A visual proof that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1

7-d Hypercube

Many people are familiar with what a 3-d cube looks like when it's projected onto 2-d space-- it's that box we all draw. This is what a 2-d representation of a 7-dimensional cube looks like.

The construction of Hilbert's space-filling-curve.

This is probably the most important image in Chaos theory- of the Lorenz attractor. It is an illustration of what many people know as the butterfly effect- the concept that a slight change could result in a hugely different outcome.

The image corresponds to the function in three dimensions of two different variables that differ by a very small amount.

This simply constructed shape known as the Hawaiin Earring causes a lot of trouble in the mathematical field of topology, where it is often a counterexample of many intuitive statements.

Each of these little critters was drawn using a very simple rule repeated over and over again (recursively). Recursion seems to pop up all over nature- whether it's coastlines that look like fractals; sea shells; or broccoli minaret. Richard Dawkins, well known for his criticism of creationism, wrote the code that generated these images. This image is printed in his book "The Blind Watchmaker" to show that as intricate and beautiful as all things in nature are, all that is required is repetition of very simple rules, weakening the intelligent design argument.

Fractals illustrate the magnificence of recursion and infinity.

"Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top."
(http://en.wikipedia.org/wiki/Algebraic_number)

Ordinals

"A graphical “matchstick” representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers."
(http://en.wikipedia.org/wiki/Ordinal_number)