Portal:Mathematics/Featured picture archive
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This page is an archive of pictures featured on the Mathematics Portal. For mathematics pictures featured elsewhere on Wikipedia see Wikipedia:Featured pictures#Mathematics.
Newest pictures at the top
[edit] 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.
[edit] May, 2009
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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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] January, 2009
In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. The fifth postulate or 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 (see 1).
In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example elliptical geometry:
- Given a line and a point not on that line, all lines drawn through that point will intersect the original line (see 2).
And 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 (see 3).
These other forms of geometry, where the parallel postulate does not hold are called Non-Euclidean geometry.
[edit] December, 2008
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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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] August, 2008
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| Credit: Inductiveload | |
The exponential function is one of the most important functions in mathematics. It can be written in the form ex, where e is a mathematical constant, sometimes known as Euler's number.
The exponential function exist for any complex number, above to the left is the real part and to the right the imaginary part of various values in the complex plane.
[edit] 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 noticable in the picture is all lines meet at exactly one point (ie there are no parallel lines).
[edit] 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 a non-zero, finite surface area and zero volume.
[edit] 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 
versus 
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.
[edit] 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. A common example is the intelligence quotient (IQ), seen above.
[edit] March, 2008
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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°.
[edit] 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.
[edit] January, 2008
In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of 
can change drastically under arbitrarily small perturbations. Above is a 3D slice of a 4D Julia set.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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 Euclidian geometry.
[edit] August, 2007
In compass and straightedge constructions an angle can be bisected, divided evenly into two, using only an unmarked ruler and a compass as seen above. Many tried and failed to trisect a general angle; Gauss proved it impossible.
[edit] 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".
[edit] 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.
[edit] 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.
[edit] April, 2007
An escape-time fractal, similar to the famous Mandelbrot set, associated with the Collatz conjecture shown near the real axis.
[edit] 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).
[edit] February, 2007
A wireframe model of an icosahedron, one of the five Platonic solids. The icosahedron is the dual of the dodecahedron.
[edit] 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.
[edit] 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.
[edit] November 08, 2006
A Klein bottle, an example of a surface that is non-orientable — one with no distinction between the "inside" and "outside".
[edit] October 22, 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).
[edit] August 24, 2006
This is the method of constructing a golden rectangle with a compass and straightedge.
[edit] August 10, 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.
[edit] May 31, 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
[edit] March 25, 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.
[edit] February 2, 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.
[edit] January 4, 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.
[edit] December 15, 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.
[edit] July 17, 2005
Part of the Mandelbrot set, an example of fractal geometry described by dynamical systems.
[edit] March 2, 2005
This picture shows the four conic sections: Circles, Ellipses, Parabolas, and Hyperbolas.
[edit] February 12, 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.
[edit] February 10, 2005
This fractal, one of the most famous fractals in mathematics, is part of the Mandelbrot set, discovered by Benoît Mandelbrot.
