Geometric patterns on the ceilings of European cathedrals. Photos by David Stephenson.

Video by NASA simulating the evolution of the moon

Hand-cut paper sculptures by artist Jen Stark

“Prismatica consists of an arrangement of pyramid-shaped crystals affixed to an LCD screen and illuminated with programmed geometric animation. The animated patterns are precisely mapped to the vertices of the crystals, illuminating them individually and in formation. The animations are further refracted through the geometry of the crystals in accordance with the shifting perspective of the observer, which in turn alters the way the illuminations appear and interact with reflections of surrounding lights within the space.”

"Coloured smoke reveals the amazing vortex airflow patterns off the wings of airplanes."

Algorithmic Art

Fourier Transform

The fractal surface of a strange attractor

This fractal consists in a combination (using complex division) of two Julia II fractals.

Conformal maps by Robert Bunney

A random tiling on the Aztec diamond.

"Four Sierpinski triangles interweave in three dimensions, each linked with, but not touching, the other three." By George Hart.

"The strange attractor of the driven pendulum at the parameter values γ= 0.052, a= 0.586, ω= 0.666. The direction of view is diagonal in the state space. The red sections are Poincaré sections at ψ= 0 (in front on the right side) and at x= -π (in front on the left side). The vertical axis is the angular velocity." By Robert Doerner.

Ducks Fractals Variation by Softology.

Sucessive generations of Asmus Schmidt’s complex continued fraction method.

Kaleidoscopic IFS Fractal
By Softology.

One Dimensional Totalistic Cellular Automaton. By Softology.

Mandelbulb

Ducks Fractals variations

Phase portrait and time series of the Moore-Spiegel Attractor

The Henon strange attractor

Two rules

This image is the result of two rules.

Inspired by Chladni patterns, Hans Jenny pioneered the field of Cymatics, a study of visible sound vibration. Tones played at fixed frequencies cause a vibration that creates patterns from light on water.

A close up of 4 of the hundredth roots of unity. By Ian Sammis.

Four elementary operation tables arranged in a pinwheel format (read more here).
By Magaret Kepner

Variations of the Mandelbox, a fractal discovered by Tom Lowe in 2010 by applying recursive spherical folding transformations to generate fractals.

An Escher-inspired painting on a sphere. By Dick Termes.

The following sextic was found by W. Barth. It has 65 ordinary double points, the maximal possible number.

Hyperbolic PDEs on the surface of a sphere
by James A. Rossmanith

Surface with constant Gaussian curvature

A 5-dimensional analogue of Rubik's cube

The Weaire-Phelan foam is a counterexample to Kelvin's conjecture
about the best partition of space into equal-volume cells.
By John M. Sullivan.

Two overlapping grids of reflective tubes produce the above Moiré pattern by Koert Feenstra.

A polyhedron made of a 7/4-Cuploid, 7 triangles, 7 squares, 1 7/3-heptagram

Inverse Julia Set calculation for √(z - c)/ √(z - d) on the (x,z) plane, extruded along the y-axis.

This surface by Xah Lee, which he calls the Borg Cube, is given by the equation
sin(xy) + sin(yz) + sin(zx) = 0

Spherical version of the 2.5 dimensional Julia Set

The basic inverse quaternion julia set.

Ikeda Attractor by Paul Bourke

This image was created by Jonathan McCabe from a cellular automata program he wrote.

"Each pixel represents the state of the 4 cells of 4 cellular automata, which are cross coupled and have their individual state transition tables. There is a "history" or "memory" of the previous states which is used as an offset into the state transition tables, resulting in update rules which depend on what has happened at that pixel in previous generations. Different regions end up in a particular state or cycle of states, and act very much like immiscible liquids with surface tension."

Circles generated by Mobius transform by Ed Pegg Jr.

3D totalistic cellular automata

John Conway's Game of Life is a cellular automaton in which randomized cells evolve according to a fixed set of rules. Usually this evolution appears as an animation, here we see the evolution through time in a still image, with height from top to bottom corresponding to subsequent generations of cells.

Quaternion Julia Fractals

"Girih tiles have interior angles that are multiples of π/5. In the examples shown here I’ve applied the girih concept to polygons with angles that are multiples of π/7."

By Joe Bartholomew

A 3-d model of tuna, anchovy and bluefish populations given certain parameters.

The Catalan numbers are a sequence of numbers, much like the Fibonacci numbers, which are given by the equation

Like the Fibonacci numbers, they too pop up all over the place, for example, the Catalan numbers correspond to the number of ways a regular n-gon can be divided into n-2 triangles.
Above is a visualization of the Catalan numbers.

By John C. Hart in 1989
Read more about quaternion fractals (and see more images) here and here.

A map of the internet by Barrett Lyon

Snowflake

Snowflake hitting the ground

Photos by Andrew Osokin using a macro lens.

A rather interesting link by artist Matt W. Moore