{
    "q_1": "The diagram above depicts an example of a potential unit root. The red line represents an observed drop in output. Green shows the path of recovery if the series has a unit root. Blue shows the recovery if there is no unit root and the series is trend-stationary. The blue line returns to meet and follow the dashed trend line while the green line remains permanently below the trend. The unit root hypothesis also holds that a spike in output will lead to levels of output higher than the past trend.",
    "q_2": "Sombrero function 3D",
    "q_3": "The divergence of different vector fields. The divergence of vectors from point $(x,y)$ equals the sum of the partial derivative-with-respect-to-x of the x-component and the partial derivative-with-respect-to-y of the y-component at that point: $\\nabla \\!\\cdot (\\mathbf {V} (x,y))={\\frac {\\partial \\,{V_{x}(x,y)}}{\\partial {x}}}+{\\frac {\\partial \\,{V_{y}(x,y)}}{\\partial {y}}}$",
    "q_4": "Vectors $\\mathbf{a}$ (purple), $\\mathbf{b}$ (cyan) and $\\mathbf{a} + \\mathbf{b}$ (blue) are shown with arrows",
    "q_5": "Venn diagram of $A\\oplus B\\oplus C$",
    "q_6": "The intersection of the unit cube with the cutting plane $x_{1}+x_{2}+x_{3}\\geq 2$. In the context of the Traveling salesman problem on three nodes, this (rather weak) inequality states that every tour must have at least two edges.",
    "q_7": "The Vámos matroid, a paving matroid of rank four; the shaded parallelograms depict its five circuits of size four",
    "q_8": "The fundamental polygon for Bring's curve is a regular hyperbolic icosagon (20-gon), shown here with dodecadodecahedral graph in green and its dual in violet. It is a quotient of the order-4 pentagonal tiling and its dual square tiling.20-gon edges marked with the same letter are equal.",
    "q_9": "The property of the bundle theorem is that the dashed red cycle exists.",
    "q_10": "The two distinguished points are examples of extreme points of a convex set that are not exposed",
    "q_11": "Symmetries of a regular decagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edges. Gyration orders are given in the center.",
    "q_12": "Square triangular number 36 depicted as a triangular number and as a square number.",
    "q_13": "An undirected graph colored based on the betweenness centrality of each vertex from least (red) to greatest (blue).",
    "q_14": "A spherical triangle",
    "q_15": "A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central ideal point.",
    "q_16": "3D visualization of a sphere and a rotation about an Euler axis (${\\hat {e}}$) by an angle of $\\theta$",
    "q_17": "A graph of logarithmic growth",
    "q_18": "The point $Z$ is the zero point, $P$ is the point such that the fraction of the circle from $Z$ to $P$ (in blue) is equal to $p$. The value of $p$ is precisely the number of blue arcs divided by the total number of arcs. If we let the first clockwise point of an arc define it, then every point on the circle defines one arc with $Z$ defining a blue arc and $P$ defining a non-blue arc. The estimate for $p$ is then one $(Z)$ more than blue points divided by two ($Z$ and $P$) more than total number of trials which is the rule of succession.",
    "q_19": "The 3-adic integers, with selected corresponding characters on their Pontryagin dual group",
    "q_20": "joined truncated icosahedron",
    "q_21": "The light gray area is the absolutely convex hull of the cross.",
    "q_22": "Three realizations of Bessel Processes.",
    "q_23": "The coefficients of the Fibonacci polynomials can be read off from a left-justified Pascal's triangle following the diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers.",
    "q_24": "Venn diagram of $P\\nleftarrow Q$(the red area is true)",
    "q_25": "Example: The matching distance between $\\ell _{1}=r+a+b$ and $\\ell _{2}=r'+a'$ is given by $d_{\\text{match}}(\\ell _{1},\\ell _{2})=\\max\\{\\delta (r,r'),\\delta (b,a'),\\delta (a,\\Delta )\\}=4$",
    "q_26": "Given three points $x_{1},x_{2},x_{3}$ in a plane as shown in the figure, the point $P$ is a convex combination of the three points, while $Q$ is not. ($Q$ is however an affine combination of the three points, as their affine hull is the entire plane.)",
    "q_27": "The Tutte fragment.",
    "q_28": "This hypercube graph is the 1-skeleton of the tesseract.",
    "q_29": "If we draw an arrow pointing from each positive integer to the sum of all its proper divisors, there will be no arrow pointing to untouchable numbers like 2 and 5.",
    "q_30": "Isoelastic utility for different values of $\\eta.$ When $\\eta>1$ the curve approaches the horizontal axis asymptotically from below with no lower bound.",
    "q_31": "Two cospectral enneahedra, the smallest possible cospectral polyhedral graphs",
    "q_32": "Hazard function. $\\alpha =1,$ values of $\\beta$ as shown in legend",
    "q_33": "Complete coloring of the Clebsch graph with 8 colors. Every pair of colors appears on at least one edge. No complete coloring with more colors exists: in any 9-coloring some color would appear only at one vertex, and there would not be enough neighboring vertices to cover all pairs involving that color. Therefore, the achromatic number of the Clebsch graph is 8.",
    "q_34": "Tschirnhausen cubic, case of $a = 1$",
    "q_35": "Illustrates combinatorial proof: ${\\binom {4}{1}}+{\\binom {4}{2}}={\\binom {5}{2}}.$",
    "q_36": "A dodecahedron is a convex body.",
    "q_37": "Important quantities in the Whewell equation",
    "q_38": "The $x$-coordinates of the red circles are stationary points; the blue squares are inflection points.",
    "q_39": "An image from inside a 3-torus. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.",
    "q_40": "A bijective function",
    "q_41": "The complete bipartite graph $K_{3,3}$ with each of its color classes drawn as parallel line segments on distinct lines.",
    "q_42": "3D model of a (uniform) triangular prism",
    "q_43": "Four pages of a cohomological spectral sequence",
    "q_44": "A minimum cut.",
    "q_45": "The function $f(x) = \\sin\\left(\\frac{1}{x}\\right)$ is not of bounded variation on the interval $[0,2/\\pi]$.",
    "q_46": "There are infinite blue rectangles each with area $\\frac{1}{2}$, yet their total area is exceeded by that of the grey bars denoting the harmonic series",
    "q_47": "The intersection point of the associated lines $k$ and $l$ describes the circle",
    "q_48": "A fourth-stage Gosper curve",
    "q_49": "A planar subdivision inside a bounding box",
    "q_50": "The ray $Aa$ is a limiting parallel to $Bb$, written: $Aa|||Bb$",
    "q_51": "Logarithmic integral function plot",
    "q_52": "The four simplexes that can be fully represented in 3D space.",
    "q_53": "Hyperbola: inscribed angle theorem",
    "q_54": "Convex and concave kites",
    "q_55": "The 3 bracelets with 3 red and 3 green beads. The one in the middle is chiral, so there are 4 necklaces. Compare box(6,9) in the triangle.",
    "q_56": "Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.",
    "q_57": "Examples of adjacent/neighboring incidences. The $^*$ above the edge $e$ closest to vertex $v$ denotes incidence $(v,e)$.",
    "q_58": "Drawings of the finite projective planes of orders 2 (the Fano plane) and 3, in grid layout, showing a method of creating such drawings for prime orders",
    "q_59": "For example, $[0,1]/\\{0,1\\}$ is homeomorphic to the circle $S^{1}.$",
    "q_60": "The ladder graphs $L_1$, $L_2$, $L_3$, $L_4$ and $L_5$.",
    "q_61": "The intersection of two planes",
    "q_62": "If the positive roots are $\\alpha _{1}$, $\\alpha _{2}$, and $\\alpha _{3}$, the shaded region is the set of points higher than $\\lambda$",
    "q_63": "More detailsConic sections of varying eccentricity sharing a focus point and directrix line, including an ellipse (red, $e = 1/2$), a parabola (green, $e = 1$), and a hyperbola (blue, $e = 2$). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity $\\infty$ is an infinitesimally separated pair of lines. A circle of finite radius has an infinitely distant directrix, while a pair of lines of finite separation have an infinitely distant focus.",
    "q_64": "Thales's theorem: if $AC$ is a diameter and $B$ is a point on the diameter's circle, the angle $\\angle ABC$ is a right angle.",
    "q_65": "Construction of equilateral triangle with compass and straightedge",
    "q_66": "The amoeba of $P(z,w)=50z^{3}+83z^{2}w+24zw^{2}+w^{3}+392z^{2}+414zw+50w^{2}-28z+59w-100.$",
    "q_67": "Eigenfaces as examples of eigenvectors",
    "q_68": "For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.",
    "q_69": "Alternation of the n-cube yields one of two n-demicubes, as in this 3-dimensional illustration of the two tetrahedra that arise as the 3-demicubes of the 3-cube.",
    "q_70": "A set of polygons in an Euler diagram",
    "q_71": "In affine geometry, one uses Playfair's axiom to find the line through $C1$ and parallel to $B1B2$, and to find the line through $B2$ and parallel to $B1C1$: their intersection $C2$ is the result of the indicated translation.",
    "q_72": "Huber loss (green, $\\delta =1$) and squared error loss (blue) as a function of $y-f(x)$",
    "q_73": "An Eisenstein triple",
    "q_74": "A cone truncated by an inclined plane",
    "q_75": "Ceva's theorem, case 1: the three lines are concurrent at a point $O$ inside $\\triangle ABC$.",
    "q_76": "Osculating circles of the Archimedean spiral, tangent to the spiral and having the same curvature at the tangent point. The spiral itself is not drawn, but can be seen as the points where the circles are especially close to each other.",
    "q_77": "According to the first meaning of permutation, each of the six rows is a different permutation of three distinct balls",
    "q_78": "The summatory function, with leading terms removed, for $x<10^{4}$",
    "q_79": "Gaussian construction of the regular heptadecagon.",
    "q_80": "Union of two sets: $~A\\cup B$",
    "q_81": "Hyperbolic trajectories followed by objects approaching central object (small dot) with same hyperbolic excess velocity (and semi-major axis $=$1) and from same direction but with different impact parameters and eccentricities. The yellow line indeed passes around the central dot, approaching it closely.",
    "q_82": "A green angle formed by two red rays on the Cartesian coordinate system",
    "q_83": "The graphic matroid of the cycle graph $C_4$, which is the uniform matroid $U{}_{4}^{3}$. More generally, the graphic matroid of $C_n$ is $U{}_{n}^{n-1}$.",
    "q_84": "Metamorphoses between two implicit surfaces: a torus and a constant distance product surface.",
    "q_85": "Cartesian coordinates identify points of the Euclidean plane with pairs of real numbers",
    "q_86": "Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy.",
    "q_87": "Comparison of the logit function with a scaled probit (i.e. the inverse CDF of the normal distribution), comparing $\\operatorname {logit} (x)$ vs. $\\Phi ^{-1}(x)/{\\sqrt {\\frac {\\pi }{8}}}$, which makes the slopes the same at the origin.",
    "q_88": "A power diagram of four circles",
    "q_89": "Polygon mesh of a circular paraboloid",
    "q_90": "The 2-dimensional cyclohedron $W_3$ and the correspondence between its vertices and edges with a cycle on three vertices",
    "q_91": "An example of a simple polygon",
    "q_92": "The dual of a cube is an octahedron. Vertices of one correspond to faces of the other, and edges correspond to each other.",
    "q_93": "The gamma function along part of the real axis",
    "q_94": "A torus as a square revolved around an axis parallel to one of its diagonals.",
    "q_95": "The {4,4} square tiling (black) with its dual (red).",
    "q_96": "If the area of the original rectangle is $\\psi$, the blue, red, and yellow copies have areas $\\frac{1}{\\psi}$, $\\frac{1}{\\psi^3}$, and $\\frac{1}{\\psi^5}$. The areas of the red and green rectangles are equal.",
    "q_97": "A projective version of a Simson line",
    "q_98": "A system of linear inequalities defines a polytope as a feasible region. The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimal solution.",
    "q_99": "Examples of four types of relations over the real numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black). 2D-plot representation is used.",
    "q_100": "A eutactic star consisting of 5 pairs of vectors in three-dimensional space $(n = 3, s = 5)$",
    "q_101": "Slope: $m={\\frac {\\Delta y}{\\Delta x}}=\\tan(\\theta )$",
    "q_102": "Eight corner points on a cube surrounding the interpolation point $C$",
    "q_103": "A lattice in the Euclidean plane. Adding any two blue lattice points produces another lattice point; the group formed by this addition operation is a free abelian group.",
    "q_104": "Cantellated cubic tiling",
    "q_105": "Example of a quasiconvex function that is not pseudoconvex. The function has a critical point at $x=0$, but this is not a minimum.",
    "q_106": "Cancelling opposite double-points.",
    "q_107": "Scatter plot of the legs $(a, b)$ of the first Pythagorean triples with $a$ and $b$ less than 6000. Negative values are included to illustrate the parabolic patterns. The rays are a result of the fact that if $(a, b, c)$ is a Pythagorean triple, then so is $(2a, 2b, 2c)$, $(3a, 3b, 3c)$ and, more generally, $(ka, kb, kc)$ for any positive integer $k$.",
    "q_108": "A set (in light blue) and its boundary (in dark blue).",
    "q_109": "The monkey saddle",
    "q_110": "The function $e^{-1/x^2}$ is not analytic at $x = 0$: the Taylor series is identically 0, although the function is not.",
    "q_111": "The leading Lyapunov vector.",
    "q_112": "Comparison of the various grading methods in a normal distribution, including: standard deviations, cumulative percentages, percentile equivalents, z-scores, T-scores",
    "q_113": "Plot of the Dawson integral function $F(z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors.",
    "q_114": "Phase space representing the assumptions $A1$ and $A2$with respect to the system (1).",
    "q_115": "A weak order on the set $\\{a,b,c,d\\}$ where $b$ and $c$ are of equal rank, $a$ is ranked below them, and $d$ is ranked above them. I) representation as a strict weak order $\\,<\\,$ where $x<y$ is shown as an arrow from $x$ to $y$; II) representation as a total preorder $\\,\\leq \\,$, shown using arrows; III) representation as an ordered partition, with the sets of the partition as dotted ellipses and the total order on these sets shown with arrows.",
    "q_116": "Illustrations of unit circles (see also superellipse) in $\\mathbb {R} ^{2}$ based on different $p$-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding $p$).",
    "q_117": "Interpolation with different polyharmonic splines that shall pass the 4 predefined points marked by a circle (the interpolation with $\\phi = r^2$ is not useful, since the linear equation system of the interpolation problem has no solution; it is solved in a least squares sense, but then does not pass the centers)",
    "q_118": "The vectors $u$ and $v$ are a basis for this two-dimensional subspace of $R^3$.",
    "q_119": "The Perles configuration, linear over the reals but not the rationals",
    "q_120": "Geometric intuition for the integral of $\\frac{1}{x}$. The three integrals from 1 to 2, from 2 to 4, and from 4 to 8 are all equal. Each region is the previous region halved vertically and doubled horizontally. Extending this, the integral from 1 to $2^{k}$ is $k$ times the integral from 1 to 2, just as $\\ln 2^{k} = k \\ln 2$.",
    "q_121": "This Hasse diagram of the lattice of subgroups of the dihedral group $Dih_4$ has no crossing edges.",
    "q_122": "This breather pseudospherical surface corresponds to a solution of a non-linear wave-equation.",
    "q_123": "Plot of the complete elliptic integral of the third kind $\\Pi(n, k)$ with several fixed values of $n$.",
    "q_124": "Interactive Csaszar polyhedron model with vertices representing nodes.",
    "q_125": "Here the counit is a universal morphism.",
    "q_126": "Illustration of the squeeze theorem",
    "q_127": "Plot of Bessel function of the first kind $J_{\\alpha }(z)$ with $\\alpha =0.5$ in the plane from $-4-4i$ to $4+4i$.",
    "q_128": "$a \\lor \\lnot b$ generates $a$ and $\\lnot b$",
    "q_129": "The universal property of the initial topology, as expressed in the corresponding categories of cones.",
    "q_130": "If the sum of the interior angles $\\alpha$ and $\\beta$ is less than $180^\\circ$, the two straight lines, produced indefinitely, meet on that side.",
    "q_131": "Triangulation of an implicit surface of genus 3",
    "q_132": "Correction factor $c_{4}$ versus sample size $n$.",
    "q_133": "A heavy anchor chain forms a catenary, with a low angle of pull on the anchor.",
    "q_134": "The complement of the Borromean rings has a non-trivial Massey product.",
    "q_135": "Plot of the scaled bump function with several choices of $\\varepsilon$",
    "q_136": "Relative error in a truncated Stirling series vs. number of terms",
    "q_137": "Symmetries of a regular icositetragon. 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.",
    "q_138": "A circular right cylinder of height $h$ and diameter $d=2r$",
    "q_139": "A p-box (probability box).",
    "q_140": "The Levi graph of the Fano plane gives rise to the Heawood graph, a bipartite graph with seven vertices on each side, 21 edges, and no 4-cycles.",
    "q_141": "Three Cassini ovals, differing by the range within which the parameter $e$ (equal to $b/a$) falls: $0 < e < 1$, $e = 1$, $1 < e < \\sqrt{2}$. Not shown: $e \\geq \\sqrt{2}$ (convex).",
    "q_142": "Plotting the normal approximation interval on an arbitrary logistic curve reveals problems of overshoot and zero-width intervals.",
    "q_143": "A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two.",
    "q_144": "The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.",
    "q_145": "The Shrikhande graph embedded on a torus. This is not a rook's graph, but is strongly regular with the same parameters as the $4\\times 4$ rook's graph.",
    "q_146": "Three squares of sides $R$ can be cut and rearranged into a dodecagon of circumradius $R$, yielding a proof without words that its area is $3R^2$",
    "q_147": "The rational points on a circle correspond, under stereographic projection, to the rational points of the line.",
    "q_148": "Three-leaved clover in Cartesian coordinates",
    "q_149": "The odd graph $O_{4}=KG(7,3)$",
    "q_150": "The regular complex polytope, $3\\{6\\}2$, represents the Cayley diagram for the binary tetrahedral group, with each red and blue triangle a directed subgraph.",
    "q_151": "Basic three-dimensional cell shapes",
    "q_152": "Ungula of a right circular cylinder.",
    "q_153": "Chernoff bound for the CDF and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom ($k=10$)",
    "q_154": "3D model of Cayley surface",
    "q_155": "A star domain is simply connected since any loop can be contracted to the center of the domain, denoted $x_{0}$.",
    "q_156": "A Möbius band is a non-orientable I-bundle. The dark line is the base for a set of transversal lines that are homeomorphic to the fiber and that each touch the edge of the band twice.",
    "q_157": "Geometrical basis for calculating the distance to the horizon, tangent-secant theorem",
    "q_158": "A solution circle (pink) in the second family encloses encloses the inner given circle (black). Twice the solution radius $r_s$ equals the sum $r_{outer} + r_{inner}$ of the inner and outer radii, while twice its center distance $d_s$ equals their difference.",
    "q_159": "The witch of Agnesi (curve MP) with labeled points",
    "q_160": "A golden rectangle with long side $a + b$ and short side $a$ can be divided into two pieces: a similar golden rectangle (shaded red, right) with long side $a$ and short side $b$, and a square (shaded blue, left) with sides of length $a$. This illustrates the relationship $\\frac{a + b}{a} = \\frac{a}{b} = \\varphi$..",
    "q_161": "Illustration of the scenarios of monomorphism.",
    "q_162": "Generation of the involute of the cycloid unwrapping a tense wire placed on half cycloid arc (red marked)",
    "q_163": "Slope: $m={\\frac {\\Delta y}{\\Delta x}}=\\tan(\\theta)$",
    "q_164": "Truncations of the cube beyond rectification",
    "q_165": "Coordinate surfaces of parabolic cylindrical coordinates. Parabolic cylinder functions occur when separation of variables is used on Laplace's equation in these coordinates",
    "q_166": "Ellipse with the points $F_{1}$ and $F_{2}$ as foci passing through $P_{1}$ and $P_{2}$",
    "q_167": "Two square-counting puzzles: There are 14 squares in a $3 \\times 3$ grid of squares (top), but as a $4 \\times 4$ grid of points it has six more off-axis squares (bottom) for a total of 20.",
    "q_168": "The 42 possible triangulations for a convex heptagon (7-sided convex polygon). This number is given by the 5th Catalan number.",
    "q_169": "Heighway dragon curve",
    "q_170": "A circle of radius a compressed to an ellipse.",
    "q_171": "Evolution of an advected contour",
    "q_172": "Illustration of a run of the Walk-on-spheres algorithm on an arbitrary domain $\\Omega$ with an $\\varepsilon$-shell",
    "q_173": "The sides of the squares used to construct a silver spiral are the Pell numbers",
    "q_174": "Cobweb plot of the Gauss map for $\\alpha =4.90$ and $\\beta =-0.58$. This shows an 8-cycle.",
    "q_175": "Geometric interpretation of the Candido identity for sequential Fibonacci numbers. The white area equals the grey area and each of them equals half of the outer square's area.",
    "q_176": "Graphs of probability $P$ of not observing independent events each of probability $\\frac{1}{n}$ after $n$ Bernoulli trials, and $1 - P$ vs.\\ $n$; it can be observed that as $n$ increases, the probability of a $\\frac{1}{n}$-chance event never appearing after $n$ tries rapidly converges to $\\frac{1}{e}$.",
    "q_177": "8 points in a square, 5 of 12 minimal triangles shaded",
    "q_178": "Branch decomposition of a grid graph, showing an e-separation. The separation, the decomposition, and the graph all have width three.",
    "q_179": "Trefoil knot without 3-fold symmetry being unknotted by one crossing switch.",
    "q_180": "A ($10_3$) configuration that is not incidence-isomorphic to a Desargues configuration",
    "q_181": "Cissoid Curve $C_1$ Curve $C_2$ Pole $O$",
    "q_182": "A (finite) drunkard's walk is an example of an absorbing Markov chain.",
    "q_183": "Graph of a sine wave's voltage vs. time (in degrees), showing RMS, peak (PK), and peak-to-peak (PP) voltages.",
    "q_184": "A 3-dimensional foliated chart with $n = 3$ and $q = 1$. The plaques are 2-dimensional and the transversals are 1-dimensional.",
    "q_185": "The Bhargava cube corresponding to the pair of binary quadratic forms $(ax^{2}+2bxy+cy^{2}, dx^{2}+2exy+fy^{2})$.",
    "q_186": "Plot of the error function $\\mathrm{erf}(z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors",
    "q_187": "Cayley's formula counts the number of spanning trees on a complete graph. There are $2^{2-2}=1$ trees in $K_{2}$, $3^{3-2}=3$ trees in $K_{3}$, and $4^{4-2}=16$ trees in $K_{4}$.",
    "q_188": "The Hasse diagram of a semiorder. Two items are comparable when their vertical coordinates differ by at least one unit (the spacing between solid blue lines).",
    "q_189": "Open neighborhood of the Niemytzki plane, tangent to the x-axis",
    "q_190": "Equilateral pentagon built with four equal circles disposed in a chain.",
    "q_191": "Rectangular parallelepiped. The line through segment $AD$ and the line through segment $B_1B$ are skew lines because they are not in the same plane.",
    "q_192": "A geodesic triangle on the sphere.",
    "q_193": "The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.",
    "q_194": "An illustration of a convex metric space.",
    "q_195": "Fuhrmann circle with Fuhrmann triangle (red), Nagel point $N$ and orthocenter $H$ $|AP_{a}|=BP_{b}|=|CP_{c}|=2r$",
    "q_196": "Sierpinski triangle",
    "q_197": "This subspace of $R^2$ is path-connected, because a path can be drawn between any two points in the space.",
    "q_198": "A section $\\sigma$ of a bundle $\\pi \\colon E\\to B$. A section $\\sigma$ allows the base space $B$ to be identified with a subspace $\\sigma (B)$ of $E$.",
    "q_199": "A contour plot of $f(x,y)=x^{2}+y^{2}$, showing the gradient vector in black, and the unit vector $\\mathbf {u}$ scaled by the directional derivative in the direction of $\\mathbf {u}$ in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.",
    "q_200": "A Hasse diagram of Young's lattice",
    "q_201": "Dini's Surface with constants $a = 1$, $b = 0.5$ and $0 \\leq u \\leq 4\\pi$ and $0<v<1$.",
    "q_202": "Nets of a cube",
    "q_203": "An example of a partial function that is injective.",
    "q_204": "A 1-forest (a maximal pseudoforest), formed by three 1-trees",
    "q_205": "In the graph of $f(x)=x+{\\tfrac {1}{x}}$, the y-axis $(x = 0)$ and the line $y = x$ are both asymptotes.",
    "q_206": "Example of an inellipse",
    "q_207": "Illustration of the Euler method. The unknown curve is in blue, and its polygonal approximation is in red.",
    "q_208": "A lemniscate of Bernoulli and its two foci $F_1$ and $F_2$",
    "q_209": "Spiral of equilateral triangles with side lengths which follow the Padovan sequence.",
    "q_210": "Given the two red points, the blue line is the linear interpolant between the points, and the value $y$ at $x$ may be found by linear interpolation.",
    "q_211": "Generalized Pythagorean theorem for Bregman divergence.",
    "q_212": "Angle trisection with a parabola",
    "q_213": "Line graphs of linear equations on the Cartesian plane",
    "q_214": "A table of all prime knots with seven crossings or fewer (not including mirror images)",
    "q_215": "The three-dimensional cube and its dual, the octahedron, the two three-dimensional Hanner polytopes",
    "q_216": "The property $a>0\\land x<y\\Rightarrow ax<ay$",
    "q_217": "The 5-cell (4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.",
    "q_218": "A Pochhammer contour winds clockwise around one point, then clockwise around another point, then counterclockwise around the first point, then counterclockwise around the second. The exact position, curvature, etc. are in this case not essential; the sequence of windings around the two special points is.",
    "q_219": "Iterate 1 to 4 barycentric subdivisions of 2-simplices",
    "q_220": "The fundamental rectangle in the complex plane of $u$",
    "q_221": "Plot of the regularized incomplete gamma function $Q(2,z)$ in the complex plane from $-2-2i$ to $2+2i$ with colors.",
    "q_222": "Top: Angles from the $x_i$ axes to the $\\overline{x}_i$ axes. Bottom: Vice versa.",
    "q_223": "Extension of $Q$ by $N$, resulting in the group $G$. They form a short exact sequence $1\\to N\\;{\\overset {\\iota }{\\to }}\\;G\\;{\\overset {\\pi }{\\to }}\\;Q\\to 1$. The injective homomorphism $\\iota$ maps $N$ to a normal subgroup of $G$. In turn, $\\pi$ maps $G$ onto $Q$, sending each coset of $\\iota (N)$ to a different element of $Q$.",
    "q_224": "A hemisphere can represent a real projective plane by attaching opposite points on the equator together.",
    "q_225": "This plot gives the envelope of the family of lines connecting points $(t, 0)$ and $(0, k - t)$, in which $k$ takes the value $1$.",
    "q_226": "The Riemann zeta function, subject of the Riemann hypothesis",
    "q_227": "General intersection sphere-cylinder",
    "q_228": "As shown by the tick marks, lines $a$ and $b$ are parallel. This can be proved because the transversal $t$ produces congruent corresponding angles $\\theta$, shown here both to the right of the transversal, one above and adjacent to line $a$ and the other above and adjacent to line $b$.",
    "q_229": "Example of a truncated triangular prism. Its top face is truncated at an oblique angle, but it is not an oblique prism.",
    "q_230": "Permutation with one of its inversions highlighted. An inversion may be denoted by the pair of places $(2, 4)$ or the pair of elements $(5, 2)$. The inversions of this permutation using element-based notation are: $(3, 1)$, $(3, 2)$, $(5, 1)$, $(5, 2)$, and $(5,4)$.",
    "q_231": "${\\frac {2}{\\pi }}x\\leq \\sin(x)\\leq x{\\text{ for }}x\\in \\left[0,{\\frac {\\pi }{2}}\\right]$",
    "q_232": "Minkowski sum $A + B$",
    "q_233": "Grid points on a circle in taxicab geometry as the grid is made finer",
    "q_234": "A tangential quadrilateral with its incircle",
    "q_235": "intercept theorem with rays",
    "q_236": "Illustration showing how to find the angle between vectors using the dot product",
    "q_237": "A Hamiltonian path (but not cycle) in the Herschel graph",
    "q_238": "LDU decomposition of a Walsh matrix",
    "q_239": "Top: 16 sample DFT-even Hann window. Bottom: Its discrete-time Fourier transform (DTFT) and the 3 non-zero values of its discrete Fourier transform (DFT).",
    "q_240": "The number of points ($n$), chords ($c$) and regions ($r_G$) for first 6 terms of Moser's circle problem",
    "q_241": "The order-3 apeirogonal tiling",
    "q_242": "Example of a mixed graph",
    "q_243": "The Cantor pairing function assigns one natural number to each pair of natural numbers",
    "q_244": "The 8 permutations of the square, forming $D_4$",
    "q_245": "The Trott curve, shown here with 7 of its bitangents, is a quartic (degree 4) M-curve, attaining the maximum (4) components for a curve of that degree.",
    "q_246": "The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio",
    "q_247": "Multigraphs of both Königsberg Bridges and Five room puzzles have more than two odd vertices (in orange), thus are not Eulerian and hence the puzzles have no solutions.",
    "q_248": "Cube at various stages of cantellation.",
    "q_249": "With respect to the quartic Fermat curve $x^4 + y^4 = 1$, the hyperbolic lemniscate sine is analogous to the trigonometric tangent function. Unlike $\\operatorname{slh}$ and $\\operatorname{clh}$, the functions $\\sin_4$ and $\\cos_4$ cannot be analytically extended to meromorphic functions in the whole complex plane.",
    "q_250": "The Fano plane",
    "q_251": "The Cayley graph of the free group on two generators $a$ and $b$. The ends of the group are in one-to-one correspondence with the rays (infinite paths) from the identity element $e$ to the fringes of the drawing.",
    "q_252": "Comparison of mean, median and mode of two log-normal distributions with the same medians and different skewnesses.",
    "q_253": "Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.",
    "q_254": "$\\text{csch}$, $\\text{sech}$ and $\\text{coth}$",
    "q_255": "Illustration of a unit circle. The variable $t$ is an angle measure.",
    "q_256": "The Kostant partition function for the $A2$ root system",
    "q_257": "$D$ is the harmonic conjugate of $C$ with respect to $A$ and $B$. $A, D, B, C$ form a harmonic range. $KLMN$ is a complete quadrangle generating it.",
    "q_258": "The polynomial $x^{4}+x^{3}+x^{2}y$ is the Tutte polynomial of the bull graph. The red line shows the intersection with the plane $y=0$, which is essentially equivalent to the chromatic polynomial.",
    "q_259": "Poncelet's porism for bicentric quadrilaterals $ABCD$ and $EFGH$",
    "q_260": "Five sampled processes, with expected standard deviation in gray.",
    "q_261": "If each vertex angle of the outer triangle is trisected, Morley's trisector theorem states that the purple triangle will be equilateral.",
    "q_262": "A Perrin microcosm: the escape time algorithm is applied to the Newton map of the entire Perrin function $F(z)$ around critical point $z = 1.225432$ with viewport width 0.05320. The basins of attraction emanating from the centre correspond to the infinite number of real (left) and complex roots (right) $F(z) = 0$.",
    "q_263": "Archimedean spiral defined by the polar equation $r = \\frac{1}{2} \\theta + 2$",
    "q_264": "An isosceles trapezoid with variables. Here $AD$ and $BC$ are the bases of, $AC$ and $BD$ are the diagonals of, and $E$ is the intersection between two diagonals of an isosceles trapezoid.",
    "q_265": "Keplerian orbital elements.",
    "q_266": "Miquel's Pentagon Theorem",
    "q_267": "A rotation represented by an Euler axis and angle.",
    "q_268": "Axiom schema of replacement: the image $F[A]$ of the domain set $A$ under the definable class function $F$ is itself a set, $B$.",
    "q_269": "Concurrent main diagonals",
    "q_270": "Figure-eight knot of practical knot-tying, with ends joined",
    "q_271": "The primitive motifs used for power graph analysis and their corresponding diagrammatic representation: biclique, clique and star.",
    "q_272": "Spiric sections as planar sections of a torus",
    "q_273": "Pappus's hexagon theorem: Points $X$, $Y$ and $Z$ are collinear on the Pappus line. The hexagon is AbCaBc.",
    "q_274": "The rose $r = \\cos(4\\theta)$. Since $k = 4$ is an even number, the rose has $2k = 8$ petals. Line segments connecting successive peaks lie on the circle $r = 1$ and will form an octagon. Since one peak is at $(1, 0)$, the octagon makes sketching the graph relatively easy after the half-cycle boundaries (corresponding to apothems) are drawn.",
    "q_275": "The hypotenuse $c$ of $a$ right triangle with sides $a$ and $b$ is the Pythagorean sum of $a$ and $b$.",
    "q_276": "The labeled roots are a set of positive roots for the $G_2$ root system, with $\\alpha_1$ and $\\alpha_2$ being the simple roots.",
    "q_277": "Transverse curves on the surface of a sphere",
    "q_278": "T-square of order 8",
    "q_279": "A preperiodic orbit.",
    "q_280": "All non-isomorphic graphs on 3 vertices and their chromatic polynomials. The empty graph $E_3$ (red) admits a 1-coloring; the complete graph $K_3$ (blue) admits a 3-coloring; the other graphs admit a 2-coloring.",
    "q_281": "A Pappus chain",
    "q_282": "Cairo tiling as the dual of the snub square tiling",
    "q_283": "Finite affine plane of order 2, containing 4 \"points\" and 6 \"lines\". Lines of the same color are \"parallel\". The centre of the figure is not a \"point\" of this affine plane, hence the two green \"lines\" don't \"intersect\".",
    "q_284": "Finite difference estimation of derivative",
    "q_285": "The height of an equilateral triangle with sides of length 2 equals the square root of 3.",
    "q_286": "Fundamental domain of a 3-generator Schottky group",
    "q_287": "Fractional Fourier transform",
    "q_288": "3D model of an equilateral square pyramid",
    "q_289": "Concurrent perpendicular bisectors of chords between concyclic points",
    "q_290": "Geometric construction for mathematical explanation of spirograph.",
    "q_291": "Illustration of the duality between points and lines, and the double meaning of \"incidence\". If two lines $a$ and $k$ pass through a single point $Q$, then the polar $q$ of $Q$ joins the poles $A$ and $K$ of the lines $a$ and $k$, respectively.",
    "q_292": "The two triangles on the left are congruent. The third is similar to them. The last triangle is neither congruent nor similar to any of the others. Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles. The unchanged properties are called invariants.",
    "q_293": "A short proof using trigonometry for the case of an acute angle",
    "q_294": "3D model of a (uniform) pentagonal prism",
    "q_295": "Detail of the Lyapunov fractal in the form of a swallow. Iteration sequence $AB$, in the region $[3.81, 3.87] \\times [3.81, 3.87]$.",
    "q_296": "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 $n > 3$.",
    "q_297": "With 4 equally spaced points, we need 3 dimensions.",
    "q_298": "To the definition of an oval: e: exterior (passing) line, t: tangent, s: secant",
    "q_299": "Kiepert hyperbola of $\\triangle ABC$. The hyperbola passes through the vertices $A$, $B$, $C$, the orthocenter $O$ and the centroid $G$ of the triangle.",
    "q_300": "Probability density plots of gamma distributions",
    "q_301": "Line $DE$ bisects line $AB$ at $D$, line $EF$ is a perpendicular bisector of segment $AD$ at $C$, and line $EF$ is the interior bisector of right angle $AED$.",
    "q_302": "A negative Spearman correlation coefficient corresponds to a decreasing monotonic trend between $X$ and $Y$.",
    "q_303": "Example of a 2-dimensional figure with central symmetry, invariant under point reflection",
    "q_304": "Orthocentric system. Any point is the orthocenter of the triangle formed by the other three.",
    "q_305": "Plot of the density function for several members of the Student t family.",
    "q_306": "Comparison of the solutions to the single-wide (top) and multi-wide (bottom) block-stacking problem with three blocks",
    "q_307": "Ellipse: pole-polar relation",
    "q_308": "Curvilinear (top), affine (right), and Cartesian (left) coordinates in two-dimensional space",
    "q_309": "A ring torus with a selection of circles on its surface",
    "q_310": "Three symmetric increasingly leptokurtic probability density functions; their intersections are indicated by vertical lines.",
    "q_311": "A bijective function, $f: X \\rightarrow Y$, where set $X = {1, 2, 3, 4}$ and set $Y = {A, B, C, D}$. For example, $f(1) = D$.",
    "q_312": "3D model of a regular dodecahedron",
    "q_313": "Illustration of connected sum.",
    "q_314": "A unit circle with trigonometric functions.",
    "q_315": "Fundamental domains in a cubic element of three groups.",
    "q_316": "Measuring the width of a Reuleaux triangle as the distance between parallel supporting lines. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width.",
    "q_317": "The branches of the unit hyperbola evolve as the points $(\\cosh a,\\sinh a)$ and $(-\\cosh a,-\\sinh a)$ depending on the hyperbolic angle parameter $a$.",
    "q_318": "A part of an infinite sequence of real numbers (in blue), indexed by a natural number $\\textstyle n$. This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded (by red dashed lines).",
    "q_319": "The first six triangular numbers (not starting with $T_0$, but rather, $T_1$)",
    "q_320": "An equidiagonal quadrilateral, showing its equal diagonals, Varignon rhombus, and perpendicular bimedians",
    "q_321": "Epigraph of a function",
    "q_322": "The sides of parallelogram $ABCD$ are shown in blue and the diagonals in red. The sum of the areas of the blue squares equal that of the red ones.",
    "q_323": "A drawing of the Petersen graph with slope number 3",
    "q_324": "$M$: circumcenter of $\\triangle ABC$, orthocenter of $\\triangle DEF$; $N$: incenter of $\\triangle ABC$, Nagel point of $\\triangle DEF$; $S$: centroid of $\\triangle ABC$ and $\\triangle DEF$",
    "q_325": "A triangle (black), its orthocenter (blue), its centroid (red), and its orthocentroidal disk (yellow)",
    "q_326": "The graph $G = (5, 5)$ illustrates that the Fiedler vector (in red) bisects the graph into two communities: one with vertices $\\{1, 2, 3\\}$, which have positive entries in the vector space, and the other with vertices $\\{4, 5\\}$, which have negative vector space entries.",
    "q_327": "Lines of curvature on an ellipsoid showing umbilic points (red).",
    "q_328": "Proof of Butterfly theorem",
    "q_329": "Example of a four-colored map",
    "q_330": "The hypercube graph $Q_4$",
    "q_331": "The best-known packing of equal-sized regular pentagons on a plane is a double lattice structure which covers $92.131%$ of the plane.",
    "q_332": "The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. $2\\pi$. The corresponding sequence for inscribed polygons has the same limit.",
    "q_333": "Villarceau circles as intersection of a torus and a plane",
    "q_334": "Visual representation of well-separated pair",
    "q_335": "The truncated tetrahedron looks similar; but its hexagons correspond to the 4 faces, not to the 6 edges, of the yellow tetrahedron, i.e. to the 4 vertices, not to the 6 edges, of the red tetrahedron.",
    "q_336": "Rotation of an object in two dimensions around a point $O$.",
    "q_337": "A Pythagorean \"Hugieia Pentagram\"",
    "q_338": "An injective function, which is not also surjective.",
    "q_339": "Klein quartic tiled with 24 heptagons (Klein map)",
    "q_340": "Young diagram of shape $(5, 4, 1)$, English notation",
    "q_341": "Input points before kernel PCA",
    "q_342": "A Morin surface, an immersion used in sphere eversion.",
    "q_343": "The cardinal sine function $sinc(z)$ plotted in the complex plane from $-2-2i$ to $2+2i$",
    "q_344": "For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The result matrix has the number of rows of the first and the number of columns of the second matrix.",
    "q_345": "A saddle point",
    "q_346": "Example of Exact Coloring with 7 colors and 14 vertices",
    "q_347": "Addition of two complex numbers can be done geometrically by constructing a parallelogram.",
    "q_348": "The gradient of the function $f(x,y) = -\\left(\\cos^2 x + \\cos^2 y\\right)^2$ depicted as a projected vector field on the bottom plane.",
    "q_349": "Four of the methods for approximating the area under curves. Left and right methods make the approximation using the right and left endpoints of each subinterval, respectively. Upper and lower methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. The values of the sums converge as the subintervals halve from top-left to bottom-right.",
    "q_350": "Illustration of the close-packing of equal spheres in both hexagonal close-packed (left) and face-centered cubic (right) lattices",
    "q_351": "Plot of the sigma function using Domain coloring.",
    "q_352": "Pascal line $GHK$ of self-crossing hexagon $ABCDEF$ inscribed in ellipse. Opposite sides of hexagon have the same color.",
    "q_353": "Diophantus II.VIII: Intersection of the line $CB$ and the circle gives a rational point $(x_0,y_0)$.",
    "q_354": "3D model of a regular tetrahedron",
    "q_355": "The four Hermite basis functions. The interpolant in each subinterval is a linear combination of these four functions.",
    "q_356": "A simplicial arrangement formed by 20 lines, the sides and symmetry axes of a regular decagon. Adding the line at infinity produces another simplicial arrangement with 21 lines.",
    "q_357": "The Lorenz attractor arises in the study of the Lorenz oscillator, a dynamical system.",
    "q_358": "The four ways of gluing a square to make a closed surface: glue single arrows together and glue double arrows together so that the arrowheads point in the same direction.",
    "q_359": "Quasiregular polyhedra are generated from all 3 corners of the fundamental domain for Schwarz triangles that have no right angles: $q \\mid 2p$, $p \\mid 2q$, $2 \\mid pq$.",
    "q_360": "Graphs are drawn as rods connected by rotating hinges. The cycle graph $C_4$ drawn as a square can be tilted over by the blue force into a parallelogram, so it is a flexible graph. $K_3$, drawn as a triangle, cannot be altered by any force that is applied to it, so it is a rigid graph.",
    "q_361": "A section of the Mandelbrot set following a logarithmic spiral",
    "q_362": "Symmetries of a regular enneagon. 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.",
    "q_363": "Circles of Apollonius; isodynamic points $S$ and $S'$ at their intersections Interior angle bisectors, used to construct the circles Exterior angle bisectors, also used to construct the circles",
    "q_364": "A diagram illustrating the great-circle distance (in cyan) and the straight-line distance (in red) between two points $P$ and $Q$ on a sphere.",
    "q_365": "The function $f(x)$ (in blue) is approximated by a linear function (in red).",
    "q_366": "Convex cone that is not a conic hull of finitely many generators.",
    "q_367": "A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable.",
    "q_368": "Laguerre-plane: stereographic projection of the x-z-plane onto a cylinder",
    "q_369": "An intersection graph of disks, with at most $k=5$ disks covering any point of the plane",
    "q_370": "Regular triacontagon with given circumcircle. $D$ is the midpoint of $AM$, $DC = DF$, and $CF$, which is the side length of the regular pentagon, is $E_{25}E_1$. Since $\\frac{1}{30} = \\frac{1}{5} - \\frac{1}{6}$, the difference between the arcs subtended by the sides of a regular pentagon and hexagon ($E_{25}E_1$ and $E_{25}A$) is that of the regular triacontagon, $AE_1$.",
    "q_371": "Example of bilinear interpolation on the unit square with the $z$ values 0, 1, 1 and 0.5 as indicated. Interpolated values in between represented by color.",
    "q_372": "Torricelli's geometrical solution of the Fermat triangle problem.",
    "q_373": "Gluing a manifold along manifold edges (in green) may create non-pseudomanifold edges (in red). A decomposition is possible cutting (in blue) at a singular edge",
    "q_374": "An Apollonian gasket is an example of a limit set of a Kleinian group",
    "q_375": "Diagram of Aristotle's wheel as described in Mechanica.",
    "q_376": "First four components of Pascal's tetrahedron.",
    "q_377": "As the volume is subdivided into smaller parts, the ratio of the flux $\\Phi (V_{\\text{i}})$ out of each volume to the volume $|V_{\\text{i}}|$ approaches $\\operatorname {div} \\mathbf {F}$",
    "q_378": "Kissing number in two dimensions is 6",
    "q_379": "Comparison of mean, median and mode of two log-normal distributions with different skewness",
    "q_380": "Trefoil knot linked with a circle.",
    "q_381": "Klein's j-invariant in the complex plane",
    "q_382": "Randolph diagram that represents the logical statement $P\\lor Q$ (disjunction).",
    "q_383": "Brownian bridge as a union of excursions.",
    "q_384": "Nephroid as envelope of a pencil of circles",
    "q_385": "Illustration of proof of the HSEAT",
    "q_386": "$E_8$ 2d projection with thread made by hand",
    "q_387": "A lattice in the Euclidean plane",
    "q_388": "The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions.",
    "q_389": "Diagram for differential proof",
    "q_390": "How to measure a mountain's height",
    "q_391": "Geometry of the first isogonic center.",
    "q_392": "p-value of chi-squared distribution for different number of degrees of freedom",
    "q_393": "A regular skew octagon seen as edges of a square antiprism, symmetry $D_{4d}$, $[2^+,8]$, $(2*4)$, order 16.",
    "q_394": "In the middle, the fitted straight line represents the best balance between the points above and below this line. The dotted straight lines represent the two extreme lines, considering only the variation in the slope. The inner curves represent the estimated range of values considering the variation in both slope and intercept. The outer curves represent a prediction for a new measurement.",
    "q_395": "The result of fitting a set of data points with a quadratic function",
    "q_396": "The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points.",
    "q_397": "Oloid structure, showing the two 240-degree circular sectors and the convex hull",
    "q_398": "Isosceles triangle with equal sides $\\overline{AB} = \\overline{AC}$ divided into two right triangles by an altitude drawn from one of the two base angles.",
    "q_399": "The Trott curve with all 28 bitangents.",
    "q_400": "The logistic curve"
}