Stability and chaos are analyzed by computing the Lyaponuv exponent which is plotted above.

A generalization of the Cornu spiral, this is an example of a polynomial spiral described by Dillen (1990) for which the curvature is a polynomial function of the arc length.

Klein's j-invariant in the complex plane

A plot of z = |W(x+iy)|, where W is the Lambert W function

Icosahedron by Richard Sweeney

Point collections by Zach Kron

Zach Kron writes an interesting post about this image, which is created by using several layers of a very basic shape repeated in a pattern and then varying the size and angles, and seems to show up a lot on ceilings and walls of buildings.

"Spiral waves in colonies of the mold Dictyostelium discoideum. These patterns are created when the mold forms aggregates in response to some external 'stress', such as a lack of moisture or nutrients." (BALL, 1994)

An interesting new look at the Menger sponge: slicing the sponge at an angle produces six-sided stars. One of the proposed exhibits for the Museum of Mathematics, MOMATH.

Using different values of a and b, the equations
x(t) = sin(t/2) - a*sin(bt)cos(t)
y(t) = cos(t/2) - a*sin(bt)sin(t)
produce images such as the ones above.

By Herbert Franke

Triply Orthogonal system by Daniel Piker

By Daniel Piker

Conformal circle packing by Daniel Piker

Bipolar coordinates

Newton Fractal

A moiré pattern is an interference pattern created by overlaying two or more grids at an angle. The moiré pattern above is by Jacob Yerex.

The Dedekind Tessellation: a tessellation of the half-plane in hyperbolic triangles.

The fourth stellation of an icosahedron

Above are Cayley graphs of a finite Coxeter group on four generators embedded into the 3-sphere and then projected onto Euclidean space. If we could see these images in the hypersphere, the curved lines would be straight. More here.

Particles colliding in a particle accelerator

A closed meander of order 120: a closed curve that intersects a line at 120 points, without intersecting itself.

The Real part of z^z^z

A polyhedron constructed by folding along the edges of a Spidron - a figure in the plane made up of two alternating sequences of isosceles triangles.

A plot of the complex valued function (z-2)²(z+1-2i)(z+2+2i)/z³ by Hans Lundmark

Penrose tiling

Plot of the equation Sin(xSin(y)) = Cos(yCos(x)) by Xah Lee

The scintillating grid illusion

Real part of the j-invariant as a function of the nome q on the unit disk

Mobius strip and knot by Rob Scharein (from The Mobius Strip by Clifford Pickover)

A strange attractor by Nathan Selikoff resembling the Eagle Nebula.

Part of a tessellation of the plane generated using Möbius transforms. By Doug Hensley.

Some examples of Chladni patterns

A rhombic triacontahedron and its net

This is a Klein diagram by Francisco Lara-Dammer that represents A5, the group of symmetries of the icosahedron.

A strange attractor by Nathan Selikoff.

Impossible Geometry by Jos Leys

A pattern showing the result of six levels of recursion using the "inflation rules" of Penrose tiles. By L. Kerry Mitchell.

Kürschák's Dodecagon

A visual proof by Kürschák that a regular 12-sided polygon inscribed in the unit circle has area 3. This means that estimating π to be 3 is like saying a circle is a Dodecagon.

"The Easy Way Up" from Impossible Geometry by Jos Leys

A Mobius-like object with holes by Tom Longtin and Rinus Roelofs.

The interactions between sub-atomic particles at the particle accelerator at CERN in Geneva.

Circle packings

Circle packing is a study of arranging a maximal number of different sized circles (or spheres) so that none overlap. In the picture above, circles of the same size have the same color. See more here.

A beautiful spot in the 3-d Mandelbrot set. To see more check out this gallery.

Conchoids of a 5 cusped epicycloid with pole at center.

Boundary formed by trochoids of different parameters.

Parallels of the Witch of Agnesi by Xah Lee

A fractal dodecahedral subdivision of the sphere

The colors represent the amplitude of the ground's horizontal motion 6 seconds into a 4.4-magnitude earthquake. This image produced on a supercomputer in Pittsburgh is one of several such simulations whose accuracy provides architects with localized information on the kind of intensity buildings may have to withstand.
This is one of many beautiful supercomputer simulations compiled by Discover.

An Apollonian gasket