersection of two distinct two-dimensional subspaces is one-dimensional",
    "Linearity_1": "Hasse diagram of a linear Boolean function",
    "Linearization_1": "An approximation of $f(x) = x^2$ at $(x, f(x))$",
    "Link_(knot_theory)_2": "The Hopf link is cobordant to the unlink.",
    "Link_(knot_theory)_3": "$(2,8)$ torus link",
    "List_of_aperiodic_sets_of_tiles_1": "A periodic tiling with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. In order to do this, the basic triangle needs to be rotated 180 degrees in order to fit it edge-to-edge to a neighboring triangle. Thus a triangular tiling of fundamental units will be generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be translated to form an infinite tiling of the plane. It is not necessary to rotate this patch in order to achieve this.",
    "List_of_aperiodic_sets_of_tiles_2": "Aperiodic monotile construction diagram, based on Smith (2023)",
    "List_of_aperiodic_sets_of_tiles_3": "Aperiodic monotile construction diagram, based on Smith (2023)",
    "List_of_aperiodic_sets_of_tiles_4": "Aperiodic monotile from a paper by Mampusti and Whittaker",
    "List_of_aperiodic_sets_of_tiles_5": "Base tiles for Penrose's P3 tiling",
    "List_of_unsolved_problems_in_mathematics_2": "The free Burnside group $B(2,3)$ is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups $B(m,n)$ are finite remains open.",
    "List_of_unsolved_problems_in_mathematics_3": "A detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.",
    "List_of_unsolved_problems_in_mathematics_4": "In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.",
    "List_of_unsolved_problems_in_mathematics_5": "The area of the blue region converges to the Euler-Mascheroni constant, which may or may not be a rational number.",
    "List_of_unsolved_problems_in_mathematics_6": "Goldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.",
    "List_of_unsolved_problems_in_mathematics_7": "The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.",
    "List_of_unsolved_problems_in_mathematics_8": "In the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.",
    "Littlewood_polynomial_1": "Roots of all the Littlewood polynomials of degree 15.",
    "Lituus_(mathematics)_1": "Branch for positive $r$",
    "Locally_connected_space_1": "In this topological space, $V$ is a neighbourhood of $p$ and it contains a connected open set (the dark green disk) that contains $p$.",
    "Loewner's_torus_inequality_1": "Shortest loop on a torus",
    "Log-normal_distribution_1": "Relation between normal and log-normal distribution. If $Y=\\mu +\\sigma Z$ is normally distributed, then $X\\sim e^{Y}$ is log-normally distributed.",
    "Log-normal_distribution_2": "Overview of parameterizations of the log-normal distributions.",
    "Log-normal_distribution_3": "Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting",
    "Log-normal_distribution_4": "Comparison of mean, median and mode of two log-normal distributions with different skewness.",
    "Log-normal_distribution_5": "a. $y$ is a log-normal variable with $\\mu = 1$, $\\sigma = 0.5$. $p(\\sin y > 0)$ is computed by transforming to the normal variable $x = \\ln y$, then integrating its density over the domain defined by $\\sin(e^{x}) > 0$ (blue regions), using the numerical method of ray-tracing. b & c. The pdf and cdf of the function $\\sin y$ of the log-normal variable can also be computed in this way.",
    "Log-normal_distribution_6": "Plot of the Lognormal PDF",
    "Log-normal_distribution_7": "Plot of the Lognormal CDF",
    "Logarithmic_decrement_1": "The logarithmic decrement can be obtained e.g. as $\\ln(x_1/x_3)$.",
    "Logarithmic_distribution_1": "Plot of the logarithmic PMF",
    "Logarithmic_distribution_2": "Plot of the logarithmic CDF",
    "Logarithmic_integral_function_2": "Plot of the logarithmic integral function $\\mathrm{li}(z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors.",
    "Logarithmic_integral_function_3": "Logarithmic integral function plot",
    "Logarithmic_mean_1": "Three-dimensional plot showing the values of the logarithmic mean.",
    "Logarithmic_spiral_2": "Definition of slope angle and sector",
    "Logarithmic_spiral_3": "Examples for $a=1,2,3,4,5$",
    "Logarithmic_spiral_4": "Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral. The plotted spiral (dashed blue curve) is based on growth rate parameter $b=0.1759$, resulting in a pitch of $\\arctan b\\approx 10^{\\circ }$.",
    "Logarithmic_spiral_5": "Logarithmic spiral (pitch $10^\\circ$)",
    "Logical_matrix_1": "Multiplication of two logical matrices using Boolean algebra.",
    "Logistic_function_1": "Standard logistic function where $L=1,k=1,x_{0}=0$",
    "Logit_1": "Plot of $logit(x)$ in the domain of 0 to 1, where the base of the logarithm is $e$.",
    "Lp_space_2": "Astroid, unit circle in $p={\\tfrac {2}{3}}$ metric",
    "Lucas_number_1": "The Lucas spiral, made with quarter-arcs, is a good approximation of the golden spiral when its terms are large. However, when its terms become very small, the arc's radius decreases rapidly from 3 to 1 then increases from 1 to 2.",
    "Luttinger's_theorem_1": "Luttinger's theorem relates a Fermi liquid's particle density to the volume of its Fermi surface.",
    "Lyapunov_exponent_2": "Points inside and outside Mandelbrot set colored by Lyapunov exponent.",
    "Lyapunov_fractal_2": "Generalized Lyapunov logistic fractal with iteration sequence AABAB, in the region $[2, 4] \\times [2, 4]$.",
    "Lyapunov_fractal_3": "Standard Lyapunov logistic fractal with iteration sequence $AB$, in the region $[2, 4] \\times [2, 4]$.",
    "M-spline_1": "An M-spline family of order three with four interior knots.",
    "Magic_square_1": "The smallest (and unique up to rotation and reflection) non-trivial case of a magic square, order 3",
    "Magic_tee_1": "A magic T consisting of four rectangular waveguides meeting in a single three-dimensional junction",
    "Mandelbrot_set_1": "The Mandelbrot set within a continuously colored environment",
    "Manin_conjecture_1": "Rational points of bounded height outside the 27 lines on Clebsch's diagonal cubic surface.",
    "Mapping_cone_(topology)_1": "An illustration of a mapping cone; that is, a cone is glued to a space along some function $f\\colon X\\to Y$.",
    "Maps_of_manifolds_2": "Riemann surface for the function $f(z) = \\sqrt{z}$, shown as a ramified covering space of the complex plane.",
    "Maps_of_manifolds_3": "The right-handed trefoil knot.",
    "Martingale_(probability_theory)_2": "Software-created martingale series",
    "Massey_product_2": "Non-trivial Brunnian links correspond to non-vanishing Massey products.",
    "Matrix_(mathematics)_1": "An $m \\times n$ matrix: the $m$ rows are horizontal and the $n$ columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, $a_{2,1}$ represents the element at the second row and first column of the matrix.",
    "Matrix_(mathematics)_2": "Schematic depiction of the matrix product AB of two matrices A and B",
    "Matrix_addition_1": "Illustration of the addition of two matrices.",
    "Matrix_multiplication_2": "The computation of the bottom left entry of $\\mathbf {AB}$ corresponds to the consideration of all paths (highlighted) from basic commodity $b_{4}$ to final product $f_{1}$ in the production flow graph.",
    "Matroid_partitioning_1": "A partition of the edges of the complete bipartite graph $K_{4,4}$ into three forests, showing that it has arboricity at most three",
    "Maximal_independent_set_1": "The graph of the cube has six different maximal independent sets, shown as the red vertices. Only two of these, the ones with four vertices, are maximum independent sets.",
    "Maximal_independent_set_2": "The top two $P_3$ graphs are maximal independent sets while the bottom two are independent sets, but not maximal. The maximum independent set is represented by the top left.",
    "Maximal_independent_set_3": "Two independent sets for the star graph $S_8$ show how vastly different in size two maximal independent sets (the right being maximum) can be.",
    "Maximal_independent_set_4": "Left is a maximal independent set. Middle is a clique, $K_{3}$, on the graph complement. Right is a vertex cover on the maximal independent set complement.",
    "Mean_curvature_1": "A rendering of Costa's minimal surface.",
    "Mean_width_1": "The definition of the \"width\" of body B in direction ${\\hat {n}}$ in 2 dimensions.",
    "Median_(geometry)_1": "The triangle medians and the centroid.",
    "Median_(geometry)_2": "Triangle with Centroid $O$ and three Medians",
    "Median_(geometry)_3": "medians of a tetrahedron ${\\begin{aligned}&{\\frac {|AS|}{|SS_{BCD}|}}={\\frac {|BS|}{|SS_{ACD}|}}={\\frac {|CS|}{|SS_{ABD}|}}\\\\=&{\\frac {|DS|}{|SS_{ABC}|}}={\\frac {3}{1}}\\end{aligned}}$",
    "Median_graph_1": "The median of three vertices in a median graph",
    "Median_graph_2": "The median of three vertices in a tree, showing the subtree formed by the union of shortest paths between the vertices.",
    "Median_graph_3": "The graph of a distributive lattice, drawn as a Hasse diagram.",
    "Median_graph_4": "Retraction of a cube onto a six-vertex subgraph.",
    "Median_graph_5": "Converting a triangle-free graph into a median graph.",
    "Median_graph_6": "The Buneman graph for five types of mouse.",
    "Median_graph_7": "The Cartesian product of graphs forms a median graph from two smaller median graphs.",
    "Mediant_(mathematics)_1": "Determining the mediant of two rational numbers graphically. The slopes of the blue and red segments are two rational numbers; the slope of the green segment is their mediant.",
    "Mercator_series_1": "Polynomial approximation to logarithm with $n=1, 2, 3,$ and $10$ in the interval $(0,2)$.",
    "Metallic_mean_1": "Metallic Ratios in Primitive Pythagorean Triangles",
    "Metallic_mean_2": "Relationship between the silver ratio and the octagon.",
    "Metallic_mean_3": "If one removes $n$ largest possible squares from a rectangle with ratio length/width equal to the $n$th metallic mean, one gets a rectangle with the same ratio length/width (in the figures, $n$ is the number of dotted lines).",
    "Metallic_mean_4": "Bronze spiral in the bronze rectangle, approximated by quarter circles.",
    "Metallic_mean_5": "Approximation to the silver spiral by quarter circles inscribed in squares whose side lengths are Pell numbers.",
    "Metalog_distribution_1": "Three-term metalog distributions",
    "Metalog_distribution_2": "Four-term metalog distribution when $a_{3}=0$",
    "Metalog_distribution_3": "Weibull distributions (blue) closely approximated by nine-term semi-bounded metalog distributions (dashed, yellow)",
    "Method_of_steepest_descent_1": "An illustration of Complex Morse lemma",
    "Method_of_steepest_descent_2": "An illustration to the derivation of equation ${\\displaystyle I(\\lambda )\\equiv \\int _{I_{x}}f(x)e^{\\lambda S(x)}dx=\\left({\\frac {2\\pi }{\\lambda }}\\right)^{\\frac {n}{2}}e^{\\lambda S(x^{0})}\\left(f(x^{0})+O\\left(\\lambda ^{-1}\\right)\\right)\\prod _{j=1}^{n}(-\\mu _{j})^{-{\\frac {1}{2}}},\\qquad \\lambda \\to \\infty ,}$",
    "Midpoint_method_1": "Illustration of the midpoint method assuming that $y_{n}$ equals the exact value $y(t_{n}).$ The midpoint method computes $y_{n+1}$ so that the red chord is approximately parallel to the tangent line at the midpoint (the green line).",
    "Midpoint_method_2": "Illustration of numerical integration for the equation $y'=y,y(0)=1.$ Blue: the Euler method, green: the midpoint method, red: the exact solution, $y=e^{t}.$ The step size is $h=1.0.$",
    "Midpoint_method_3": "The same illustration for $h=0.25.$ It is seen that the midpoint method converges faster than the Euler method.",
    "Minimum-weight_triangulation_1": "Three possible triangulations of the same polygon. The central triangulation has less weight (sum of perimeters).",
    "Minkowski_addition_2": "The red figure is the Minkowski sum of blue and green figures.",
    "Minkowski_plane_1": "Minkowski plane: minimal model",
    "Misiurewicz_point_2": "Principal Misiurewicz point of the wake $\\frac{1}{31}$",
    "Mollifier_1": "A mollifier (top) in dimension one. At the bottom, in red is a function with a corner (left) and sharp jump (right), and in blue is its mollified version.",
    "Mollifier_2": "The function $\\varphi(x)$ in dimension one",
    "Mollweide's_formula_1": "Any cyclic quadrilateral satisfies a generalization of Mollweide's formula.",
    "Monkey_saddle_2": "Smelt petal: $x + y + z + xyz = 0$",
    "Monodromy_1": "The imaginary part of the complex logarithm. Trying to define the complex logarithm on $\\mathbb {C} -\\{0\\}$ gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of $\\mathbb {C} -\\{0\\}$ by a helicoid (an example of a Riemann surface).",
    "Monodromy_2": "A path in the base has paths in the total space lifting it. Pushing along these paths gives the monodromy action from the fundamental groupoid.",
    "Monodromy_theorem_1": "Illustration of analytic continuation along a curve (only a finite number of the disks $U_{t}$ are shown).",
    "Monodromy_theorem_2": "Homotopy with fixed endopoints is necessary for the monodromy theorem to hold.",
    "Monoid_1": "Example monoid homomorphism $x \\mapsto 2x$ from $(\\mathbb{N}, +, 0)$ to $(\\mathbb{N}, \\times, 1)$. It is injective, but not surjective.",
    "Monotone_cubic_interpolation_1": "Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set.",
    "Montgomery_curve_1": "A Montgomery curve of equation ${\\frac{1}{4}}y^{2}=x^{3}+{\\frac{5}{2}}x^{2}+x$",
    "Moore_plane_2": "Moore Plane graphic representation",
    "Mordell_curve_1": "$y^2 = x^3 + 1$, with solutions at $(-1, 0)$, $(0, 1)$ and $(0, -1)$",
    "Morley's_trisector_theorem_2": "Elementary proof of Morley's trisector theorem",
    "Morley_centers_1": "Diagram showing the first Morley center of a trinagle $ABC$.",
    "Morley_centers_2": "Diagram showing the second Morley center of a triangle $ABC$.",
    "Morrie's_law_1": "Regular nonagon $ABCDEFGHI$ with $O$ being the center of its circumcircle. Computing of the angles: ${\\begin{aligned}40^{\\circ }&={\\frac {360^{\\circ }}{9}}\\\\70^{\\circ }&={\\frac {180^{\\circ }-40^{\\circ }}{2}}\\\\\\alpha &=180^{\\circ }-90^{\\circ }-70^{\\circ }=20^{\\circ }\\\\\\beta &=180^{\\circ }-90^{\\circ }-(70^{\\circ }-\\alpha )=40^{\\circ }\\\\\\gamma &=140^{\\circ }-\\beta -\\alpha =80^{\\circ }\\end{aligned}}$",
    "Morse_theory_2": "Contour lines around a saddle point",
    "Morse_theory_3": "A cylinder (upper right), formed by $M^{a}$ when $f(q)<a<f(r)$, is homotopy equivalent to a 1-cell attached to a disk (lower left).",
    "Morse_theory_4": "A torus with a disk removed (upper right), formed by $M^{a}$ when $f(r)<a<f(s)$, is homotopy equivalent to a 1-cell attached to a cylinder (lower left).",
    "Motion_(geometry)_1": "A glide reflection is a type of Euclidean motion.",
    "Moving_average_1": "Smoothing of a noisy sine (blue curve) with a moving average (red curve).",
    "Multibrot_set_1": "Multibrot 3 at the bottom-left of the main part.",
    "Multibrot_set_2": "Detail of Multijulia 8.",
    "Multibrot_set_3": "Enlarged first quadrant of the multibrot set for the iteration $z \\mapsto z^{-2} + c$ rendered with the Escape Time algorithm.",
    "Multibrot_set_4": "Enlarged first quadrant of the multibrot set for the iteration $z \\mapsto z^{-2} + c$ rendered using the Lyapunov exponent of the sequence as a stability criterion rather than using the Escape Time algorithm. Periodicity checking was used to colour the set according to the period of the cycles of the orbits.",
    "Multicollinearity_1": "A depiction of multicollinearity.",
    "Multicollinearity_2": "In a linear regression, the true parameters are $a_{1}=2,a_{2}=4$ which are reliably estimated in the case of uncorrelated $X_{1}$ and $X_{2}$ (black case) but are unreliably estimated when $X_{1}$ and $X_{2}$ are correlated (red case).",
    "Multigraph_1": "A multigraph with multiple edges (red) and several loops (blue). Not all authors allow multigraphs to have loops.",
    "Multiple_gamma_function_1": "Plot of the Barnes G aka double gamma function $G(z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors",
    "Multiple_integral_1": "Integral as area between two curves.",
    "Multiplication_1": "Multiplication can also be thought of as scaling. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result.",
    "Multiplication_2": "3 by 4 is 12.",
    "Multiplication_3": "A complex number in polar coordinates",
    "Multiplication_4": "Four bags with three marbles per bag gives twelve marbles ($4 \\times 3 = 12$).",
    "Multiplication_5": "Product of 45 and 256. Note the order of the numerals in 45 is reversed down the left column. The carry step of the multiplication can be performed at the final stage of the calculation (in bold), returning the final product of $45 \\times 256 = 11520$. This is a variant of Lattice multiplication.",
    "Multiplicative_inverse_2": "The reciprocal function: $y = \\frac{1}{x}$. For every $x$ except $0$, $y$ represents its multiplicative inverse. The graph forms a rectangular hyperbola.",
    "Multiply_perfect_number_1": "Demonstration, with Cuisenaire rods, of the 2-perfection of the number 6",
    "Multivariate_normal_distribution_1": "Bivariate normal distribution centered at $(1,3)$ with a standard deviation of 3 in roughly the $(0.878,0.478)$ direction and of 1 in the orthogonal direction.",
    "Multivariate_normal_distribution_2": "Top: the probability of a bivariate normal in the domain $x\\sin y-y\\cos x>1$ (blue regions). Middle: the probability of a trivariate normal in a toroidal domain. Bottom: converging Monte-Carlo integral of the probability of a 4-variate normal in the 4d regular polyhedral domain defined by $\\sum _{i=1}^{4}\\vert x_{i}\\vert <1$. These are all computed by the numerical method of ray-tracing.",
    "Multivariate_normal_distribution_3": "a: Probability density of a function $\\cos x^{2}$ of a single normal variable $x$ with $\\mu =-2$ and $\\sigma =3$. b: Probability density of a function $x^{y}$ of a normal vector $(x,y)$, with mean ${\\boldsymbol {\\mu }}=(1,2)$, and covariance $\\mathbf {\\Sigma } ={\\begin{bmatrix}.01&.016\\\\.016&.04\\end{bmatrix}}$. c: Heat map of the joint probability density of two functions of a normal vector $(x,y)$, with mean ${\\boldsymbol {\\mu }}=(-2,5)$, and covariance  $\\mathbf {\\Sigma } ={\\begin{bmatrix}10&-7\\\\-7&10\\end{bmatrix}}$. d: Probability density of a function $\\sum _{i=1}^{4}\\vert x_{i}\\vert$ of 4 iid standard normal variables. These are computed by the numerical method of ray-tracing.",
    "Mylar_balloon_(geometry)_1": "Profile of the mylar balloon in the $xz$ plane",
    "N-curve_1": "n-curve with $N=53$",
    "N-ellipse_1": "Examples of 3-ellipses for three given foci. The progression of the distances is not linear.",
    "NK_model_1": "Visualization of two dimensions of a NK fitness landscape. The arrows represent various mutational paths that the population could follow while evolving on the fitness landscape",
    "NK_model_2": "Illustration of tunable topology in the NK model. Nodes are individual binary strings, edges connect strings with a Hamming distance of exactly one. (left) $N = 5, K = 0$. (centre) $N = 5, K = 1$. (right) $N = 5, K = 2$. The colour of a node denotes its fitness, with redder values having higher fitness. The embedding of the hypercube is chosen so that the fitness maximum is at the centre. Notice that the $K = 0$ landscape appears smoother than the higher-K cases.",
    "Napkin_ring_problem_1": "If a hole of height $h$ is drilled straight through the center of a sphere, the volume of the remaining band does not depend on the size of the sphere. For a larger sphere, the band will be thinner but longer.",
    "Napkin_ring_problem_2": "Finding the measurements of the ring that is the horizontal cross-section.",
    "Napoleon's_theorem_1": "Napoleon-Barlotti theorem for a pentagon",
    "Napoleon's_theorem_2": "Equilateral triangles on the sides of an arbitrary hexagon: When $A_1 = A_4$, $A_2 = A_5$, and $A_3 = A_6$, this theorem becomes the Napoleon theorem",
    "Narayana_number_1": "The 6 ordered rooted trees of 4 edges and 2 leaves, corresponding to the Narayana number $N(4, 2)$",
    "Narayana_number_2": "The 1,6,6,1 non-crossing partitions with 1,2,3,4 blocks of a 4-element set",
    "Neighbourhood_(mathematics)_1": "A closed rectangle $V$ is not a neighbourhood of any of its corners or its boundary since there is no open set in $V$ containing any corner or edge point.",
    "Neighbourhood_(mathematics)_2": "A set $S$ in the plane and a uniform neighbourhood $V$ of $S.$",
    "Neighbourhood_(mathematics)_3": "The epsilon neighbourhood of a number $a$ on the real number line.",
    "Neighbourhood_(mathematics)_4": "A set $V$ in the plane is a neighbourhood of a point $p$ if a small disc around $p$ is contained in $V$.} The small disc around $p$ is an open set $U$.",
    "Nested_intervals_1": "4 members of a sequence of nested intervals",
    "Newton's_method_1": "$x_n+1$ is a better approximation than $x_n$ for the root $x$ of the function $f$ (blue curve)",
    "Newton's_method_2": "An illustration of Newton's method.",
    "Newton's_theorem_(quadrilateral)_1": "$P$ lies on the Newton line $EF$",
    "Nine-point_center_1": "A triangle showing its circumcircle and circumcenter (black), altitudes and orthocenter (red), and nine-point circle and nine-point center (blue)",
    "Nine-point_circle_1": "The nine points",
    "Nine-point_circle_2": "The nine-point circle is tangent to the incircle and excircles.",
    "Nine-point_circle_3": "9 point circle and circumcircle.",
    "Nine-point_circle_4": "$ABCD$ is a cyclic quadrilateral. $\\triangle EFG$ is the diagonal triangle of $ABCD$. The point $T$ of intersection of the bimedians of $ABCD$ belongs to the nine-point circle of $\\triangle EFG$.",
    "Nine-point_circle_5": "The nine point circle and the 16 tangent circles of the orthocentric system",
    "Non-analytic_smooth_function_1": "The smooth transition $g$ from 0 to 1 defined here.",
    "Non-analytic_smooth_function_2": "Approximation of the smooth-everywhere, but nowhere-analytic function mentioned here. This partial sum is taken from $k = 2^0$ to $2^{500}$.",
    "Non-analytic_smooth_function_3": "The function $\\Psi_1(x)$ in one dimension.",
    "Non-negative_matrix_factorization_1": "Illustration of approximate non-negative matrix factorization: the matrix $V$ is represented by the two smaller matrices $W$ and $H$, which, when multiplied, approximately reconstruct $V$.",
    "Nonagon_2": "Regular nonagon with annotation.",
    "Norm_(mathematics)_1": "Illustrations of unit circles in different norms.",
    "Numerical_differentiation_2": "Example showing the difficulty of choosing $h$ due to both rounding error and formula error",
    "Obstacle_problem_1": "Solution of a one-dimensional obstacle problem. Notice how the solution stays superharmonic (concave down in 1-D), and matches derivatives with the obstacle (which is the $C^{1,1}$ condition)",
    "Octagon_2": "The diagonals of the green quadrilateral are equal in length and at right angles to each other",
    "Octagon_3": "The area of a regular octagon can be computed as a truncated square.",
    "Octagon_4": "building a regular octagon by folding a sheet of paper",
    "Octagon_5": "Meccano octagon construction.",
    "Octagon_6": "A regular skew octagon seen as edges of a square antiprism, symmetry $D_{4d},\\ [2^+, 8],\\ (2^*4)$, order 16.",
    "Octagon_7": "Octagon at a given circumcircle",
    "Octahedron_1": "3D model of regular octahedron",
    "Octahedron_2": "The graph of a regular octahedron",
    "Octahedron_3": "The octahedron represents the central intersection of two tetrahedra",
    "Octahedron_4": "A regular faced convex polyhedron, the gyrobifastigium.",
    "Odd_graph_2": "The Kneser graph $KG_{5,2}$, one way of constructing the Petersen graph",
    "Odds_ratio_1": "A graph showing how the log odds ratio relates to the underlying probabilities of the outcome $X$ occurring in two groups, denoted $A$ and $B$. The log odds ratio shown here is based on the odds for the event occurring in group $B$ relative to the odds for the event occurring in group $A$. Thus, when the probability of $X$ occurring in group $B$ is greater than the probability of $X$ occurring in group $A$, the odds ratio is greater than 1, and the log odds ratio is greater than 0.",
    "Oloid_2": "The plane shape of a developed oloid surface",
    "One-sided_limit_1": "The function $f(x)=x^{2}+\\operatorname {sign} (x),$ where $\\operatorname {sign} (x)$ denotes the sign function, has a left limit of $-1,$ a right limit of $+1,$ and a function value of $0$ at the point $x=0.$",
    "One-sided_limit_2": "Plot of the function $1/(1+2^{-1/x}).$",
    "Orbit_portrait_1": "Julia set with external rays landing on period-3 orbit",
    "Orbit_portrait_2": "Julia set with period-two parabolic orbit. The associated orbit portrait has characteristic arc $I = \\left(\\frac{22}{63}, \\frac{25}{63}\\right)$ and valence $v = 3$ rays per orbit point.",
    "Orbit_portrait_3": "3 external rays of period 3 cycle: ${\\mathcal {R}}_{\\frac {1}{7}},{\\mathcal {R}}_{\\frac {2}{7}},{\\mathcal {R}}_{\\frac {4}{7}}\\,$, which land on fixed point $z=\\alpha _{c}\\,$",
    "Order_of_integration_(calculus)_1": "Integration over the triangular area can be done using vertical or horizontal strips as the first step. This is an overhead view, looking down the $z$-axis onto the $xy$-plane. The sloped line is the curve $y = x$.",
    "Order_theory_1": "Hasse diagram of the set of all divisors of 60, partially ordered by divisibility",
    "Order_type_1": "Three well-orderings on the set of natural numbers with distinct order types (top to bottom): $\\omega$, $\\omega +5$, and $\\omega +\\omega$.",
    "Ordered_field_2": "The property $x<y\\Rightarrow a+x<a+y$",
    "Ordered_vector_space_1": "A point $x$ in $\\mathbb {R} ^{2}$ and the set of all $y$ such that $x\\leq y$ (in red). The order here is $x\\leq y$ if and only if $x_{1}\\leq y_{1}$ and $x_{2}\\leq y_{2}.$",
    "Orthocentric_system_2": "Common nine-point circle, where $N$, $O_4$, $A_4$ are the nine-point center, circumcenter, and orthocenter respectively of the triangle formed from the other three orthocentric points $A_1$, $A_2$, $A_3$.",
    "Orthocentric_system_3": "Orthocentric system. Where $O_1$, $O_2$, $O_3$, $O_4$ are the circumcenters of the four possible triangles formed from the orthocentric points $A_1$, $A_2$, $A_3$, $A_4$.",
    "Orthocentroidal_circle_2": "Orthocentroidal circle bounded by the orthocenter ($H$) and centroid ($S$) Euler line, on which the circumcenter ($O$) and nine-point center ($N$) both lie along with $H$ and $S$ $F_1$ and $F_2$: Fermat points $F$: Feuerbach point $I$: Incenter $G$: Gergonne point $U$: Symmedian point",
    "Orthodiagonal_quadrilateral_1": "An orthodiagonal quadrilateral (yellow). According to the characterization of these quadrilaterals, the two red squares on two opposite sides of the quadrilateral have the same total area as the two blue squares on the other pair of opposite sides.",
    "Orthodiagonal_quadrilateral_2": "An orthodiagonal quadrilateral $ABCD$ (in blue). The Varignon parallelogram (in green) formed by the midpoints of the edges of $ABCD$ is a rectangle. Additionally, the four midpoints (grey) and the four feet of the maltitudes (red) are cocyclic on the 8-point-circle.",
    "Orthodiagonal_quadrilateral_3": "A second 8-point circle can be constructed from an orthodiagonal quadrilateral $ABCD$ (in blue). The lines perpendicular to each side through the intersection of the diagonals intersect the sides in 8 different points, which are all cocyclic.",
    "Orthodiagonal_quadrilateral_4": "$ABCD$ is an orthodiagonal quadrilateral, $P_{1}X_{1}Z_{1}Y_{1}$ and $P_{2}X_{2}Z_{2}Y_{2}$ are rectangles whose sides are parallel to the diagonals of the quadrilateral.",
    "Orthodiagonal_quadrilateral_5": "$ABCD$ is an orthodiagonal quadrilateral. $P_{1}$ and $Q_{1}$ are Pascal points formed by the circle $\\omega _{1}$, $\\sigma _{P_{1}Q_{1}}$ is Pascal-points circle which defines the rectangle $P_{1}V_{1}Q_{1}W_{1}$. $P_{2}$ and $Q_{2}$ are Pascal points formed by the circle $\\omega _{2}$, $\\sigma _{P_{2}Q_{2}}$ is Pascal-points circle which defines the rectangle $P_{2}V_{2}Q_{2}W_{2}$.",
    "Orthogonal_coordinates_1": "A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained.",
    "Orthogonal_coordinates_2": "Visualization of 2D orthogonal coordinates. Curves obtained by holding all but one coordinate constant are shown, along with basis vectors. Note that the basis vectors aren't of equal length: they need not be, they only need to be orthogonal.",
    "Orthogonal_trajectory_1": "Concentric circles with orthogonal trajectories",
    "Orthogonal_trajectory_2": "Parabolas with orthogonal trajectories",
    "Orthogonal_trajectory_3": "Isogonal trajectories of concentric circles for $\\alpha =45^{\\circ}$",
    "Orthogonality_1": "The line segments $AB$ and $CD$ are perpendicular to each other.",
    "Orthogonality_2": "Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle $\\varphi$, right: in Minkowski spacetime through hyperbolic angle $\\varphi$ (red lines labelled $c$ denote the worldlines of a light signal; a vector is orthogonal to itself if it lies on this line).",
    "Orthopole_1": "The triangle's Orthopole",
    "Osculating_circle_1": "An osculating circle",
    "Osculating_circle_2": "Osculating circles of the Archimedean spiral, nested by the Tait-Kneser theorem. The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other.",
    "Osculating_circle_3": "The osculating circle of the parabola at its vertex has radius 0.5 and fourth order contact.",
    "Outer_automorphism_group_1": "The symmetries of the Dynkin diagram, $D_4$, correspond to the outer automorphisms of Spin(8) in triality.",
    "Oval_(projective_plane)_2": "projective conic in inhomogeneous coordinates: parabola plus point at infinity of the axis",
    "Oval_(projective_plane)_3": "projective conic in inhomogeneous coordinates: hyperbola plus points at infinity of the asymptotes",
    "Ovoid_(projective_geometry)_1": "To the definition of an ovoid: t tangent, s secant line",
    "P-rep_1": "P-rep function (in log scale)",
    "Packing_problems_1": "Spheres or circles packed loosely (top) and more densely (bottom)",
    "Packing_problems_2": "The hexagonal packing of circles on a 2-dimensional Euclidean plane.",
    "Packing_problems_3": "Packing nine L tricubes into a cube",
    "Packing_problems_4": "The optimal packing of 10 circles in a circle",
    "Packing_problems_5": "The optimal packing of 15 circles in a square",
    "Packing_problems_6": "The optimal packing of 10 squares in a square",
    "Padovan_sequence_2": "Triangles with sides in ratio of $1/?$ form a closed spiral",
    "Pairing_function_2": "Graph of the Cantor pairing function",
    "Pairing_function_3": "A diagonally incrementing \"snaking\" function, from same principles as Cantor's pairing function, is often used to demonstrate the countability of the rational numbers.",
    "Pappus's_hexagon_theorem_2": "Pappus's theorem: affine form $Ab\\parallel aB,Bc\\parallel bC\\Rightarrow Ac\\parallel aC$",
    "Pappus's_hexagon_theorem_3": "Pappus theorem: proof",
    "Pappus's_hexagon_theorem_4": "Triangles $XcC$ and $BbY$ are perspective from $A$ and $a$, and so, also from $Z$.",
    "Parabolic_coordinates_1": "In green, confocal parabolae opening upwards, $2y={\\frac {x^{2}}{\\sigma ^{2}}}-\\sigma ^{2}$ In red, confocal parabolae opening downwards, $2y=-{\\frac {x^{2}}{\\tau ^{2}}}+\\tau ^{2}$",
    "Parabolic_cylinder_function_2": "Plot of the parabolic cylinder function $D_\\nu(z)$ with $\\nu = 5$ in the complex plane from $-2 - 2i$ to $2 + 2i$.",
    "Paraboloid_2": "A hyperbolic paraboloid with lines contained in it",
    "Paraboloid_3": "A hyperbolic paraboloid with hyperbolas and parabolas",
    "Paraboloid_4": "STL hyperbolic paraboloid model",
    "Paraboloid_5": "elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid",
    "Parallel_(geometry)_2": "The problem: Draw a line through a parallel to $l$.",
    "Parallel_(geometry)_3": "Intersecting, parallel and ultra parallel lines through $a$ with respect to $l$ in the hyperbolic plane. The parallel lines appear to intersect $l$ just off the image. This is just an artifact of the visualisation. On a real hyperbolic plane the lines will get closer to each other and 'meet' in infinity.",
    "Parallel_(geometry)_4": "On the sphere there is no such thing as a parallel line. Line $a$ is a great circle, the equivalent of a straight line in spherical geometry. Line $c$ is equidistant to line a but is not a great circle. It is a parallel of latitude. Line $b$ is another geodesic which intersects a in two antipodal points. They share two common perpendiculars (one shown in blue).",
    "Parallel_projection_1": "Parallel projection terminology and notations. The two blue parallel line segments to the right remain parallel when projected onto the image plane to the left.",
    "Parallel_transport_1": "Parallel transport of a vector around a closed loop (from $A$ to $N$ to $B$ and back to $A$) on the sphere. The angle by which it twists, $\\alpha$, is proportional to the area inside the loop.",
    "Parallel_transport_2": "Two rungs of Schild's ladder. The segments $A1_X_1$ and $A_2X_2$ are an approximation to first order of the parallel transport of $A_0X_0$ along the curve.",
    "Parallelepiped_1": "Parallelepiped, generated by three vectors",
    "Parallelogram_1": "A parallelogram can be rearranged into a rectangle with the same area.",
    "Parallelogram_2": "The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram",
    "Parallelogram_3": "Proof without words of Varignon's theorem",
    "Parallelogram_law_2": "Vectors involved in the parallelogram law.",
    "Parametric_equation_1": "A Lissajous curve where $k_x = 3$ and $k_y = 2$.",
    "Parametric_equation_2": "The butterfly curve can be defined by parametric equations of $x$ and $y$.",
    "Parametric_surface_1": "Torus, created with equations: ${\\begin{aligned}x&=r\\sin v\\\\y&=(R+r\\cos v)\\sin u\\\\z&=(R+r\\cos v)\\cos u\\end{aligned}}$",
    "Parametric_surface_2": "Parametric surface forming a trefoil knot.",
    "Partial_correlation_1": "Geometrical interpretation of partial correlation for the case of $N = 3$ observations and thus a 2-dimensional hyperplane",
    "Partial_derivative_2": "The volume of a cone depends on height and radius",
    "Partial_derivative_3": "A slice of the graph above showing the function in the $xz$-plane at $y = 1$. The two axes are shown here with different scales. The slope of the tangent line is 3.",
    "Partial_derivative_4": "A graph of $z = x^2 + xy + y^2$. For the partial derivative at $(1, 1)$ that leaves y constant, the corresponding tangent line is parallel to the $xz$-plane.",
    "Partial_function_2": "An example of a function that is not injective.",
    "Partially_ordered_set_1": "The Hasse diagram of the set of all subsets of a three-element set $\\{x, y, z\\}$, ordered by inclusion. Sets connected by an upward path, like $\\emptyset$ and $\\{x, y\\}$, are comparable, while, e.g., $\\{x\\}$ and $\\{y\\}$ are not.",
    "Partially_ordered_set_2": "Graph of the divisibility of numbers from 1 to 4. This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship from 2 to 3 or 3 to 4",
    "Partially_ordered_set_3": "Nonnegative integers, ordered by divisibility",
    "Partially_ordered_set_4": "Order isomorphism between the divisors of $120$ (partially ordered by divisibility) and the divisor-closed subsets of $\\{2, 3, 4, 5, 8\\}$ (partially ordered by set inclusion).",
    "Partially_ordered_set_5": "The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are all maximal elements, and the bottom row are all minimal elements, but there is no greatest and no least element.",
    "Partially_ordered_set_6": "Reflexive closure of strict direct product order on $\\mathbb{N} \\times \\mathbb{N}$. Elements covered by $(3, 3)$ and covering $(3, 3)$ are highlighted in green and red, respectively.",
    "Partition_of_an_interval_1": "A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red.",
    "Partition_of_unity_1": "A partition of unity on a circle with four functions. The circle is unrolled to a line segment (the bottom solid line) for graphing purposes. The dashed line on top is the sum of the functions in the partition.",
    "Pascal's_pyramid_1": "Pascal's pyramid's first five layers. Each face (orange grid) is Pascal's triangle. Arrows show derivation of two example terms.",
    "Pascal's_pyramid_2": "Derivation of the first five levels of Pascal's pyramid - where multiple values point to a number, the values are summed",
    "Pascal's_theorem_2": "Self-crossing hexagon $ABCDEF$, inscribed in a circle. Its sides are extended so that pairs of opposite sides intersect on Pascal's line. Each pair of extended opposite sides has its own color: one red, one yellow, one blue. Pascal's line is shown in white.",
    "Pascal's_theorem_3": "The intersections of the extended opposite sides of simple cyclic hexagon $ABCDEF$ (right) lie on the Pascal line $MNP$ (left).",
    "Pascal's_theorem_4": "Pascal's theorem: degenerations",
    "Pascal's_triangle_1": "Pascal's version of the triangle",
    "Pascal's_triangle_2": "Visualisation of binomial expansion up to the 4th power",
    "Pascal's_triangle_3": "The numbers of compositions of $n+1$ into $k+1$ ordered partitions form Pascal's triangle.",
    "Pascal's_triangle_4": "Fibonacci sequence in Pascal's triangle",
    "Pascal's_triangle_5": "A level-4 approximation to a Sierpinski triangle obtained by shading the first 32 rows of a Pascal triangle white if the binomial coefficient is even and black if it is odd.",
    "Path_(graph_theory)_1": "A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black",
    "Path_(graph_theory)_2": "Trail from A to E, but not path (B vertex repeated)",
    "Path_(topology)_1": "The points traced by a path from $A$ to $B$ in $\\mathbb {R} ^{2}.$ However, different paths can trace the same set of points.",
    "Path_(topology)_2": "A homotopy between two paths.",
    "Pell's_equation_1": "Pell's equation for $n = 2$ and six of its integer solutions",
    "Pell_number_2": "Rational approximations to regular octagons, with coordinates derived from the Pell numbers.",
    "Pell_number_3": "Integer right triangles with nearly equal legs, derived from the Pell numbers.",
    "Penning_trap_1": "A classical trajectory in the radial plane for $\\omega _{+}/\\omega _{-}=8$",
    "Pentadecagon_1": "The symmetries of a regular pentadecagon as shown with colors on edges and vertices. Lines of reflections are blue. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions.",
    "Pentagon_2": "Symmetries of a regular pentagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.",
    "Pentagon_3": "Side ($t$), circumradius ($R$), inscribed circle radius ($r$), height ($R+r$), width/diagonal ($\\varphi t$)",
    "Pentagon_4": "Equilateral pentagon built with four equal circles disposed in a chain.",
    "Pentagonal_number_1": "Proof without words that the nth pentagonal number can be decomposed into three triangular numbers and the number n.",
    "Pentagonal_prism_2": "The pentagonal stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism.",
    "Pentagram_2": "Drawing of a pentagram with notations of its angles.",
    "Pentagram_3": "A regular pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.",
    "Pentagram_4": "A golden triangle (red), and large (blue) and small (green) golden gnomons in a regular pentagram.",
    "Perifocal_coordinate_system_1": "The perifocal coordinate system (with unit vectors p, q, w), against the reference coordinate system (with unit vectors I, J, K)",
    "Perimeter_1": "cardioid $\\gamma :[0,2\\pi ]\\to \\mathbb {R} ^{2}$(drawing with $a=1$) $x(t)=2a\\cos(t)(1+\\cos(t))$ $y(t)=2a\\sin(t)(1+\\cos(t))$ $L=\\int _{0}^{2\\pi }{\\sqrt {x'(t)^{2}+y'(t)^{2}}}\\,\\mathrm {d} t=16a$",
    "Perimeter_2": "Perimeter is the distance around a two-dimensional shape, the length of the shape's boundary.",
    "Periodic_function_1": "An illustration of a periodic function with period $P$.",
    "Periodic_function_2": "A graph of the sine function, showing two complete periods",
    "Periodic_function_3": "A plot of $f(x)=\\sin(x)$ and $g(x)=\\cos(x)$; both functions are periodic with period $2\\pi$.",
    "Permutation_2": "Permutations without repetition on the left, with repetition to their right",
    "Permutation_3": "Composition of permutations corresponding to a multiplication of permutation matrices.",
    "Permutation_graph_1": "The permutation graph and the matching diagram for the permutation $(4,3,5,1,2)$",
    "Perpendicular_bisector_construction_of_a_quadrilateral_1": "First iteration of the perpendicular bisector construction",
    "Perspective_(geometry)_1": "Two perspective triangles, with their perspective axis and center",
    "Perspective_(geometry)_2": "Two triply perspective triangles $BbY$ and $CcX$",
    "Perspectivity_1": "A perspectivity: $ABCD\\doublebarwedge A'B'C'D',$",
    "Petersen's_theorem_1": "A perfect matching (red edges), in the Petersen graph. Since the Petersen graph is cubic and bridgeless, it meets the conditions of Petersen's theorem.",
    "Petersen's_theorem_2": "A cubic (but not bridgeless) graph with no perfect matching, showing that the bridgeless condition in Petersen's theorem cannot be omitted",
    "Phase_line_(mathematics)_1": "A plot of $f(y)$ (left) and its phase line (right). In this case, a and c are both sinks and b is a source.",
    "Philo_line_1": "The philo line of a point $P$ and angle $DOE$, and the defining equality of distances from $P$ and $Q$ to the ends of $DE$, where $Q$ is the base of a perpendicular from the apex of the angle",
    "Pinched_torus_1": "A Pinched Torus",
    "Planar_graph_1": "Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either $K_{5}$ (top) or $K_{3,3}$ (bottom) subgraphs",
    "Planar_graph_2": "An example of a graph with no $K_{5}$ or $K_{3,3}$ subgraph. However, it contains a subdivision of $K_{3,3}$ and is therefore non-planar.",
    "Planar_graph_3": "A Schlegel diagram of a regular dodecahedron, forming a planar graph from a convex polyhedron.",
    "Planar_graph_4": "A planar graph and its dual",
    "Planar_graph_5": "The Goldner-Harary graph is maximal planar. All its faces are bounded by three edges.",
    "Planar_ternary_ring_1": "Coordinates of a projective plane to establish a planar ternary ring",
    "Plane_partition_1": "A plane partition of 30 represented as stacks of unit cubes",
    "Plane_partition_2": "A symmetric plane partition, sum 35",
    "Plane_partition_3": "A cyclically symmetric plane partition",
    "Plane_partition_4": "A totally symmetric plane partition",
    "Plane_partition_5": "A self-complementary plane partition",
    "Plane_partition_6": "A cyclically symmetric self-complementary plane partition",
    "Plane_partition_7": "A totally symmetric self-complementary plane partition",
    "Plastic_ratio_1": "Triangles in ratio of the plastic number () in a three armed counter clockwise spiral. If the first (middle) triangle has a side length of 6, the next thee triangles have a side length of . If continued to infinity the spiral fills the space completely. This is the sixteenth iteration approximated at two decimal points.",
    "Plastic_ratio_2": "Squares with sides in ratio $\\rho$ form a closed spiral.",
    "Plastic_ratio_3": "Rectangles in aspect ratios $\\rho$, $\\rho^2$, $\\rho^3$ (top) and $\\rho^2$, $\\rho$, $\\rho^3$ (bottom row) tile the square.",
    "Plastic_ratio_4": "A Rauzy fractal associated with the plastic ratio-cubed. The central tile and its three subtiles have areas in the ratios $\\rho^5 : \\rho^2 : \\rho : 1$.",
    "Plastic_ratio_5": "A plastic Rauzy fractal: the combined surface and the three separate tiles have areas in the ratios $\\rho^5 : \\rho^2 : \\rho : 1$.",
    "Plastic_ratio_6": "Three partitions of a square into similar rectangles, $1 = 3 \\cdot \\tfrac{1}{3} = \\tfrac{2}{3} + 2 \\cdot \\tfrac{1}{6} = \\tfrac{1}{\\rho^2} + \\tfrac{1}{\\rho^4} + \\tfrac{1}{\\rho^8}$.",
    "Plastic_ratio_7": "The relation $x_c + t = \\sqrt{\\rho}$ in a $\\rho^2$ rectangle.",
    "Pochhammer_contour_2": "Pochhammer cycle is homologous to zero: it is the boundary of the green area minus the boundary of the red one.",
    "Point_reflection_2": "The composition of two offset point reflections in 2-dimensions is a translation.",
    "Polar_space_1": "Generalized quadrangle with three points per line; a polar space of rank 2",
    "Polite_number_1": "Young diagram representing visually $15 = 4 + 5 + 6$",
    "Polydivisible_number_1": "Graph of number of $n$-digit polydivisible numbers in base 10 $F_{10}(n)$ vs estimate of $F_{10}(n)$",
    "Polygon_2": "Some different types of polygon",
    "Polygon_3": "Partitioning an n-gon into $n - 2$ triangles",
    "Polygon_4": "Coordinates of a non-convex pentagon",
    "Polygon_triangulation_2": "A polygon ear",
    "Polygon_triangulation_3": "Breaking a polygon into monotone polygons",
    "Polygonal_chain_1": "A simple polygonal chain",
    "Polygonal_chain_2": "A self-intersecting polygonal chain",
    "Polygonal_chain_3": "A closed polygonal chain",
    "Polygonal_chain_4": "A set of $n=17$ points has a polygonal path with 4 same-sign slopes",
    "Polyharmonic_spline_2": "Polyharmonic basis functions",
    "Polyhedron_1": "The cube is regular polyhedron, because its faces, edges, and vertices are transitive to another, and the appearance is unchanged.",
    "Polyhedron_2": "The regular tetrahedron has full tetrahedral symmetry: three-fold rotation around axis passing both vertex and triangular face, and two-fold rotation around axis through two edges, as well as the reflection plane through two faces and one edge",
    "Polynomial_1": "The graph of a polynomial function of degree 3",
    "Polynomial_interpolation_1": "The red dots denote the data points $(x_k, y_k)$, while the blue curve shows the interpolation polynomial.",
    "Polynomial_interpolation_2": "Lozenge Diagram: geometric representation of polynomial interpolations.",
    "Polynomial_lemniscate_1": "$\\left|z^6 + z^5 + z^4 + z^3 + z^2 + z + 1\\right| = 1$",
    "Polynomial_lemniscate_2": "Erdos lemniscate of degree ten and genus six",
    "Polyomino_1": "The 18 one-sided pentominoes, including 6 mirrored pairs.",
    "Polyomino_2": "The two tiling nonominoes not satisfying the Conway criterion.",
    "Polyomino_3": "A minimal compatibility figure for the T and W pentominoes.",
    "Pompeiu's_theorem_1": "Proof of Pompeiu's theorem with Pompeiu triangle $\\triangle PCP'$",
    "Pontryagin_duality_1": "The 2-adic integers, with selected corresponding characters on their Pontryagin dual group (the Prüfer group)",
    "Post's_lattice_1": "Hasse diagram of Post's lattice",
    "Post's_lattice_2": "Central part of the lattice",
    "Post's_lattice_3": "Subset of Post's lattice showing only the 7 clones containing all constant functions",
    "Power_diagram_2": "The power of a point $P$ outside of a given circle",
    "Power_diagram_3": "The radical axis of two intersecting circles. The power diagram of the two circles is the partition of the plane into two halfplanes formed by this line.",
    "Preconditioner_1": "Illustration of gradient descent",
    "Primary_pseudoperfect_number_1": "Graphical demonstration that $1 = \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{11} + \\frac{1}{23} + \\frac{1}{31} + \\frac{1}{2\\times3\\times11\\times23\\times31}$. Therefore the product, 47058, is primary pseudoperfect.",
    "Principal_branch_1": "Principal branch of $arg(z)$",
    "Prism_(geometry)_2": "Example pentagonal frustum",
    "Prism_(geometry)_3": "A $23\\times 29$ duoprism, showing edges in stereographic projection. The squares make a $23\\times 29$ grid flat torus.",
    "Prism_(geometry)_4": "uniform hexagonal prism ($n = 6$).",
    "Prism_(geometry)_5": "net of uniform enneagonal prism ($n = 9$).",
    "Probability_box_2": "Continuous p-box showing interval probability $[0.4, 0.36]$ that $x$ is 2.5 or less.",
    "Probability_box_3": "Continuous p-box showning interval 95th percentile $[9,16]$",
    "Probability_distribution_1": "The left graph shows a probability density function. The right graph shows the cumulative distribution function. The value at $a$ in the cumulative distribution equals the area under the probability density curve up to the point $a$.",
    "Probability_distribution_2": "The probability density function (pdf) of the normal distribution, also called Gaussian or bell curve, the most important absolutely continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.",
    "Probability_distribution_3": "The probability mass function (pmf) $p(S)$ specifies the probability distribution for the sum $S$ of counts from two dice. For example, the figure shows that $p(11) = \\frac{2}{36} = \\frac{1}{18}$. The pmf allows the computation of probabilities of events such as $P(X > 9) = \\frac{1}{12} + \\frac{1}{18} + \\frac{1}{36} = \\frac{1}{6}$, and all other probabilities in the distribution.",
    "Probability_distribution_4": "The probability mass function of a discrete probability distribution. The probabilities of the singletons $\\{1\\}$, $\\{3\\}$, and $\\{7\\}$ are respectively $0.2$, $0.5$, and $0.3$. A set not containing any of these points has probability zero.",
    "Probability_mass_function_2": "The probability mass function of a fair die. All the numbers on the die have an equal chance of appearing on top when the die stops rolling.",
    "Probability_theory_1": "The Poisson distribution, a discrete probability distribution",
    "Probability_theory_2": "The normal distribution, a continuous probability distribution",
    "Probit_2": "Plot of probit function",
    "Product_rule_1": "Geometric illustration of a proof of the product rule",
    "Projection_(linear_algebra)_1": "The transformation $P$ is the orthogonal projection onto the line $m$.",
    "Projective_harmonic_conjugate_2": "Midpoint and infinity are harmonic conjugates.",
    "Projective_line_over_a_ring_1": "Six lines through the origin in $\\mathbb{F}_{25}$, each corresponding to a point in the projective line $\\mathbb{P}(\\mathbb{F}_5)$.",
    "Projective_line_over_a_ring_2": "Six colors illustrate the projective line over Galois field $GF(5)$",
    "Projective_line_over_a_ring_3": "Eight colors illustrate the projective line over Galois field $GF(7)$",
    "Projective_line_over_a_ring_4": "Eight colors illustrate the projective line over Galois field $GF(7)$",
    "Projective_line_over_a_ring_5": "Six colors illustrate the projective line over Galois field $GF(5)$",
    "Projective_line_over_a_ring_6": "Six lines through the origin in $F\\_25$, each corresponding to a point in the projective line $P(F\\_5)$.",
    "Projective_polyhedron_1": "The hemi-cube is a regular projective polyhedron with 3 square faces, 6 edges, and 4 vertices.",
    "Projective_polyhedron_2": "The tetrahemihexahedron is a projective polyhedron, and the only uniform projective polyhedron that immerses in Euclidean 3-space.",
    "Projectively_extended_real_line_1": "The projectively extended real line can be visualized as the real number line wrapped around a circle (by some form of stereographic projection) with an additional point at infinity.",
    "Proof_without_words_1": "Proof without words of the Nicomachus theorem (Gulley (2010)) that the sum of the first $n$ cubes is the square of the $n$th triangular number",
    "Proof_without_words_2": "A proof without words for the sum of odd numbers theorem",
    "Proof_without_words_3": "A graphical proof of Jensen's inequality",
    "Proof_without_words_4": "Rearrangement proof of the Pythagorean theorem. The uncovered area of gray space remains constant before and after the rearrangement of the triangles: on the left it is shown to equal $c^2$, and on the right $a^2+b^2$.",
    "Proportionality_(mathematics)_1": "The variable $y$ is directly proportional to the variable $x$ with proportionality constant $\\sim 0.6$.",
    "Pseudoconvex_function_2": "Pseudoconvex function that is not convex.",
    "Pseudoconvex_function_3": "Quasiconvex function that is not convex, nor pseudoconvex.",
    "Pseudoconvex_function_4": "Functions $x^3$ (quasiconvex but not pseudoconvex) and $x^3 + x$ (pseudoconvex and thus quasiconvex). None of them is convex.",
    "Pseudoconvex_function_5": "Functions $x^3$ (quasiconvex but not pseudoconvex) and $x^3 + x$ (pseudoconvex and thus quasiconvex). None of them is convex.",
    "Pseudoconvex_function_6": "Example of a quasiconvex function that is not pseudoconvex. The function has a critical point at $x = 0$, but this is not a minimum.",
    "Pseudoconvex_function_7": "Pseudoconvex function that is not convex.",
    "Pseudoconvex_function_8": "Quasiconvex function that is not convex, nor pseudoconvex.",
    "Pseudoforest_2": "A function from the set $\\{0,1,2,3,4,5,6,7,8\\}$ to itself, and the corresponding functional graph",
    "Pseudoforest_3": "The butterfly graph (left) and diamond graph (right), forbidden minors for pseudoforests",
    "Pseudomanifold_2": "The non pseudomanifold surface on the left can be decomposed into an orientable manifold (central) or into a non-orientable one (on the right).",
    "Pseudomanifold_3": "Two 3-pseudomanifolds with singularities (in red) that cannot be broken into manifold parts only by cutting at singularities.",
    "Pseudomanifold_4": "A pinched torus",
    "Pseudosphere_1": "The pseudosphere and its relation to three other models of hyperbolic geometry",
    "Puiseux_series_1": "Truncated Puiseux expansions for the cubic curve $y^{2}=x^{3}+x^{2}$ at the double point $x=y=0$. Darker colors indicate more terms.",
    "Pursuit_curve_1": "A simple pursuit curve in which $P$ is the pursuer and $A$ is the pursuee",
    "Pursuit_curve_2": "Curves of pursuit with different parameters",
    "Pyramid_(geometry)_1": "Parts of a pyramid",
    "Pythagorean_addition_2": "Pythagorean addition finds the length of the body diagonal of a rectangular cuboid, or equivalently the length of the vector sum of orthogonal vectors.",
    "Pythagorean_means_1": "A geometric construction of the quadratic mean and the Pythagorean means (of two numbers $a$ and $b$). Harmonic mean denoted by $H$, geometric by $G$, arithmetic by $A$ and quadratic mean (also known as root mean square) denoted by $Q$.",
    "Pythagorean_means_2": "Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are asymptotes for the harmonic means.",
    "Pythagorean_quadruple_1": "All four primitive Pythagorean quadruples with only single-digit values",
    "Pythagorean_triple_2": "The primitive Pythagorean triples. The odd leg $a$ is plotted on the horizontal axis, the even leg $b$ on the vertical. The curvilinear grid is composed of curves of constant $m - n$ and of constant $m + n$ in Euclid's formula.",
    "Pythagorean_triple_3": "A plot of triples generated by Euclid's formula maps out part of the $z^2 = x^2 + y^2$ cone. A constant $m$ or $n$ traces out part of a parabola on the cone.",
    "Pythagorean_triple_4": "Stereographic projection of the unit circle onto the $x$-axis. Given a point $P$ on the unit circle, draw a line from $P$ to the point $N = (0, 1)$ (the north pole). The point $P'$ where the line intersects the $x$-axis is the stereographic projection of $P$. Inversely, starting with a point $P'$ on the x-axis, and drawing a line from $P'$ to $N$, the inverse stereographic projection is the point $P$ where the line intersects the unit circle.",
    "Pythagorean_triple_5": "A scatter plot of the legs $(a,b)$ of the first Pythagorean triples with $a$ and $b$ less than 4500.",
    "Q-function_1": "A plot of the Q-function.",
    "Q-function_2": "the Q-function plotted in the complex plane",
    "Q-function_3": "Q-factor vs. bit error rate (BER).",
    "QR_decomposition_1": "Householder reflection for QR-decomposition: The goal is to find a linear transformation that changes the vector $\\mathbf {x}$ into a vector of the same length which is collinear to $\\mathbf {e} _{1}$. We could use an orthogonal projection (Gram-Schmidt) but this will be numerically unstable if the vectors $\\mathbf {x}$ and  $\\mathbf {e} _{1}$ are close to orthogonal. Instead, the Householder reflection reflects through the dotted line (chosen to bisect the angle between $\\mathbf {x}$ and  $\\mathbf {e} _{1}$). The maximum angle with this transform is 45 degrees.",
    "Quadrisecant_1": "Three quadrisecants of a trefoil knot",
    "Quadrisecant_2": "The Schläfli double six",
    "Quantile_function_1": "The probit is the quantile function of the normal distribution.",
    "Quantile_function_2": "The cumulative distribution function (shown as $F(x)$) gives the $p$ values as a function of the $q$ values. The quantile function does the opposite: it gives the $q$ values as a function of the $p$ values. Note that the portion of $F(x)$ in red is a horizontal line segment.",
    "Quantum_graph_1": "A metric graph embedded in the plane with three open edges. The dashed line denotes the metric distance between two points $x$ and $y$.",
    "Quasi-isometry_1": "This lattice is quasi-isometric to the plane.",
    "Quasicircle_1": "The Douady rabbit is composed of quasicircles with Hausdorff dimension approximately 1.3934",
    "Quasiconvex_function_1": "A quasiconvex function that is not convex",
    "Quasiconvex_function_2": "A function that is not quasiconvex: the set of points in the domain of the function for which the function values are below the dashed red line is the union of the two red intervals, which is not a convex set.",
    "Quasiconvex_function_3": "The probability density function of the normal distribution is quasiconcave but not concave.",
    "Quasiconvex_function_4": "The bivariate normal joint density is quasiconcave.",
    "Quasiconvex_function_5": "A quasilinear function is both quasiconvex and quasiconcave.",
    "Quasiconvex_function_6": "The graph of a function that is both concave and quasiconvex on the nonnegative real numbers.",
    "Quasidihedral_group_1": "Cayley graph of the quasidihedral group of order 16",
    "Quasidihedral_group_2": "Cayley graph of the modular maximal-cyclic group of order 16",
    "Quasidihedral_group_3": "Cayley graph of the dihedral group of order 16",
    "Quasiregular_polyhedron_2": "Regular ($p \\mid 2q$) and quasiregular polyhedra ($2 \\mid pq$) are created from a Wythoff construction with the generator point at one of 3 corners of the fundamental domain. This defines a single edge within the fundamental domain.",
    "Quaternion_1": "Cayley Q8 graph showing the six cycles of multiplication by $i$, $j$ and $k$.",
    "Quotient_group_1": "The cosets of the fourth roots of unity $N$ in the twelfth roots of unity $G$.",
    "Quotient_space_(topology)_2": "Characteristic property of the quotient topology",
    "Quotient_space_(topology)_3": "Commutative diagram illustrating the universal property of quotient spaces",
    "Racah_W-coefficient_1": "Angular momenta in the Racah W coefficients. The top is a 2d plane projection as a quadrilateral, the bottom is a 3d tetrahedral arrangement.",
    "Radian_1": "Some common angles, measured in radians. All the large polygons in this diagram are regular polygons.",
    "Rado_graph_1": "The Rado graph, as numbered by Ackermann (1937) and Rado (1964).",
    "Rado_graph_2": "The extension property of the Rado graph: for every two disjoint finite sets of vertices $U$ and $V$, there exists another vertex $x$ connected to everything in $U$, and to nothing in $V$",
    "Radon's_theorem_1": "Two sets of four points in the plane (the vertices of a square and an equilateral triangle with its centroid), the multipliers solving the system of three linear equations for these points, and the Radon partitions formed by separating the points with positive multipliers from the points with negative multipliers.",
    "Rainbow_coloring_1": "Rainbow coloring of a wheel graph, with three colors. Every two non-adjacent vertices can be connected by a rainbow path, either directly through the center vertex (bottom left) or by detouring around one triangle to avoid a repeated edge color (bottom right).",
    "Ramification_(mathematics)_2": "Using the Riemann surface of the square root",
    "Ramp_function_1": "Graph of the ramp function",
    "Randolph_diagram_2": "A Randolph diagram for five sets",
    "Randolph_diagram_3": "The Randolph diagram representation of $r$.",
    "Randolph_diagram_4": "The truth values for every position on a Randolph diagram comparing 3 truth values.",
    "Randolph_diagram_5": "A Randolph diagram with numbers that match up to a truth table.",
    "Random_walk_1": "Five eight-step random walks from a central point. Some paths appear shorter than eight steps where the route has doubled back on itself. (animated version)",
    "Random_walk_2": "All possible random walk outcomes after 5 flips of a fair coin",
    "Random_walk_3": "Random walk in two dimensions (animated version)",
    "Random_walk_4": "Random walk in two dimensions with 25 thousand steps (animated version)",
    "Random_walk_5": "Random walk in two dimensions with two million even smaller steps. This image was generated in such a way that points that are more frequently traversed are darker. In the limit, for very small steps, one obtains Brownian motion.",
    "Random_walk_6": "Three random walks in three dimensions",
    "Random_walk_7": "Simulated steps approximating a Wiener process in two dimensions",
    "Rate_of_convergence_1": "Log-linear plots of the example sequences $a_k$, $b_k$, $c_k$, and $d_k$ that exemplify linear, linear, superlinear (quadratic), and sublinear rates of convergence, respectively.",
    "Ratio_distribution_1": "The ratio distribution of correlated complex variables, $\\rho = 0.7 e^{i \\pi/4}$.",
    "Ratio_distribution_2": "Evaluating the cumulative distribution of a ratio",
    "Ratio_distribution_3": "Contours of the correlated bivariate Gaussian distribution (not to scale) giving ratio $\\frac{x}{y}$",
    "Ratio_distribution_4": "PDF of the Gaussian ratio $z$ and a simulation (points) for $\\sigma_x = \\sigma_y = 1$, $\\mu_x = 0$, $\\mu_y = 0.5$, $\\rho = 0.975$.",
    "Reachability_1": "A suitable digraph for Kameda's method with $s$ and $t$ added.",
    "Reachability_2": "The same graph as above after Kameda's algorithm has run, showing the DFS labels for each vertex",
    "Real_coordinate_space_2": "Both hypercube and cross-polytope in $R^2$ are squares, but coordinates of vertices are arranged differently",
    "Real_coordinate_space_3": "Cube (the hypercube) and octahedron (the cross-polytope) of $R^3$. Coordinates are not shown",
    "Real_coordinate_space_4": "The $n$-simplex (see below) is the standard convex set, that maps to every polytope, and is the intersection of the standard $(n + 1)$ affine hyperplane (standard affine space) and the standard $(n + 1)$ orthant (standard cone).",
    "Rectangle_1": "A perfect rectangle of order 9",
    "Rectangle_2": "A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a rectangular cuboid, with a unique minimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces.",
    "Rectangular_function_1": "Rectangular function with $a = 1$",
    "Rectangular_function_2": "Plot of normalized $\\operatorname {sinc} (x)$ function (i.e. $\\operatorname {sinc} (\\pi x)$) with its spectral frequency components.",
    "Rectangular_function_3": "Rectangular function with $a = 1$",
    "Rectified_Gaussian_distribution_1": "A comparison of Gaussian distribution, rectified Gaussian distribution, and truncated Gaussian distribution.",
    "Reflection_(mathematics)_1": "A reflection through an axis.",
    "Reflection_(mathematics)_2": "Point $Q$ is the reflection of point $P$ through the line $AB$.",
    "Regular_dodecahedron_2": "The regular icosahedron inside the regular dodecahedron",
    "Regular_dodecahedron_3": "Five tetrahedra inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.",
    "Regular_grid_1": "Example of a regular grid",
    "Regular_polyhedron_1": "The hexagonal tiling honeycomb, $\\{6,3,3\\}$, has hexagonal tiling, $\\{6,3\\}$, facets with vertices on a horosphere. One such facet is shown in as seen in this Poincaré disk model.",
    "Regularization_(mathematics)_1": "The green and blue functions both incur zero loss on the given data points. A learned model can be induced to prefer the green function, which may generalize better to more points drawn from the underlying unknown distribution, by adjusting $\\lambda$, the weight of the regularization term.",
    "Regularization_(mathematics)_2": "A comparison between the $L_1$ ball and the $L_2$ ball in two dimensions gives an intuition on how $L_1$ regularization achieves sparsity.",
    "Regularization_(mathematics)_3": "Elastic net regularization",
    "Relaxed_intersection_1": "$q$-intersection of 6 sets for $q=2$ (red), $q=3$ (green), $q=4$ (blue), $q=5$ (yellow).",
    "Relaxed_intersection_2": "Set of all parameter vectors consistent with exactly 6-q data bars (painted red), for $q=1,2,3,4,5$.",
    "Relaxed_intersection_3": "$q$-intersection of 6 sets for $q=2$ (red), $q=3$ (green), $q= 4$ (blue), $q= 5$ (yellow).",
    "Relaxed_intersection_4": "Set-membership function associated to the 6 intervals.",
    "Relaxed_intersection_5": "The red box corresponds to the 4-relaxed intersection of the 6 boxes",
    "Resolution_of_singularities_1": "Strong desingularization of $X:=(x^{2}-y^{3}=0)\\subset W:=\\mathbf {R} ^{2}.$ Observe that the resolution does not stop after the first blowing-up, when the strict transform is smooth, but when it is simple normal crossings with the exceptional divisors.",
    "Resolution_of_singularities_2": "Conical singularity $x^2 + y^2 = z^2$",
    "Restriction_(mathematics)_1": "The function $x^{2}$ with domain $\\mathbb {R}$ does not have an inverse function. If we restrict $x^{2}$ to the non-negative real numbers, then it does have an inverse function, known as the square root of $x.$",
    "Rhombic_dodecahedron_1": "3D model of a rhombic dodecahedron",
    "Rhombic_dodecahedron_2": "In a perfect vertex-first projection two of the tesseract's vertices (marked in pale green) are projected exactly in the center of the rhombic dodecahedron",
    "Rhombic_dodecahedron_3": "The rhombic dodecahedron as a parallelohedron",
    "Rhombic_dodecahedron_4": "The rhombic dodecahedron can tessellate space by translational copies of itself",
    "Rhombic_dodecahedron_5": "A rhombic dodecahedron but with square and rhombic faces",
    "Rhombic_triacontahedron_1": "A topological rhombic triacontahedron in truncated octahedron",
    "Rhombus_1": "A rhombus. Each angle marked with a black dot is a right angle. The height h is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengths $p$ and $q$ are the red dotted line segments.",
    "Ribbon_knot_1": "A 3-dimensional rendering of the ribbon knot $8_{20}$, showing the ribbon property",
    "Ricci_flow_1": "Several stages of Ricci flow on a 2D manifold.",
    "Richmond_surface_1": "Richmond surface for $m=2$.",
    "Riemann_sum_2": "Left Riemann sum of $x \\mapsto x^3$ over $[0, 2]$ using 4 subintervals.",
    "Riemann_sum_3": "Right Riemann sum of $x \\mapsto x^3$ over $[0, 2]$ using 4 subintervals.",
    "Riemann_sum_4": "Middle Riemann sum of $x \\mapsto x^3$ over $[0, 2]$ using 4 subintervals.",
    "Riemann_sum_5": "Trapezoidal sum of $x \\mapsto x^3$ over $[0, 2]$ using 4 subintervals.",
    "Riemann_sum_6": "The value of the right Riemann sum of $x \\mapsto x^2$ over $[0, 2]$. As the number of rectangles increases, it approaches the exact area of $\\dfrac{8}{3}$.",
    "Right_conoid_1": "A right conoid as a ruled surface.",
    "Right_triangle_1": "A right triangle $\\triangle ABC$ with its right angle at $C$, hypotenuse $c$, and legs $a$ and $b$",
    "Right_triangle_2": "The diagram for Euclid's proof of the Pythagorean theorem: each smaller square has area equal to the rectangle of corresponding color.",
    "Right_triangle_3": "Median of a right angle of a triangle",
    "Rook's_graph_2": "8-coloring of a chessboard graph obtained from a Cayley table of a finite group",
    "Rook's_graph_3": "$8\\times 8$ rook's graph: graph of possible moves for a standard chess rook",
    "Rook's_graph_4": "The $3\\times 3$ rook's graph (the graph of the 3-3 duoprism), colored with three colors and showing a clique of three vertices. In this graph and each of its induced subgraphs the chromatic number equals the clique number, so it is a perfect graph.",
    "Root_of_unity_1": "The 5th roots of unity (blue points) in the complex plane",
    "Rooted_product_of_graphs_1": "The rooted product of graphs.",
    "Ropelength_1": "A numeric approximation of an ideal trefoil.",
    "Rosenbrock_function_1": "Plot of the Rosenbrock function of two variables. Here $a=1,b=100$, and the minimum value of zero is at $(1,1)$.",
    "Rosenbrock_function_2": "Rosenbrock roots exhibiting hump structures",
    "Rotating_calipers_1": "Sequence of probes around the convex hull of a polygon to determine its diameter using Rotating Caliper method.",
    "Rotating_calipers_2": "An antipodal pair of vertex and their supporting parallel lines.",
    "Rotating_calipers_3": "Rotating calipers, finding a bridge between two convex polygons",
    "Rotation_(mathematics)_2": "A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation.",
    "Rule_of_three_(statistics)_1": "Comparison of the rule of three to the exact binomial one-sided confidence interval with no positive samples",
    "Rule_of_twelfths_1": "A regular dodecahedron (grey) approximated with a lattice polygon using the rule of twelfths (red)",
    "Ruled_surface_1": "Definition of a ruled surface: every point lies on a line",
    "Ruled_surface_2": "Ruled surface generated by two Bézier curves as directrices (red, green)",
    "Ruled_surface_3": "hyperboloid of one sheet for $\\varphi =63^{\\circ }$",
    "Ruled_surface_4": "Developable connection of two ellipses and its development",
    "Runcinated_tesseracts_1": "Vertex figure for the omnisnub tesseract",
    "Runcinated_tesseracts_2": "Runcinated tesseract Schlegel diagram, solid tetrahedron cells",
    "Runcinated_tesseracts_3": "Runcinated tesseract stereo projection wireframe",
    "Runcinated_tesseracts_4": "Runcinated tesseract stereo projection wireframe + tetrehedra",
    "Runcinated_tesseracts_5": "Runcinated tesseract stereo projection wireframe + triangular prisms",
    "Runcinated_tesseracts_6": "Uniform polychoron image created from beta release of Stella4D software by Rob Webb",
    "Runcinated_tesseracts_7": "Vertex figure for the bialternatosnub 16-cell",
    "Runcinated_tesseracts_8": "Schlegel diagram of Runcitruncated tesseract and some faces",
    "Runge's_phenomenon_1": "Demonstration of the Runge phenomenon in which the oscillations near the interval's boundaries increase with higher order polynomial interpolations The function ${\frac {1}{1+25x^{2}}}$ A fifth order polynomial interpolation (exact replication of the red curve at 6 points) A ninth order polynomial interpolation (exact replication of the red curve at 10 points)",
    "Rytz's_construction_1": "Rytz construction: start - end",
    "Rytz's_construction_2": "Cube with circles: military projection",
    "Rytz's_construction_3": "Rytz's construction in 6 steps. Given: center $C$ and two conjugate half diameters $CP$, $CQ$ of an ellipse. sought: the semi-axes and the vertices of the ellipse.",
    "Rytz's_construction_4": "Rytz: left turn of point $P$",
    "STAR_model_1": "Exponential transition function for the ESTAR model with $z_{t}$ varying from -10 to +10 and $\\zeta$ - from 0 to 1.",
    "STAR_model_2": "Exponential transition function for the ESTAR model with $z_{t}$ varying from -10 to +10, $\\zeta$ from 0 to 1 and two exponential roots ($c_{1}$ and $c_{2}$) equal to -7 and +3.",
    "STAR_model_3": "Logistic transition function for the ESTAR model with $z_{t}$ varying from -10 to +10 and $\\zeta$ - from 0 to 1. Calculated using GNU R package.",
    "Sagitta_(geometry)_1": "Visualization of the sagitta, denoted as $s$",
    "Salem_number_1": "Plot of the roots of Lehmer's polynomial, with the corresponding Salem number near $x=1.17628$ in gold.",
    "Salinon_1": "The salinon (red) and the circle (blue) have the same area.",
    "Sample_space_1": "A visual representation of a finite sample space and events. The red oval is the event that a number is odd, and the blue oval is the event that a number is prime.",
    "Sawtooth_wave_1": "Sine, square, triangle, and sawtooth waveforms",
    "Scalar_(mathematics)_1": "Scalars are real numbers used in linear algebra, as opposed to vectors. This image shows a Euclidean vector. Its coordinates $x$ and $y$ are scalars, as is its length, but $v$ is not a scalar.",
    "Scaling_(geometry)_1": "Each iteration of the Sierpinski triangle contains triangles related to the next iteration by a scale factor of $\\frac{1}{2}$",
    "Scherk_surface_1": "STL unit cell of the first Scherk surface",
    "Scherk_surface_2": "Five unit cells placed together",
    "Scherk_surface_3": "Scherk's second surface",
    "Scherk_surface_4": "STL unit cell of the second Scherk surface",
    "Schild's_ladder_1": "A curve in $M$ with a \"vector\" $X_0$ at $A_0$, identified here as a geodesic segment.",
    "Schild's_ladder_2": "Select $A_1$ on the original curve. The point $P_1$ is the midpoint of the geodesic segment $X_0A_1$.",
    "Schild's_ladder_3": "The point $X_1$ is obtained by following the geodesic $A_0P_1$ for twice its parameter length.",
    "Schild's_ladder_4": "Two rungs of Schild's ladder. The segments $A_1X_1$ and $A_2X_2$ are an approximation to first order of the parallel transport of $A_0X_0$ along the curve.",
    "Schlegel_diagram_1": "A tesseract projected into 3-space as a Schlegel diagram. There are 8 cubic cells visible: the outer cell into which the others are projected, one below each of the six exterior faces, and one in the center.",
    "Schoch_line_1": "The Schoch line (cyan) passes through the point A1.",
    "Schottky_anomaly_1": "Schottky anomalous heat capacity $C$ for a two-level system. $T$ is the absolute temperature, $k_{\\mathrm {B}}$ is the Boltzmann constant and $\\varepsilon$ is the energy difference between the levels.",
    "Schottky_group_2": "Schottky (Kleinian) group limit set in plane",
    "Scorer's_function_1": "Graph of $\\mathrm {Gi} (x)$ and $\\mathrm {Hi} (x)$",
    "Scorer's_function_2": "Plot of the Scorer function $\\mathrm {Gi} (z)$ in the complex plane from $-2 - 2i$ to $2 + 2i$ with colors.",
    "Scorer's_function_3": "Plot of the derivative of the Scorer function $\\mathrm {Hi}' (z)$ in the complex plane from $-2 - 2i$ to $2 + 2i$ with colors.",
    "Scorer's_function_4": "Plot of the derivative of the Scorer function $\\mathrm {Gi}' (z)$ in the complex plane from $-2 - 2i$ to $2 + 2i$ with colors.",
    "Scorer's_function_5": "Plot of the Scorer function $\\mathrm {Hi} (z)$ in the complex plane from $-2 - 2i$ to $2 + 2i$ with colors.",
    "Second_derivative_1": "The second derivative of a quadratic function is constant.",
    "Second_fundamental_form_1": "Definition of second fundamental form",
    "Section_(fiber_bundle)_2": "A vector field on $\\mathbb {R} ^{2}$. A section of a tangent vector bundle is a vector field.",
    "Seifert_fiber_space_1": "A standard fibered solid torus corresponding to $(5,2)$ is obtained by gluing the bottom of the cylinder to the top by a $\\frac{2}{5}$ rotation clockwise.",
    "Seifert_surface_1": "A Seifert surface bounded by a set of Borromean rings.",
    "Seifert_surface_2": "An illustration of (curves isotopic to) the pushoffs of a homology generator a in the positive and negative directions for a Seifert surface of the figure eight knot.",
    "Semi-differentiability_1": "This function does not have a derivative at the marked point, as the function is not continuous there. However, it has a right derivative at all points, with $\\partial _{+}f(a)$ constantly equal to 0.",
    "Semicircle_1": "Construction of a square with the same area as a given oblong",
    "Semicircle_2": "Proof without words of the AM--GM inequality: $PR$ is the diameter of a circle centered on $O$; its radius $AO$ is the arithmetic mean of $a$ and $b$. Using the geometric mean theorem, triangle $PGR$'s altitude $GQ$ is the geometric mean. For any ratio $a:b$, $AO \\geq GQ$.",
    "Semicircle_3": "Comparison of Ford circles and a Farey diagram with semicircles for $n$ from 1 to 9. Each semicircle intersects its corresponding circles at right angles.",
    "Semicircle_4": "Semicircle - contains the curve only. Not to be confused with semicircular region, where diameter is also included.",
    "Semicubical_parabola_1": "Semicubical parabola for various $a$.",
    "Semicubical_parabola_2": "Tangent at a semicubical parabola",
    "Semicubical_parabola_3": "Relation between a semicubical parabola and a cubic function (green)",
    "Semiorder_2": "Forbidden: two mutually incomparable two-point linear orders",
    "Semiorder_3": "Forbidden: three linearly ordered points and a fourth incomparable point",
    "Semiperimeter_1": "In any triangle, the distance along the boundary of the triangle from a vertex to the point on the opposite edge touched by an excircle equals the semiperimeter.",
    "Serpentine_curve_1": "The serpentine curve for $a = b = 1$.",
    "Set_(mathematics)_2": "This set equals the one depicted above since both have the very same elements.",
    "Set_(mathematics)_3": "All standard number systems are infinite sets",
    "Set_(mathematics)_4": "The intersection of $A$ and $B$, denoted $A \\cap B$",
    "Set_(mathematics)_5": "The union of $A$ and $B$, denoted $A \\cup B$",
    "Set_(mathematics)_6": "The set difference $A \\setminus B$",
    "Set_(mathematics)_7": "The complement of $A$ in $U$",
    "Set_(mathematics)_8": "The symmetric difference of $A$ and $B$",
    "Shannon_multigraph_1": "This nine-edge Shannon multigraph requires nine colors in any edge coloring; its vertex degree is six and its multiplicity is three.",
    "Shape_1": "A variety of polygonal shapes.",
    "Shear_mapping_1": "Horizontal shearing of the plane, transforming the blue into the red shape. The black dot is the origin.",
    "Shear_mapping_2": "Horizontal shear of a square into parallelograms with factors $\\cot(60^{\\circ })=\\tan(30^{\\circ })\\approx 0.58$ and $\\cot(45^{\\circ })=\\tan(45^{\\circ })=1$",
    "Shekel_function_1": "A Shekel function in 2 dimensions and with 10 maxima",
    "Shoelace_formula_1": "Shoelace scheme for determining the area of a polygon with point coordinates $(x_{1},y_{1}),...,(x_{n},y_{n})$",
    "Shoelace_formula_2": "Basic idea: Any polygon edge determines the signed area of a trapezoid. All these areas sum up to the polygon area.",
    "Shoelace_formula_3": "Horizontal shoelace form for the example.",
    "Shoelace_formula_4": "Deriving the trapezoid formula",
    "Shoelace_formula_5": "Triangle form: The color of the edges indicate, which triangle area is positive (green) and negative (red) respectively.",
    "Shoelace_formula_6": "Manipulations of a polygon",
    "Sierpinski_triangle_2": "The evolution of the Sierpinski triangle",
    "Sierpinski_triangle_3": "Arrowhead construction of the Sierpinski gasket",
    "Simplicial_complex_1": "A simplicial 3-complex.",
    "Simpson's_rule_1": "Simpson's rule can be derived by approximating the integrand $f(x)$ (in blue) by the quadratic interpolant $P(x)$ (in red).",
    "Sinc_function_2": "The local maxima and minima (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue cosine function.",
    "Sinc_function_3": "Plot of the graphs of (red) and (blue) using cubic Bézier curves.",
    "Sine-Gordon_equation_1": "Static 1-soliton solution $4\\arctan e^{x}$",
    "Singular_point_of_a_curve_1": "A curve with a triple point at the origin: $x(t) = \\sin(2t) + \\cos(t)$, $y(t) = \\sin(t) + \\cos(2t)$",
    "Singular_point_of_a_curve_2": "Three limaçons illustrating the types of double point. When converted to Cartesian coordinates as $(x^{2}+y^{2}-x)^{2}=(1.5)^{2}(x^{2}+y^{2}),$ the left curve acquires an acnode at the origin, which is an isolated point in the plane. The central curve, the cardioid, has a cusp at the origin. The right curve has a crunode at the origin and the curve crosses itself to form a loop.",
    "Singular_point_of_a_curve_3": "A cusp in the semicubical parabola $y^{2}=x^{3}$",
    "Singular_point_of_a_curve_4": "A curve with a triple point at the origin: $x(t) = \\sin(2t) + \\cos(t)$, $y(t) = \\sin(t) + \\cos(2t)$.",
    "Sitnikov_problem_1": "Configuration of the Sitnikov problem",
    "Skew-symmetric_graph_1": "An implication graph. Its skew symmetry can be realized by rotating the graph through a 180 degree angle and reversing all edges.",
    "Skew_lines_2": "$PQ$, the shortest distance between two skew lines $AB$ and $CD$ is perpendicular to both $AB$ and $CD$",
    "Skew_lines_3": "A fibration of projective space by skew lines on nested hyperboloids.",
    "Skew_partition_1": "A skew partition of a chordal graph. On the left side of the partition, the top and bottom parts are disconnected from each other. On the right side of the partition, all possible edges from top to bottom exist, forming a graph whose complement is disconnected.",
    "Skinny_triangle_1": "Isosceles skinny triangle",
    "Slerp_1": "Oblique vector rectifies to slerp factor.",
    "Slope_2": "Slope illustrated for $y = (3/2)x - 1.$",
    "Slope_3": "Slopes of parallel and perpendicular lines",
    "Slope_4": "The illusion of a paradox of area is dispelled by comparing slopes where blue and red triangles meet.",
    "Slope_number_2": "The method of Keszegh, Pach & Pálvölgyi (2011) for combining circle packings and quadtrees to achieve bounded slope number for planar graphs with bounded degree",
    "Smallest-circle_problem_1": "Some instances of the smallest bounding circle.",
    "Smallest-circle_problem_2": "Run of Megiddo's algorithm phase, discarding from point set $A, B, \\ldots, U$ needless points $E$, $T$.",
    "Smoothed_octagon_1": "A smoothed octagon.",
    "Smoothed_octagon_2": "Construction of the smoothed octagon (black), the tangent hyperbola (red), the asymptotes of this hyperbola (green), and the tangent sides to the hyperbola (blue)",
    "Smoothness_1": "A bump function is a smooth function with compact support.",
    "Snub_cube_1": "3D model of a snub cube",
    "Snub_cube_2": "The graph of a snub cube",
    "Snub_dodecahedron_1": "3D model of a snub dodecahedron",
    "Snub_dodecahedron_2": "The snub dodecahedron has no point symmetry, so the vertex in the front does not correspond to an opposite vertex in the back.",
    "Soddy's_hexlet_1": "An annular hexlet.",
    "Soddy's_hexlet_2": "A Dupin cyclide, through which the hexlet spheres rotate, always touching. The cyclide is tangent to an inner sphere, an outer sphere and two spheres above and below the \"hole\" in the \"doughnut\".",
    "Solid_angle_1": "Visual representation of a solid angle",
    "Solid_angle_2": "Any area on a sphere which is equal in area to the square of its radius, when observed from its center, subtends precisely one steradian.",
    "Solid_angle_3": "Diagram showing a section through the centre of a cone (1) subtending a solid angle of 1 steradian in a sphere of radius $r$, along with the spherical cap (2). The external surface area $A$ of the cap equals $r^2$ only if the solid angle of the cone is exactly 1 steradian. Hence, in this figure $\\theta = A/2$ and $r = 1$.",
    "Solid_angle_4": "Archimedes' theorem that surface area of the region of sphere below horizontal plane $H$ in given diagram is equal to area of a circle of radius $t$.",
    "Solvency_cone_1": "Sample solvency cone with no transaction costs",
    "Solvency_cone_2": "Sample solvency cone with transaction costs",
    "Sorgenfrey_plane_1": "An illustration of the anti-diagonal and an open rectangle in the Sorgenfrey plane that meets the anti-diagonal at a single point.",
    "Space-filling_curve_1": "Six iterations of the Hilbert curve construction, whose limiting space-filling curve was devised by mathematician David Hilbert.",
    "Space_diagonal_1": "$AC'$ (shown in blue) is a space diagonal, while $AC$ (shown in red) is a face diagonal.",
    "Spanning_tree_2": "A spanning tree (blue heavy edges) of a grid graph",
    "Spectral_sequence_2": "The $E_2$ sheet of a cohomological spectral sequence",
    "Spherical_cap_1": "An example of a spherical cap in blue (and another in red)",
    "Spherical_cap_2": "Rotating the green area creates a spherical cap with height $h$ and sphere radius $r$.",
    "Spherical_mean_1": "The spherical mean of a function $u$ (shown in red) is the average of the values $u(y)$ (top, in blue) with $y$ on a \"sphere\" of given radius around a given point (bottom, in blue).",
    "Spherical_sector_1": "A spherical sector (blue)",
    "Spherical_sector_2": "A spherical sector",
    "Spherical_trigonometry_1": "The octant of a sphere is a spherical triangle with three right angles.",
    "Spherical_trigonometry_2": "Eight spherical triangles defined by the intersection of three great circles.",
    "Spherical_trigonometry_3": "The basic triangle on a unit sphere.",
    "Spherical_trigonometry_4": "The polar triangle $\\triangle A'B'C'$",
    "Spheroid_1": "The assignment of semi-axes on a spheroid. It is oblate if $c < a$ (left) and prolate if $c > a$ (right).",
    "Spider_diagram_1": "Logical disjunction superimposed on Euler diagram",
    "Spiral_1": "Spirals generated by 6 mathematical relationships between radius and angle.",
    "Spiral_2": "Hyperbolic spiral as central projection of a helix",
    "Spiral_3": "Definition of sector (light blue) and polar slope angle $\\alpha$",
    "Spiral_4": "Bounded spirals: $r=a\\arctan(k\\varphi )$ (left), $r=a(\\arctan(k\\varphi )+\\pi /2)$ (right)",
    "Spiral_5": "An Archimedean spiral (black), a helix (green), and a conical spiral (red)",
    "Spiral_6": "Conical spiral with Archimedean spiral as floor plan",
    "Spiral_7": "Clelia curve with $c=8$",
    "Spirograph_2": "Geometric construction for mathematical explanation of spirograph.",
    "Spline_interpolation_1": "Interpolation with cubic splines between eight points. Hand-drawn technical drawings for shipbuilding are a historical example of spline interpolation; drawings were constructed using flexible rulers that were bent to follow pre-defined points.",
    "Spline_interpolation_2": "Interpolation with cubic \"natural\" splines between three points",
    "Split-octonion_1": "A mnemonic for the products of the split octonions.",
    "Split_graph_1": "A split graph, partitioned into a clique and an independent set.",
    "Square_knot_(mathematics)_1": "The square knot, drawn as a ribbon knot",
    "Square_knot_(mathematics)_2": "Square knot = trefoil + trefoil reflection. Sticks depicted.",
    "Square_pyramid_2": "Tetrakis hexahedra, a construction of polyhedra by augmentation involving square pyramids",
    "Square_pyramid_3": "Symmetric net for folding the square pyramid, from Kurana, Rohit (2010)",
    "Square_root_of_3_2": "The space diagonal of the unit cube is $\\sqrt{3}$.",
    "Square_root_of_3_3": "The height of an equilateral triangle with edge length $2$ is $\\sqrt{3}$. Also, the long leg of a $30$-$60$-$90$ triangle with hypotenuse $2$.",
    "Square_root_of_3_4": "And, the height of a regular hexagon with sides of length 1.",
    "Square_root_of_5_1": "The ${\\frac {\\sqrt {5}}{2}}$ diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.",
    "Square_root_of_5_2": "Decomposition of a $1\\times 2$ right triangle into five similar triangles, the basis for the aperiodic pinwheel tiling.",
    "Square_root_of_5_3": "Jay Hambidge's construction of \"root rectangles\"",
    "Square_root_of_5_4": "Distances between vertices of a double unit cube are square roots of the first six natural numbers. ($\\sqrt {7}$ is not possible due to Legendre's three-square theorem.)",
    "Squared_triangular_number_1": "A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010). The $n$th coloured region shows $n$ squares of dimension $n$ by $n$ (the rectangle is 1 evenly divided square), hence the area of the $n$th region is $n$ times $n \\times n$.",
    "Squared_triangular_number_2": "All $36 (= (1+2+3)^2 = 1^3 + 2^3 + 3^3)$ rectangles, including $14 (= 1^2 + 2^2 + 3^2)$ squares (red), in a $3 \\times 3$ square ($4 \\times 4$ vertex) grid",
    "Squared_triangular_number_3": "Visual demonstration that the square of a triangular number equals a sum of cubes.",
    "Squeeze_mapping_1": "A squeeze mapping moves one purple hyperbolic sector to another with the same area. It also squeezes blue and green rectangles.",
    "Squeeze_mapping_2": "Euclidean orthogonality is preserved by rotation in the left diagram; hyperbolic orthogonality with respect to hyperbola (B) is preserved by squeeze mapping in the right diagram",
    "Squeeze_theorem_2": "When a sequence lies between two other converging sequences with the same limit, it also converges to this limit.",
    "Squeeze_theorem_3": "$x^{2}\\sin \\left({\\tfrac {1}{x}}\\right)$ being squeezed in the limit as $x$ goes to 0",
    "Squeeze_theorem_4": "Comparing areas: ${\\begin{array}{cccccc}&A(\\triangle ADB)&\\leq &A({\\text{sector }}ADB)&\\leq &A(\\triangle ADF)\\\\[4pt]\\Rightarrow &{\\frac {1}{2}}\\cdot \\sin x\\cdot 1&\\leq &{\\frac {x}{2\\pi }}\\cdot \\pi &\\leq &{\\frac {1}{2}}\\cdot \\tan x\\cdot 1\\\\[4pt]\\Rightarrow &\\sin x&\\leq &x&\\leq &{\\frac {\\sin x}{\\cos x}}\\\\[4pt]\\Rightarrow &{\\frac {\\cos x}{\\sin x}}&\\leq &{\\frac {1}{x}}&\\leq &{\\frac {1}{\\sin x}}\\\\[4pt]\\Rightarrow &\\cos x&\\leq &{\\frac {\\sin x}{x}}&\\leq &1\\end{array}}$",
    "Squircle_1": "Squircle centred on the origin $(a = b = 0)$ with minor radius $r = 1: x^4 + y^4 = 1$",
    "Squircle_2": "A squircle (blue) compared with a rounded square (red).",
    "Squircle_3": "Various forms of a truncated circle",
    "Stable_distribution_1": "Log-log plot of symmetric centered stable distribution PDFs showing the power law behavior for large $x$. The power law behavior is evidenced by the straight-line appearance of the PDF for large $x$, with the slope equal to $-(\\alpha +1)$. (The only exception is for $\\alpha =2$, in black, which is a normal distribution.)",
    "Stable_distribution_2": "Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large $x$. Again the slope of the linear portions is equal to $-(\\alpha +1)$",
    "Stable_distribution_3": "Skewed centered stable distributions",
    "Stable_distribution_4": "Symmetric stable distributions",
    "Stadium_(geometry)_1": "Parameters of a stadium",
    "Stadium_(geometry)_2": "The Bunimovich stadium, a chaotic dynamical system based on the stadium shape",
    "Standard_score_2": "The z-score for Student A was 1, meaning Student A was 1 standard deviation above the mean. Thus, Student A performed in the 84.13 percentile on the SAT.",
    "Standard_score_3": "The z-score for Student B was 0.6, meaning Student B was 0.6 standard deviation above the mean. Thus, Student B performed in the 72.57 percentile on the SAT.",
    "Star_coloring_1": "The star chromatic number of Dyck graph is 4, while the chromatic number is 2.",
    "Star_domain_1": "An annulus is not a star domain.",
    "Star_domain_2": "A star domain (equivalently, a star-convex or star-shaped set) is not necessarily convex in the ordinary sense.",
    "Star_of_David_theorem_1": "The Star of David theorem (the rows of the Pascal triangle are shown as columns here).",
    "Stationary_point_2": "A graph in which local extrema and global extrema have been labeled.",
    "Stationary_point_3": "The roots, stationary points, inflection point and concavity of a cubic polynomial $x^3 - 6x^2 + 9x - 4$ (solid black curve) and its first (dashed red) and second (dotted orange) derivatives.",
    "Steiner_ellipse_1": "The Steiner ellipse of an isosceles triangle. The three line segments inside the triangle are the triangle's medians, each bisecting a side. The medians coincide at the triangle's centroid, which is also the center of the Steiner ellipse.",
    "Steiner_ellipse_2": "Steiner ellipse of an equilateral (left) and isosceles triangle",
    "Steiner_ellipse_3": "Steiner ellipse of a triangle including the axes and verices (purple)",
    "Steiner_ellipse_4": "Steiner ellipse as example for equation",
    "Steiner_tree_problem_1": "Steiner tree for three points $A$, $B$, and $C$ (note there are no direct connections between $A$, $B$, $C$). The Steiner point $S$ is located at the Fermat point of the triangle $ABC$.",
    "Steiner_tree_problem_2": "Solution for four points-there are two Steiner points, $S_1$ and $S_2$",
    "Steinmetz_solid_1": "The generation of a bicylinder",
    "Steinmetz_solid_2": "Calculating the volume of a bicylinder",
    "Steinmetz_solid_3": "Generating the surface of a tricylinder: At first two cylinders (red, blue) are cut. The so generated bicylinder is cut by the third (green) cylinder.",
    "Stewart's_theorem_1": "Diagram of Stewart's theorem",
    "Stirling's_approximation_1": "Comparison of Stirling's approximation with the factorial",
    "Stirling's_approximation_2": "The relative error in a truncated Stirling series vs. the number of terms used",
    "Stirling's_approximation_3": "The relative error in a truncated Stirling series vs. $n$, for 0 to 5 terms. The kinks in the curves represent points where the truncated series coincides with $\\Gamma(n + 1)$.",
    "Stirling_numbers_of_the_second_kind_1": "Parity of Stirling numbers of the second kind.",
    "Stirling_numbers_of_the_second_kind_2": "The 15 partitions of a 4-element set ordered in a Hasse diagram There are $S(4,1), ..., S(4, 4) = 1, 7, 6, 1$ partitions containing 1, 2, 3, 4 sets.",
    "Stokes_phenomenon_1": "Stokes lines and anti-Stokes lines for the Airy function",
    "Strong_product_of_graphs_1": "The king's graph, a strong product of two path graphs",
    "Strophoid_1": "Construction of a strophoid. Given curve $C$ Variable line $L$ which rotates about the pole $O$; intersects $C$ at point $K$ Circle centered at $K$ whose size is constrained by the fixed point $A$; intersects $L$ at $P_1$ and $P_2$ Inner portion of the strophoid curve, traced by $P_1$ as $L$ rotates Outer portion of the strophoid curve, traced by $P_2$ as $L$ rotates",
    "Strophoid_2": "A right strophoid",
    "Struve_function_1": "Plot of the modified Struve function $L_n(z)$ with $n=2$ in the complex plane from $-2-2i$ to $2+2i$ with colors created with Mathematica 13.1 function ComplexPlot3D",
    "Sturm_separation_theorem_1": "The zeros of two linearly independent solutions of the Airy equation $y''-xy=0$ alternate, as predicted by the Sturm separation theorem.",
    "Sturmian_word_1": "The Fibonacci word is an example of a Sturmian word. The start of the cutting sequence shown here illustrates the start of the word 0100101001.",
    "Subgroup_1": "Symmetric group $S_4$",
    "Subgroup_2": "Alternating group $A_4S$",
    "Submanifold_1": "Immersed manifold straight line with self-intersections",
    "Submanifold_2": "This image of the open interval (with boundary points identified with the arrow marked ends) is an immersed submanifold.",
    "Subset_1": "$A\\subseteq B$ and $B\\subseteq C$ implies $A\\subseteq C.$",
    "Successive_over-relaxation_1": "Spectral radius $\\rho (C_{\\omega })$ of the iteration matrix for the SOR method $C_{\\omega }$. The plot shows the dependence on the spectral radius of the Jacobi iteration matrix $\\mu :=\\rho (C_{\\text{Jac}})$.",
    "Superellipse_1": "Examples of superellipses for $a=1,\\ b=0.75$",
    "Supergolden_ratio_2": "A triangle with side lengths $\\psi$, $1$, and $\\frac{1}{\\psi}$ has an angle of exactly $120^\\circ$.",
    "Supergolden_ratio_3": "Rectangles in aspect ratios $\\psi$, $\\psi^2$, and $\\psi^3$ (from left to right) tile the square.",
    "Supergolden_ratio_4": "A Rauzy fractal associated with the supergolden ratio-cubed. The central tile and its three subtiles have areas in the ratios $\\psi^4 : \\psi^2 : \\psi : 1$.",
    "Supergolden_ratio_5": "A supergolden Rauzy fractal of type $a \\mapsto ab$, with areas as above. The fractal boundary has box-counting dimension $1.50$.",
    "Supergolden_ratio_6": "Nested supergolden rectangles with perpendicular diagonals and side lengths in powers of $\\psi$.",
    "Supergolden_ratio_7": "Supergolden spirals with different initial radii on a $\\psi$- rectangle.",
    "Surface_1": "A sphere is the surface of a solid ball, here having radius $r$",
    "Surface_of_general_type_1": "Chern numbers of minimal complex surfaces",
    "Surface_of_revolution_2": "A toroid generated from a square",
    "Surface_of_revolution_3": "A portion of the curve $x = 2 + \\cos(z)$ rotated around the $z$-axis.",
    "Surface_triangulation_2": "Triangulation of a parametric surface (Monkey Saddle)",
    "Surface_triangulation_3": "Triangulation: cylinder, surface $x^4 + y^4 + z^4 = 1$",
    "Surface_triangulation_4": "Torus: triangulated by the marching method",
    "Surface_triangulation_5": "Torus: polygonized by the cutting cube method",
    "Suspension_(topology)_1": "Suspension of a circle. The original space is in blue, and the collapsed end points are in green.",
    "Sylvester's_sequence_1": "Graphical demonstration of the convergence of the sum $\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{43}+\\dots$ to 1. Each row of k squares of side length $\\frac{1}{k}$ has total area $\\frac{1}{k}$, and all the squares together exactly cover a larger square with area 1. Squares with side lengths $\\frac{1}{1807}$ or smaller are too small to see in the figure and are not shown.",
    "Symmetric_decreasing_rearrangement_1": "Function and its symmetric decreasing rearrangement preserve the measure of level sets.",
    "Symmetric_derivative_1": "Graph of the absolute value function. Note the sharp turn at $x = 0$, leading to non-differentiability of the curve at $x = 0$. The function hence possesses no ordinary derivative at $x = 0$. The symmetric derivative, however, exists for the function at $x = 0$.",
    "Symmetric_derivative_2": "Graph of $y = \\frac{1}{x^2}$. Note the discontinuity at $x = 0$. The function hence possesses no ordinary derivative at $x = 0$. The symmetric derivative, however, exists for the function at $x = 0$.",
    "Symmetric_difference_1": "Venn diagram of $~(A\\Delta B)\\Delta C$",
    "Symmetric_matrix_1": "Symmetry of a $5 \\times 5$ matrix",
    "Symmetrical_components_1": "Napoleon's theorem: If the triangles centered on L, M, and N are equilateral, then so is the green triangle.",
    "Symmetry_set_1": "An ellipse (red), its evolute (blue), and its symmetry set (green and yellow). the medial axis is just the green portion of the symmetry set. One bi-tangent circle is shown.",
    "Syntractrix_1": "The syntractrix for $a=0.5$ and $b=1.$",
    "Syntractrix_2": "The syntractrix for $a=1.5$ and $b=1.$",
    "System_of_linear_equations_1": "A linear system in three variables determines a collection of planes. The intersection point is the solution.",
    "System_of_linear_equations_2": "The solution set for the equations $x - y = -1$ and $3x + y = 9$ is the single point $(2, 3)$.",
    "System_of_linear_equations_3": "The solution set for two equations in three variables is, in general, a line.",
    "System_of_linear_equations_4": "Three lines on the $xy$ plane. Each pair of lines intersect, but the three lines have no single point in common.",
    "System_of_linear_equations_5": "Two lines in the $xy$ plane, intersecting at one point.",
    "System_of_linear_equations_6": "A line in the $xy$ plane.",
    "System_of_linear_equations_7": "The equations $x - 2y = -1$, $3x + 5y = 8$, and $4x + 3y = 7$ are linearly dependent.",
    "System_of_linear_equations_8": "The equations $3x + 2y = 6$ and 3$x + 2y = 12$ are inconsistent.",
    "T-coloring_1": "Two $T$-colorings of a graph for $T = \\{0, 1, 4\\}$",
    "T-square_(fractal)_2": "Golden squares with T-branching",
    "T-square_(fractal)_3": "Square branches, related by the $1/\\varphi$.",
    "T-square_(fractal)_4": "Squares branches, related by $1/2$",
    "Tangent_1": "Tangent plane to a sphere",
    "Tangent_2": "A tangent, a chord, and a secant to a circle",
    "Tangent_3": "The limaçon trisectrix: a curve with two tangents at the origin.",
    "Tangent_4": "Two pairs of tangent circles. Above internally and below externally tangent",
    "Tangent_cone_1": "$y^2 = x^3 + x^2$ (red) with tangent cone (blue)",
    "Tangent_developable_1": "The tangent developable of a helix",
    "Tangent_developable_2": "Tangent developable of a curve with zero torsion.",
    "Tangent_half-angle_substitution_1": "The tangent half-angle substitution relates an angle to the slope of a line.",
    "Tangential_trapezoid_1": "A tangential trapezoid.",
    "Tangential_trapezoid_2": "A right tangential trapezoid.",
    "Tangential_trapezoid_3": "Every isosceles tangential trapezoid is bicentric.",
    "Taylor's_theorem_1": "The exponential function $y=e^{x}$ (red) and the corresponding Taylor polynomial of degree four (dashed green) around the origin.",
    "Taylor's_theorem_2": "Graph of $f(x)=e^{x}$ (blue) with its linear approximation $P_{1}(x)=1+x$ (red) at $a=0$.",
    "Taylor's_theorem_3": "Graph of $f(x)=e^{x}$ (blue) with its quadratic approximation $P_{2}(x)=1+x+{\\dfrac {x^{2}}{2}}$ (red) at $a=0$. Note the improvement in the approximation.",
    "Telephone_number_(mathematics)_1": "The complete graph $K_4$ has ten matchings, corresponding to the value $T(4) = 10$ of the fourth telephone number.",
    "Ternary_plot_1": "Derivation of a ternary plot from Cartesian coordinates",
    "Tesseract_1": "The Dalí cross, a net of a tesseract",
    "Tesseract_2": "Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)",
    "Tetradecagon_1": "Symmetries of a regular tetradecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.",
    "Tetradecagon_2": "A regular tetradecagon",
    "Tetrahedron_2": "Regular tetrahedron $ABCD$ and its circumscribed sphere",
    "Tetrahedron_3": "The proper rotations, (order-3 rotation on a vertex and face, and order-2 on two edges) and reflection plane (through two faces and one edge) in the symmetry group of the regular tetrahedron",
    "Tetrahedron_4": "A central cross section of a regular tetrahedron is a square.",
    "Tetrahedron_5": "A tetragonal disphenoid viewed orthogonally to the two green edges.",
    "Tetration_1": "Domain coloring of the holomorphic tetration ${}^{z}e$, with hue representing the function argument and brightness representing magnitude",
    "Tetration_2": "Tetration by period",
    "Tetration_3": "Tetration by escape",
    "Tetration_4": "$\\textstyle \\lim _{n\\rightarrow \\infty }{}^{n}x$ of the infinitely iterated exponential converges for the bases $\\textstyle \\left(e^{-1}\\right)^{e}\\leq x\\leq e^{\\left(e^{-1}\\right)}$",
    "Tetration_5": "The function $\\left|{\\frac {\\mathrm {W} (-\\ln {z})}{-\\ln {z}}}\\right|$ on the complex plane, showing the real-valued infinitely iterated exponential function (black curve)",
    "Tetration_6": "$\\,{}^{x}e$ using linear approximation",
    "Thales's_theorem_2": "Thales's theorem and reflections",
    "Thales's_theorem_3": "Constructing a tangent using Thales's theorem.",
    "Thales's_theorem_4": "Illustration of the use of Thales's theorem and a right angle to find the centre of a circle",
    "Thales's_theorem_5": "Geometric method to find $\\sqrt{p}$ using the geometric mean theorem $h = \\sqrt{pq}$ with $q = 1$.",
    "Thomsen's_theorem_1": "Thomsen's theorem, $P_{7}=P_{1}$",
    "Threshold_graph_1": "An example of a threshold graph.",
    "Thue_number_1": "The Thue number of the 5-cycle is four.",
    "Time_dilation_1": "Left: Observer at rest measures time $\\frac{2L}{c}$ between co-local events of light signal generation at A and arrival at A. Right: Events according to an observer moving to the left of the setup: bottom mirror A when signal is generated at time $t'=0$, top mirror B when signal gets reflected at time $t'=\\frac{D}{c}$, bottom mirror A when signal returns at time $t'=\\frac{2D}{c}$",
    "Torsion_(algebra)_1": "The 4-torsion subgroup of an elliptic curve over the complex numbers.",
    "Torus_2": "A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle.",
    "Torus_3": "Poloidal direction (red arrow) and toroidal direction (blue arrow)",
    "Total_variation_distance_of_probability_measures_1": "Total variation distance is half the absolute area between the two curves: Half the shaded area above.",
    "Trace_operator_1": "A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red).",
    "Tractrix_1": "Tractrix with object initially at (4, 0)",
    "Transitive_closure_1": "Transitive closure constructs the output graph from the input graph.",
    "Transitive_closure_2": "A cluster graph, the transitive closure of an undirected graph",
    "Transversality_(mathematics)_2": "Non-transverse curves on the surface of a sphere",
    "Transversality_(mathematics)_3": "Transversality depends on ambient space. The two curves shown are transverse when considered as embedded in the plane, but not if we consider them as embedded in a plane in three-dimensional space",
    "Trapezoid_1": "general trapezoid/trapezium:parallel sides: $a,\\,b$ with $a<b$ legs: $c,\\,d $diagonals:$q,\\,p$ midsegment: $m$ height/altitude:$h$",
    "Trapezoid_2": "trapezoid/trapezium with opposing triangles $S,\\,T$ formed by the diagonals",
    "Trapezoid_3": "An example of trapezoidal number: $15 = 4 + 5 + 6$",
    "Trapezoid_4": "Trapezoid special cases. The orange figures also qualify as parallelograms.",
    "Trapezoid_5": "Explanatory diagram of numerical integration with the trapezoidal rule.",
    "Trapezoid_graph_1": "Trapezoid representation of graph $G$.",
    "Trapezoidal_rule_2": "Illustration of \"chained trapezoidal rule\" used on an irregularly-spaced partition of $[a,b]$.",
    "Tree_decomposition_2": "Two different tree-decompositions of the same graph",
    "Trefoil_knot_1": "Overhand knot becomes a trefoil knot by joining the ends.",
    "Trefoil_knot_2": "The trefoil knot is tricolorable.",
    "Trefoil_knot_3": "Form of trefoil knot without visual three-fold symmetry",
    "Triacontagon_2": "The symmetries of a regular triacontagon as shown with colors on edges and vertices. Lines of reflections are blue through vertices, and purple through edges. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions. Subgroup symmetries are connected by colored lines, index 2, 3, and 5.",
    "Triacontagon_3": "30-gon with 420 rhombs",
    "Triacontagon_4": "A regular triacontagon",
    "Triangle_1": "A triangular bipyramid can be constructed by attaching two tetrahedra. This polyhedron can be said to be a simplicial polyhedron because all of its faces are triangles. More specifically, when the faces are equilateral, it is categorized as a deltahedron.",
    "Triangle_2": "The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal).",
    "Triangle_3": "The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle.",
    "Triangle_4": "Applying trigonometry to find the altitude $h$",
    "Triangle-free_graph_2": "The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors",
    "Triangle-free_graph_3": "The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors",
    "Triangle-free_graph_4": "The triangle-free graphs with the most edges for their vertices are balanced complete bipartite graphs. Many triangle-free graphs are not bipartite, for example any cycle graph $C_n$ for odd $n > 3$.",
    "Triangular_matrix_1": "Binary lower unitriangular Toeplitz matrices, multiplied using $F_2$ operations. They form the Cayley table of $Z_4$ and correspond to powers of the 4-bit Gray code permutation.",
    "Triangular_number_2": "Triangular Numbers Plot",
    "Triangular_number_3": "The fourth triangular number equals the third tetrahedral number as the nth k-simplex number equals the kth n-simplex number due to the symmetry of Pascal's triangle, and its diagonals being simplex numbers; similarly, the fifth triangular number (15) equals the third pentatope number, and so forth",
    "Triangular_number_4": "Proof without words that all hexagonal numbers are odd-sided triangular numbers",
    "Triangular_prism_2": "A crossed triangular antiprism shares its vertex arrangement with a triangular prism as a faceting, with lateral isosceles triangles.",
    "Triangular_prism_3": "Truncated right triangular prism",
    "Triangular_tiling_1": "A 2-uniform triangular tiling, 4 colored triangles, related to the geodesic polyhedron as $\\{3,6_{+}\\}_{2,0}$.",
    "Triangular_tiling_2": "The $A^*_2$ lattice as three triangular tilings:",
    "Tridecagon_1": "Symmetries of a regular tridecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices and edge. Gyration orders are given in the center.",
    "Trident_curve_1": "trident curve with $a = b = c = d = 1$",
    "Trident_curve_2": "Trident curve at $y = \\infty$ with $a = b = c = d = 1$.",
    "Trigonometry_1": "In this right triangle: $\\sin A = a/h;$ $\\cos A = b/h;$ $\\tan A = a/b.$",
    "Trilinear_coordinates_1": "Trilinear coordinate diagram",
    "Trisectrix_of_Maclaurin_1": "The trisectrix of Maclaurin showing the angle trisection property",
    "Tropical_geometry_1": "A tropical cubic curve",
    "True_anomaly_1": "The true anomaly of point $P$ is the angle $f$. The center of the ellipse is point $C$, and the focus is point $F$.",
    "Truncated_cube_1": "3D model of a truncated cube",
    "Truncated_cube_2": "A truncated cube with its octagonal faces pyritohedrally dissected with a central vertex into triangles and pentagons, creating a topological icosidodecahedron",
    "Truncated_cube_3": "Dissected truncated cube, with elements expanded apart",
    "Truncated_dodecahedron_1": "The graph of a truncated dodecahedron",
    "Truncated_icosahedron_1": "3D model of a truncated icosahedron",
    "Truncated_icosahedron_2": "The truncated icosahedral graph",
    "Truncated_octahedron_1": "3D model of a truncated octahedron",
    "Truncated_octahedron_2": "Three different Hamiltonian cycles described by the three different LCF notations for the truncated octahedral graph",
    "Truncated_octahedron_3": "Truncated octahedron as a permutahedron of order 4",
    "Truncated_octahedron_4": "Truncated octahedra tiling space",
    "Truncated_tetrahedron_1": "3D model of a truncated tetrahedron",
    "Truncated_tetrahedron_2": "The graph of a truncated tetrahedron",
    "Truncation_(geometry)_2": "Types of truncations shown on an edge isolated from a larger polygon or polyhedron with red and blue vertices. The edge reverses direction after complete truncation.",
    "Truncation_(geometry)_3": "Regular polygon general truncation",
    "Truncus_(mathematics)_1": "A mathematical graph of the basic truncus formula, marked in blue, with domain and range both restricted to $[-5, 5]$.",
    "Tubular_neighborhood_1": "A curve, in blue, and some lines perpendicular to it, in green. Small portions of those lines around the curve are in red.",
    "Tubular_neighborhood_2": "A close up of the figure above. The curve is in blue, and its tubular neighborhood $T$ is in red. With the notation in the article, the curve is $S$, the space containing the curve is $M$, and $T=j(N).$",
    "Tubular_neighborhood_3": "A schematic illustration of the normal bundle $N$, with the zero section $N_{0}$ in blue. The transformation $j$ maps $N_{0}$ to the curve S in the figure above, and $N$ to the tubular neighbourhood of $S$.",
    "Tutte_polynomial_2": "The chromatic polynomial drawn in the Tutte plane",
    "Tutte_polynomial_3": "The Jones polynomial drawn in the Tutte plane",
    "Tutte_polynomial_4": "The partition functions for the Ising model and the 3- and 4-state Potts models drawn in the Tutte plane.",
    "Tutte_polynomial_5": "The flow polynomial drawn in the Tutte plane",
    "Tutte_polynomial_6": "The reliability polynomial drawn in the Tutte plane",
    "Tutte_polynomial_7": "The Tutte plane. Every point $(x,y)$ in the real plane corresponds to a computational problem $T_{G}(x,y)$. At any red point, the problem is polynomial-time computable; at any blue point, the problem is #P-hard in general, but polynomial-time computable for planar graphs; and at any point in the white regions, the problem is #P-hard even for bipartite planar graphs.",
    "Tverberg's_theorem_1": "A Tverberg partition of the vertices of a regular heptagon into three subsets with intersecting convex hulls.",
    "Twisted_Hessian_curves_1": "A Twisted Hessian curve of equation $10x^{3}+y^{3}+1=15xy$",
    "Two-body_problem_1": "Jacobi coordinates for two-body problem; Jacobi coordinates are ${\\boldsymbol {R}}={\\frac {m_{1}}{M}}{\\boldsymbol {x}}_{1}+{\\frac {m_{2}}{M}}{\\boldsymbol {x}}_{2}$ and ${\\boldsymbol {r}}={\\boldsymbol {x}}_{1}-{\\boldsymbol {x}}_{2}$ with $M=m_{1}+m_{2}\\$.",
    "Two-center_bipolar_coordinates_1": "Two-center bipolar coordinates.",
    "Types_of_mesh_2": "Skewness based on equilateral volume",
    "Ultraparallel_theorem_1": "Poincaré disc: The pink line is ultraparallel to the blue line and the green lines are limiting parallel to the blue line.",
    "Umbilical_point_2": "Umbilic classification, the $\\beta$-plane. The inner deltoid gives parabolic umbilics and separates elliptical and hyperbolic umbilics. Cusps on the inner deltoid correspond to cubic umbilics. The outer circle, representing the birth of umbilics, separates star and monstar configurations. The outer deltoid separates monstar and lemon configurations. The diagonals and the horizontal line indicate symmetrical umbilics with mirror symmetry.",
    "Umbilical_point_3": "A surface with an elliptical umbilic, and its focal surface.",
    "Umbilical_point_4": "A surface with a hyperbolic umbilic and its focal surface.",
    "Ungula_2": "Ungula of a right circular cone.",
    "Uniform_polytope_1": "Example tetrahedron in cubic honeycomb cell.There are 3 right dihedral angles (2 intersecting perpendicular mirrors):Edges 1 to 2, 0 to 2, and 1 to 3.",
    "Uniform_polytope_2": "Summary chart of truncation operations",
    "Uniform_polytope_3": "An alternation of a truncated cuboctahedron produces a snub cube.",
    "Uniformization_(set_theory)_1": "Uniformization of relation R (light blue) by function f (red).",
    "Union_(set_theory)_2": "Union of three sets: $~A\\cup B\\cup C$",
    "Union_(set_theory)_3": "The union of A, B, C, D, and E is everything except the white area.",
    "Unit_circle_2": "All of the trigonometric functions of the angle $\\theta$ (theta) can be constructed geometrically in terms of a unit circle centered at $O$.",
    "Unit_circle_3": "Sine function on unit circle (top) and its graph (bottom)",
    "Unit_circle_4": "The unit circle, showing coordinates of certain points",
    "Unit_circle_5": "Unit circle in complex dynamics",
    "Unit_disk_1": "An open Euclidean unit disk",
    "Unit_disk_2": "From top to bottom: open unit disk in the Euclidean metric, taxicab metric, and Chebyshev metric.",
    "Unit_distance_graph_1": "A unit distance graph with 16 vertices and 40 edges",
    "Unit_distance_graph_2": "A seven-coloring of the infinite unit distance graph formed from all points of the plane, and the four-chromatic Moser spindle embedded as a unit distance graph",
    "Unit_distance_graph_3": "The Petersen graph as a unit distance graph",
    "Unit_distance_graph_4": "The Möbius–Kantor graph as a subgraph of a unit distance graph",
    "Unit_distance_graph_5": "The hypercube graph $Q_{4}$ has 16 vertices and 32 unit distances",
    "Unit_fraction_1": "A pattern of spherical triangles with reflection symmetry across each triangle edge. Spherical reflection patterns like this with $2x$, $2y$, and $2z$ triangles at each vertex (here, $x,y,z=2,3,5$) only exist when ${\\tfrac {1}{x}}+{\\tfrac {1}{y}}+{\\tfrac {1}{z}}>1$.",
    "Universal_parabolic_constant_1": "The universal parabolic constant is the red length divided by the green length.",
    "Universal_point_set_1": "An arc diagram",
    "Universal_point_set_2": "Grid drawing of a nested triangles graph. In any drawing of this graph, at least half of the triangles must form a nested chain, which requires a bounding box of size at least $\\frac{n}{3} \\times \\frac{n}{3}$. The layout shown here comes close to this, using size approximately $\\frac{n}{3} \\times \\frac{n}{2}$",
    "Unknot_1": "Two simple diagrams of the unknot",
    "Unknot_2": "Thistlethwaite unknot",
    "Unknot_3": "One of Ochiai's unknots",
    "Unknotting_number_2": "Whitehead link being unknotted by undoing one crossing",
    "Van_Lamoen_circle_1": "The van Lamoen circle through six circumcenters $A_{b}$, $A_{c}$, $B_{c}$, $B_{a}$, $C_{a}$, $C_{b}$",
    "Van_der_Corput_sequence_1": "Illustration of the filling of the unit interval (horizontal axis) using the first $n$ terms of the decimal Van der Corput sequence, for $n$ from 0 to 999 (vertical axis)",
    "Variation_diminishing_property_1": "Sample curves (red) with their polygons (grey).",
    "Vector_projection_1": "Projection of $a$ on $b$ ($a_1$), and rejection of $a$ from $b$ ($a_2$).",
    "Vector_projection_2": "When $90^\\circ < \\theta \\leq 180^\\circ$, $a_1$ has an opposite direction with respect to $b$.",
    "Vector_projection_3": "If $0^\\circ \\leq \\theta \\leq 90^\\circ$, as in this case, the scalar projection of $\\mathbf{a}$ on $\\mathbf{b}$ coincides with the length of the vector projection.",
    "Venn_diagram_1": "Venn diagram as a truth table",
    "Verma_module_1": "Weights of Verma module for ${\\mathfrak {sl}}(3,\\mathbb {C} )$ with highest weight $\\lambda$",
    "Versine_2": "Sine, cosine, and versine of angle $\\theta$ in terms of a unit circle with radius 1, centered at $O$. This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow. If the arc $ADB$ of the double-angle $\\Delta = 2\\theta$ is viewed as a bow and the chord $AB$ as its string, then the versine $CD$ is clearly the arrow shaft.",
    "Versine_3": "The trigonometric functions can be constructed geometrically in terms of a unit circle centered at $O$.",
    "Versine_4": "Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to $2\\pi$",
    "Versine_5": "Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to $\\pi/2$",
    "Vertex_cover_1": "Example graph that has a vertex cover comprising 2 vertices (bottom), but none with fewer.",
    "Vertex_cover_2": "Examples of vertex covers",
    "Vertex_cover_3": "Examples of minimum vertex covers",
    "View_factor_1": "Nusselt analog: the projected solid angle",
    "Villarceau_circles_2": "Torus: Villarceau circlesFor the bottom picture the projection is orthogonal onto the section plane. Hence the true shape of the circles appear.",
    "Villarceau_circles_3": "Torus with two pencils of Villarceau circles",
    "Villarceau_circles_4": "Villarceau circles (magenta, green) through a given point (red). For any point there exist 4 circles on the torus containing the point.",
    "Viscosity_solution_1": "Family of solutions $u_{\\epsilon }$ converging toward $u(x)=1-|x|$.",
    "Vitali_covering_lemma_1": "On the top: a collection of balls; the green balls are the disjoint subcollection. On the bottom: the subcollection with three times the radius covers all the balls.",
    "Vitali_covering_lemma_2": "A given ball $B_{i}$ (colored red) must intersect some selected balls (colored green). Of these, pick the green ball $B_{j_{k}}$ with the lowest index $k$. It must necessarily have radius $r(B_{j_{k}})\\geq r(B_{i})$ by construction.",
    "Viviani's_curve_1": "Viviani's curve: intersection of a sphere with a tangent cylinder.",
    "Viviani's_curve_2": "The light blue part of the half sphere can be squared.",
    "Viviani's_curve_3": "With the cylinder upright.",
    "Viviani's_curve_4": "For parametric representation and the determination of the area",
    "Viviani's_curve_5": "Viviani's curve (red) as intersection of the sphere and a cone (pink)",
    "Von_Mangoldt_function_1": "The first Riemann zeta zero wave in the sum that approximates the von Mangoldt function",
    "Von_Neumann_universe_1": "An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic.",
    "Voronoi_diagram_1": "This is a slice of the Voronoi diagram of a random set of points in a 3D box. In general, a cross section of a 3D Voronoi tessellation is a power diagram, a weighted form of a 2d Voronoi diagram, rather than being an unweighted Voronoi diagram.",
    "Voronoi_diagram_2": "Approximate Voronoi diagram of a set of points. Notice the blended colors in the fuzzy boundary of the Voronoi cells.",
    "Voronoi_diagram_3": "20 points and their Voronoi cells",
    "Walk-on-spheres_method_2": "The Walk-on-spheres method is used until the process gets $\\delta$-close to the border. Then the sphere is replaced by its \"intersection\" with the boundary of the domain.",
    "Wallace_tree_1": "4 layer Wallace reduction of an $8 \\times 8$ partial product matrix, using 15 half adders (two dots) and 38 full adders (three dots). The dots in each column are bits of equal weight.",
    "Wallis's_conical_edge_1": "Wallis's Conical Edge with $a = b = c = 1$",
    "Wasserstein_metric_1": "Infimal convolution of a cone with a curve. Note how the lower envelope has slope $\\leq 1$, and how the lower envelope is equal to the curve on the parts where the curve itself has slope $\\leq 1$.",
    "Wasserstein_metric_2": "Two one-dimensional distributions $\\mu$ and $\\nu$, plotted on the $x$ and $y$ axes, and one possible joint distribution that defines a transport plan between them. The joint distribution/transport plan is not unique.",
    "Weak_ordering_2": "The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensional faces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements.",
    "Weak_ordering_3": "The 13 possible strict weak orderings on a set of three elements $\\{a,b,c,d\\}$. The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy.",
    "Weber_problem_2": "Simpson's geometrical solution of the Weber triangle problem.",
    "Weber_problem_3": "Tellier's geometrical solution of the attraction-repulsion triangle problem.",
    "Weber_problem_4": "The angles of the Weber problem.",
    "Weierstrass_functions_2": "Plot of the zeta function using Domain coloring",
    "Weierstrass_functions_3": "Plot of the p-function using Domain coloring",
    "Weierstrass_transform_1": "The graph of a function $f(x)$ (black) and its generalized Weierstrass transforms for five $t$ parameters. The standard Weierstrass transform $F(x)$ is given by the case $t=1$ (in green)",
    "Weight_(representation_theory)_2": "Algebraically integral elements (triangular lattice), dominant integral elements (black dots), and fundamental weights for $sl(3,C)$",
    "Well-quasi-ordering_1": "Hasse diagram of the natural numbers ordered by divisibility",
    "Well-quasi-ordering_2": "Hasse diagram of $\\mathbb {N} ^{2}$ with componentwise order",
    "Well-separated_pair_decomposition_2": "Visual representation of a well-separated pair computed with the bounding boxes",
    "Weyl_group_1": "The Weyl group of the $A_{2}$ root system is the symmetry group of an equilateral triangle",
    "Weyl_group_2": "The shaded region is the fundamental Weyl chamber for the base $\\{\\alpha _{1},\\alpha _{2}\\}$",
    "Wheel_graph_1": "The 7 cycles of the wheel graph $W_4$.",
    "Wheel_graph_2": "Wheel graphs.",
    "Whitney_embedding_theorem_2": "ntroducing double-point.",
    "Wigner_quasiprobability_distribution_1": "Wigner function of a so-called cat state",
    "Wigner_quasiprobability_distribution_2": "Wigner function for number states (a) $n = 0$, (b) $n = 1$, and (c) $n = 19$. Marginal distributions for $x$ and $p$ are recovered by integrating over $p$ and $x$ respectively.",
    "Winding_number_1": "This curve has winding number two around the point $p$.",
    "Wythoff_symbol_1": "Example Wythoff construction triangles with the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol.",
    "Wythoff_symbol_2": "The eight forms for the Wythoff constructions from a general triangle $(p q r)$.",
    "Young's_lattice_2": "The portion of Young's lattice lying below $1 + 1 + 1 + 1$, $2 + 2 + 2$, $3 + 3$, $4$, displayed in a way that shows its dihedral symmetry.",
    "Young's_lattice_3": "The portion of Young's lattice lying below $1 + 1 + 1 + 1$, $2 + 2 + 2$, $3 + 3$, $4$, displayed in the conventional way in which partitions of the same rank are shown at the same height.",
    "Zarankiewicz_problem_2": "A bipartite graph with 4 vertices on each side, 13 edges, and no $K_{3,3}$ subgraph, and an equivalent set of 13 points in a $4 \\times 4$ grid, showing that $z(4;3) \\geq 13$.",
    "Zariski_topology_1": "In the Zariski topology on the affine plane, this graph of a polynomial is closed.",
    "Zernike_polynomials_1": "The first 21 Zernike polynomials, ordered vertically by radial degree and horizontally by azimuthal degree",
    "Zero_object_(algebra)_1": "Morphisms to and from the zero object",
    "Zhegalkin_polynomial_1": "Graphic representation of the coefficients of the Zhegalkin polynomial for functions with different numbers of variables."
}