# Portal:Mathematics/Featured picture archive

This page is an archive of images that were featured as "Picture of the month" on the Mathematics Portal. For mathematics-related pictures featured elsewhere on Wikipedia, see Wikipedia:Featured pictures/Sciences/Mathematics.

Do not add any new images here. New images should be added to the list at Portal:Mathematics/Selected picture.

## Featured Pictures from 2014

Date Picture Description Credit Read More
February 2014 Simpson's paradox for continuous data: a positive trend appears for two separate groups (blue and red), a negative trend (black, dashed) appears when the data are combined. Simpson's paradox
January 2014 Animation showing the cocktail shaker sort. Cocktail sort

## Featured Pictures from 2013

Date Picture Description Credit Read More
December 2013 A Froebel star is a traditional German Christmas decoration. It is named after German educationist Friedrich Fröbel (1782–1852), who encouraged the use of paper folding in pre–primary education with the aim of conveying simple mathematical concepts to children. Froebel star
November 2013 A line integral is an integral where the function to be integrated, be it a scalar field as here or a vector field, is evaluated along a curve. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). line integral

August 2013 The conic sections. Conic section
June 2013 Photograph of a Klein bottle from Science Museum, London. Klein Bottle
May 2013 Animation of the act of unrolling a circle's circumference, illustrating the ratio π. Pi
March 2013 1000th row of Pascal's triangle. One coefficient per row, aligned to the right, one digit per pixel, colored in 10 shades of gray from white (digit 0) to black (digit 9). Pascal's triangle
February 2013 This spiral represents all ordinal numbers less than $\omega ^{\omega }$ . Ordinal numbers
January 2013 A generalisation of the Pythagorean Theorem. If regular pentagons or any other similar shape is drawn on each side, then A + B = C (where A, B and C are the areas of the shapes). Pythagorean Theorem#Similar figures on the three sides

## Featured Pictures from 2012

Date Picture Description Credit Read More
December 2012 An example of an Apollonian gasket, a fractal generated from multiple mutually tangent circles. Apollonian Gasket
November 2012 Francis Galton's original 1889 drawing of a "bean machine", now commonly called a "Galton box", which demonstrates the central limit theorem of probability, in particular the form of it that states that the normal distribution is a good approximation to the binomial distribution. As the "bean" falls through the machine, it can fall to the left or right of each pin it approaches. This makes the final position of the bean the sum of several Bernoulli random variables, each approximately independent of the others. A level box, as pictured, gives a probability of 0.5 to fall either way at each pin, but a tilted box results in asymmetric probabilities, and thus a skewed distribution (see a photograph of a real-world example).
Central limit theorem
October 2012 Construction and use of a hexaflexagon. Flexagon
September 2012 Homology cycles on a torus. Torus
August 2012 A Lorenz curve shows the distribution of income in a population by plotting the percentage y of total income that is earned by the bottom x percent of households. It is usually plotted with a diagonal line (reflecting a hypothetical "equal" distribution of incomes) for comparison. An example of a cumulative distribution function, the curve was developed by economist Max O. Lorenz in 1905 to describe income inequality. A derived quantity is the Gini coefficient, first published in 1912 by Corrado Gini, which is the ratio of the area between the diagonal line and the Lorenz curve (area A in this graph) to the area under the diagonal line (the sum of A and B); higher Gini coefficients reflect more income inequality. See also Pareto principle and power law. Lorenz curve
July 2012 A tetrahedron can be placed in 12 distinct but equivalent positions by rotation alone. This tetrahedral symmetry is illustrated here in the cycle graph format, along with the 180° edge rotations (blue arrows) and 120° vertex rotations (reddish arrows) that permute the tetrahedron through the positions. The 12 rotations form one symmetry group of the figure. Symmetry group
June 2012 A bubble chart is a type of chart where each plotted entity is defined in terms of three distinct numeric parameters. The first two parameters are reflected in the chart as the horizontal and vertical coordinates of the center of each plotted "disc", as in a typical scatter plot, while the third determines the size of each disc. Bubble charts can facilitate the understanding of the social, economical, medical, and other scientific relationships.
unknown user on Ukrainian Wikipedia
Bubble chart
May 2012 The knight's tour is a mathematical chess problem in which the piece called the knight is to visit each square on an otherwise empty chess board exactly once, using only legal moves. It is a special case of the more general Hamiltonian path problem in graph theory. (A closely related non-Hamiltonian problem is that of the longest uncrossed knight's path.) The tour is called closed if the knight ends on a square from which it may legally move to its starting square (thereby forming an endless cycle), and open if not. The tour shown in this animation is open (see also a static image of the completed tour). The exact number of possible open tours is still unknown, but on standard 8 × 8 board there are 26,534,728,821,064 possible closed tours (counting separately any tours that are equivalent by rotation, reflection, or reversing the direction of travel). Although the earliest known solutions to the knight's tour problem date back to the 9th century CE, the first general procedure for completing the knight's tour was Warnsdorff's rule, first described in 1823. Knight's tour
April 2012 A hand-drawn graph of the absolute value of the gamma function on the complex plane, as published in the 1909 book Tables of Higher Functions by Eugene Jahnke and Fritz Emde. Such three-dimensional graphs of complicated functions were rare before the advent of high-resolution computer graphics. Published even before applications for the complex gamma function were discovered in theoretical physics in the 1930s, Jahnke and Emde's graph "acquired an almost iconic status", according to physicist Michael Berry. See a similar computer generated image for comparison.
Gamma function
March 2012 Illustration of a failed attempt to comb the "hair" on a ball flat, leaving a tuft sticking out at each pole. The hairy ball theorem of algebraic topology proves that whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball. More precisely, it states that there is no nonvanishing continuous tangent-vector field on an even-dimensional n‑sphere (an ordinary sphere in three-dimensional space is known as a "2-sphere"). This is not true of certain other three-dimensional shapes, such as a torus (doughnut shape) which can be combed flat. The theorem was first stated by Henri Poincaré in the late 19th century, and first proved in 1912 by L. E. J. Brouwer. If one idealizes the wind in the Earth's atmosphere as a tangent-vector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a cyclone somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound. Hairy ball theorem
February 2012 The polyhedron called a truncated icosahedron (left) compared to the classic Adidas Telstar–style football (or soccer ball). The familiar 32-panel ball design, consisting of 12 black pentagonal and 20 white hexagonal panels, was first introduced by the Danish manufacturer Select Sport, based loosely on the geodesic dome designs of Buckminster Fuller; it was popularized by the selection of the Adidas Telstar as the official match ball of the 1970 FIFA World Cup. The polyhedron is also the shape of the Buckminsterfullerene (or "Buckyball") carbon molecule, discovered in 1985. Truncated icosahedron
January 2012 A hypotrochoid is a curve traced out by a point "attached" to a smaller circle rolling around inside a fixed larger circle. In this example, the hypotrochoid is the red curve that is traced out by the red point 5 units from the center of the black circle of radius 3 as it rolls around inside the blue circle of radius 5. Both hypotrochoids and epitrochoids can be created using the Spirograph drawing toy. Hypotrochoid

## Featured Pictures from 2011

Date Picture Description Credit Read More
December 2011 A bifurcation diagram for the logistic map, an archetypal example of how complex, chaotic behavior can arise from very simple non-linear dynamical equations. Bifurcation diagram
November 2011 An animation of a coffee mug morphing into a torus, a popular example of a homeomorphism in topology. Homeomorphism
October 2011 A proof of Pythagoras's theorem, showing how by rearranging triangles the areas $c^{2}$ and $a^{2}+b^{2}$ can be shown to be the same. Pythagorean theorem
September 2011 Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges
Bogdan Giuşcă
Seven Bridges of Königsberg
August 2011 All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry
July 2011 The Petersen graph, an undirected graph with 10 vertices and 15 edges, serves as a useful example or counterexample for many problems in graph theory. Petersen Graph
June 2011 Evolution of the numeral 5 from the Brahmin Indians to the Arabic numerical system. 5 (number)
May 2011 Animation of the act of unrolling a circle's circumference, illustrating the ratio π. Pi
April 2011 Illustration for the paradoxical decomposition of F2 used in the proof of the Banach-Tarski paradox. Banach-Tarski paradox
March 2011 A chart of all prime knots with seven or fewer crossings, not including mirror-images. The knots are labeled with Alexander-Briggs notation. Originally studied to create a list of possible real-world knots, knot theory has become quite theoretical in nature and, at the same time, has been applied to concrete problems in organic chemistry, including the behavior of DNA. Knot theory
February 2011 In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-term behavior under repeated iteration of $f\,$ can change drastically under arbitrarily small perturbations. Above is a 3D slice of a 4D Julia set. Julia set
January 2011 A Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles, where n is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usual closed Steiner chains, the first and last circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively. Steiner chain

## Featured Pictures from 2010

Date Picture Description Credit Read More
December 2010 When produced, corresponding sides of two perspective triangles meet at collinear points along the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues' theorem guarantees that the truth of the first condition is necessary and sufficient for the truth of the second. Desargues' theorem
November 2010 In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance one from each other have the same color. The answer is unknown, but has been narrowed down to one of the numbers 4, 5, 6 or 7. Hadwiger–Nelson problem
October 2010 The trefoil knot is the simplest non-trivial knot, that is the simplest knot not equal to the unknot, and is the only knot with three crossings. It is also the simplest tricolorable knot, as illustrated in the above diagram: all three colours are used at each of the three crossings. Trefoil knot
September 2010 The Pythagorean theorem, or Pythagoras's theorem, relates the lengths of the three sides of a right triangle, and states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

There are many proofs of the theorem, dating back to Ancient Greek mathematician Pythagoras, who is credited with its discovery. The above diagram shows a geometric proof.

Pythagorean theorem
August 2010 There are an infinite number of uniform tilings on the hyperbolic plane based on the (p q r) where 1/p + 1/q + 1/r < 1. p,q,r are each orders of reflection symmetry at three points of the fundamental domain triangle – the symmetry group is a hyperbolic triangle group. Uniform tilings in hyperbolic plane
July 2010 Mandelbulb, a 3-dimensional analog of the Mandelbrot set, constructed by Daniel White and Paul Nylander, using a hypercomplex algebra based on spherical coordinates. The above picture is of a power 9 Mandelbulb. Mandelbulb
June 2010 Mandelbulb, a 3-dimensional analog of the Mandelbrot set, constructed by Daniel White and Paul Nylander, using a hypercomplex algebra based on spherical coordinates. The above picture is of a power 9 Mandelbulb. Mandelbulb
May 2010 Oxyrhynchus papyrus showing fragment of Euclid's Elements

Euclid's Elements (Greek: {{polytonic|Στοιχεῖα}} Stoicheia) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The Elements is one of the oldest extant Greek mathematical treatises, the oldest extant axiomatic deductive treatment of mathematics, and has proven instrumental in the development of logic and modern science. It is also the most successful and influential textbook ever written.

Euclid's Elements
April 2010 Oxyrhynchus papyrus showing fragment of Euclid's Elements

Euclid's Elements (Greek: {{polytonic|Στοιχεῖα}} Stoicheia) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The Elements is one of the oldest extant Greek mathematical treatises, the oldest extant axiomatic deductive treatment of mathematics, and has proven instrumental in the development of logic and modern science. It is also the most successful and influential textbook ever written.

Euclid's Elements
March 2010 The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures. It depicts two arrangements of shapes, each of which apparently forms a 13×5 right-angled triangle, but one of which has a 1×1 hole in it.

The puzzle works as the blue and red triangles are not similar so their hypotenuses are not parallel. In one arrangement the line bends one way, in one arrangement it bends the other way, and the gap between these bent lines is exactly one unit.

Missing square puzzle
February 2010 Menger sponge after four iterations.

The Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge. It is a three-dimensional extension of the Cantor set and Sierpinski carpet. It was first described by Karl Menger (1926) while exploring the concept of topological dimension.

Menger sponge
January 2010 Orthographic projection of the Great grand 120-cell a star polychoron with Schläfli symbol {5,5/2,3}. It is one of 10 regular Schläfli-Hess polychora.

The Schläfli-Hess polychora are the complete set of 10 regular self-intersecting star polychora (four-dimensional polytopes). They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {p,q,r} in which one of the numbers is 5/2. They are thus analogous to the regular nonconvex Kepler-Poinsot polyhedra.

Great grand 120-cell

## Featured Pictures from 2009

Date Picture Description Credit Read More
December 2009 A Bézier curve is a parametric curve important in computer graphics and related fields.

Widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies, the curves were first developed in 1959 by Paul de Casteljau using de Casteljau's algorithm.

In the animation above, a quartic Bézier curve is constructed using control points P0 through P4. The green line segments join points moving at a constant rate from one control point to the next; the parameter t shows the progress over time. Meanwhile, the blue line segments join points moving in a similar manner along the green segments, and the magenta line segment points along the blue segments. Finally, the black point moves at a constant rate along the magenta line segment, tracing out the final curve in red. The curve is a fourth-degree function of its parameter, t.

Quadratic and cubic Bézier curves are most common since higher-degree curves are more computationally costly to evaluate. When more complex shapes are needed, low-order Bézier curves are patched together.

Bézier curve
November 2009 A Bézier curve is a parametric curve important in computer graphics and related fields.

Widely publicized in 1962 by the French engineer Pierre Bézier, who used them to design automobile bodies, the curves were first developed in 1959 by Paul de Casteljau using de Casteljau's algorithm.

In the animation above, a quartic Bézier curve is constructed using control points P0 through P4. The green line segments join points moving at a constant rate from one control point to the next; the parameter t shows the progress over time. Meanwhile, the blue line segments join points moving in a similar manner along the green segments, and the magenta line segment points along the blue segments. Finally, the black point moves at a constant rate along the magenta line segment, tracing out the final curve in red. The curve is a fourth-degree function of its parameter, t.

Quadratic and cubic Bézier curves are most common since higher-degree curves are more computationally costly to evaluate. When more complex shapes are needed, low-order Bézier curves are patched together.

Bézier curve
October 2009 The Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. When computed and graphed on the complex plane the Mandelbrot Set is seen to have an elaborate boundary which does not simplify at any given magnification. Above is a magnification of the boundary. Mandelbrot set
September 2009 Villarceau circles are a pair of circles produced by cutting a torus diagonally through the center at the correct angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in the plane (containing the point) parallel to the equatorial plane of the torus. Another is perpendicular to it. The other two are Villarceau circles. The cut that produces Villarceau circles is shown above in the animation. They are named after the French astronomer and mathematician Yvon Villarceau (1813–1883). Villarceau circles
August 2009 Villarceau circles are a pair of circles produced by cutting a torus diagonally through the center at the correct angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in the plane (containing the point) parallel to the equatorial plane of the torus. Another is perpendicular to it. The other two are Villarceau circles. The cut that produces Villarceau circles is shown above in the animation. They are named after the French astronomer and mathematician Yvon Villarceau (1813–1883). Villarceau circles
July 2009 Conway's Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. The "game" is a zero-player game, meaning that its evolution is determined by its initial state, requiring no further input from humans. One interacts with the Game of Life by creating an initial configuration and observing how it evolves. From a very simple set of rules extremely complex patterns can emerge. Above is an example of a breeder, which creates guns, which in turn create gliders. Conway's Game of Life
June 2009 Fractals are geometric shapes that are, either as a whole or in part, self-similar. As well as being interesting mathematical objects, they also occur in nature. Above is a photograph of a Romanesco broccoli set against a black background. Self-similarity can be observed in the inflorescence (the bud), which is approximately a scaled version of the broccoli as a whole, each of which are made up of yet more scaled versions of the broccoli. In addition, the branched meristems making a logarithmic spiral, a feature that can be seen in many other fractals. Fractal
May 2009 A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of prototiles named after Roger Penrose, who investigated these sets in the 1970s.

Among the infinitely many possible tilings there are two that possess both reflection symmetry and fivefold rotational symmetry, as in the diagram, and the term Penrose tiling usually refers to both.

Penrose tiling
April 2009 Pi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. In the above animation the circle has a diameter of 1 giving it a circumference of π. The rolling shows the distance a point moves linearly in one revolution of the circle, which is equal to its circumference. Pi is an irrational number and so can not be expressed as the ratio of two integer numbers; the decimal expansion of pi is 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 to 50 decimal places. Pi
March 2009 The Sieve of Eratosthenes was one of the first methods developed for finding prime numbers. The method is simple but is still effective for finding all the primes to a given range. Sieve of Eratosthenes
February 2009 A fractal is a geometric shape that is, either a whole or in part, self-similar. The above fractal was generated using a Sterling program. Fractal
January 2009 In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. His parallel postulate is equivalent to:
Given a line and a point not on that line, exactly one line can be drawn through that point which does not intersect the original line. (case 1)

In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example in elliptical geometry:

Given a line and a point not on that line, all lines drawn through that point will intersect the original line. (case 2)

And in hyperbolic geometry:

Given a line and a point not on that line, an infinite number of lines can be drawn through the point that do not intersect the original line. (case 3)

These other forms of geometry, where the parallel postulate does not hold are called non-Euclidean geometries.

Parallel postulate

## Featured Pictures from 2008

Date Picture Description Credit Read More
December 2008 The Pythagoras tree is a plane fractal constructed from squares. It is named after Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. This one has been specially coloured to give a more tree like and more 3 dimensional appearance. Pythagoras tree
November 2008 The Lorenz attractor, named for Edward N. Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterfly shape. The map shows how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern. Lorenz attractor
October 2008 Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after Blaise Pascal in much of the western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy.

The rows of Pascal's triangle are conventionally enumerated starting with row zero, and the numbers in odd rows are usually staggered relative to the numbers in even rows. A simple construction of the triangle proceeds in the following manner. On the zeroth row, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. The above animation shows the procedure for doing this for the first 5 rows.

Pascal's triangle
September 2008 The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without or possibly with changing their shape. However, the pieces themselves are extremely complicated: they are usually not solids but infinite scatterings of points. Banach–Tarski paradox
August 2008 The exponential function is one of the most important functions in mathematics. It can be written in the form $e^{x}$ , where e is a mathematical constant, sometimes known as Euler's number.

The exponential function exists for any complex number. Plotted above are the real part (left) and the imaginary part (right) of various values of the exponential function in the complex plane.

Exponential function
July 2008 In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:
In a projective space, two triangles are in perspective axially if and only if they are in perspective centrally.

The above picture illustrates Desargues' theorem. Another important feature of projective geometry noticeable in the picture is all lines meet at exactly one point (ie there are no parallel lines).

Desargues' theorem
June 2008 The Sierpiński pyramid is a higher dimension analog of the Sierpiński triangle. It is a fractal formed by repeatedly shrinking a regular pyramid to one half its original height, putting together five copies of this pyramid with corners touching, and then repeating the process. The Sierpiński pyramid has infinite surface area but zero volume. Sierpiński triangle
May 2008 The Riemann zeta function along the critical line, all complex numbers with a real part of a half. That is, it is a graph of $\Re (\zeta (it+1/2))$ versus $\Im (\zeta (it+1/2))$ for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin. The zeros of the Riemann zeta function are central to the Riemann hypothesis. Riemann zeta function
April 2008 The normal distribution, also called the Laplace-Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem. Many psychological measurements and physical phenomena (like noise) can be approximated well by the normal distribution. An example frequently given is the intelligence quotient (IQ), seen above. Note that this graph has been artificially generated and is not based on any real measurements. Normal distribution
March 2008 Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy. On a sphere, the sum of the angles of a triangle is not equal to 180°. Spherical geometry
February 2008
The Möbius strip is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. It is also a ruled surface. Möbius strip
January 2008 In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-term behavior under repeated iteration of $f$ can change drastically under arbitrarily small perturbations. Above is a 3D slice of a 4D Julia set. Julia set

## Featured Pictures from 2007

Date Picture Description Credit Read More
December 2007 Borromean rings consist of three topological circles which are linked and form a Brunnian link. Put more simply removing any ring results in two unlinked rings. Borromean rings
November 2007 Leonardo da Vinci's illustrations in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his own paintings. Some suggest that his Mona Lisa, for example, employs the golden ratio in its geometric equivalents.
Juan Ángel Paniagua Sánchez
Golden ratio
October 2007 An attractor is a set to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed.
Nicolas Desprez
Attractor
September 2007 A triangle in three different geometries. The top is a spherical triangle in spherical geometry, the middle shows a hyperbolic triangle in hyperbolic geometry and the bottom is a triangle in Euclidean geometry. Triangle
August 2007 In compass and straightedge constructions an angle can be bisected, divided evenly in to two, using only an unmarked ruler and a compass as seen above. Many tried and failed to trisect a general angle in this way; Gauss proved it impossible. compass and straightedge constructions
July 2007 Partial view of the Mandelbrot set, step 7 of a sequence of pictures showing increasing levels of zoom. Each of the crowns consists of similar "seahorse tails". Mandelbrot set
June 2007 This animation shows a zoom sequence in the fractal known as the Mandelbrot set. Fractals such as this contain an infinite amount of detail. Mandelbrot set
May 2007 It is often suggested that a topologist cannot tell the difference between a coffee cup and a doughnut. This is because these objects when thought of as topological spaces are homeomorphic. The above picture depicts a continuous deformation of a coffee cup into a doughnut such that at each stage the object is homeomorphic to the original. Topology
April 2007 An escape-time fractal, similar to the famous Mandelbrot set, associated with the Collatz conjecture shown near the real axis. Collatz conjecture
March 2007 A 3D projection of a rotating tesseract, the 4D version of the cube, and one of the six convex regular polychora. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions). Tesseract
February 2007 A wireframe model of an icosahedron, one of the five Platonic solids. The icosahedron is the dual of the dodecahedron. Icosahedron
January 2007 The Morin surface is a half-way model of a particular sphere eversion (turning a sphere inside out in 3-space, allowing self-intersection but no creasing). It is named after its discoverer, Bernard Morin. Morin surface

## Featured Pictures from 2006

Date Picture Description Credit Read More
December 2006 The circle map is a chaotic map showing a number of interesting chaotic behaviors. This figure shows the average Poincaré recurrence time for the iterated circle map modulo 1. Circle map
November 2006 A Klein bottle, an example of a surface that is non-orientable — one with no distinction between the "inside" and "outside". Klein bottle
October 2006 A Penrose tiling, an example of a tiling that can completely cover an infinite plane, but only in a pattern which is non-repeating (aperiodic). Penrose tiling
September 2006 This is the method of constructing a golden rectangle with a compass and straightedge. Golden rectangle
August 2006 A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex.
Dodecahedron
June 2006 The tesseract, also known as a hypercube, is the 4-dimensional analog of the cube. That is, the tesseract is to the cube as the cube is to the square Tesseract
April 2006 These are all the connected Dynkin diagrams, which classify the irreducible root systems, which themselves classify simple complex Lie algebras and simple complex Lie groups. These diagrams are therefore fundamental throughout Lie group theory. Dynkin diagram
February 2006 The Lorenz attractor is a non-linear dynamical system derived from the simplified equations of convection rolls in certain atmospheric equations. For a certain set of parameters the system exhibits chaotic behavior and forms what is called a strange attractor. Lorenz attractor
January 2006 A logarithmic spiral is a special kind of spiral curve which often appears in nature. This is a cutaway of a Nautilus shell showing the chambers arranged in an approximately logarithmic spiral. Logarithmic spiral

## Featured Pictures from 2005

Date Picture Description Credit Read More
December 2005 A cuboctahedron is a polyhedron and an Archimedean solid. It is quasi-regular because although its faces are not all identical, its vertices and edges are. It gets its name from the fact that it is both a rectified cube and a rectified octahedron. Cuboctahedron

July 2005 Part of the Mandelbrot set, an example of fractal geometry described by dynamical systems. Mandelbrot set

March 2005 This picture shows the four conic sections: Circles, Ellipses, Parabolas, and Hyperbolas. Conic section
February 2005 This fractal, a Buddhabrot iteration, is believed by many to have a resemblance to the Buddha. The fractal is special rendering of the Mandelbrot set, discovered by Benoît Mandelbrot. Buddhabrot
January 2005 This fractal, one of the most famous fractals in mathematics, is part of the Mandelbrot set, discovered by Benoît Mandelbrot. Mandelbrot set