"Koch Curve 85degrees: Generalizing the von Koch curve with an angle a chosen between 0 and 90°. The fractal dimension is then


http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Irreducible fractions come up all the time in math problems, but who knew they looked so beautiful all together?
This picture, from Wolfram, is of irreducible fractions in the Complex plane.

These pictures, drawn by a harmonograph, may better be described as pictures of music. A harmonograph is a mechanical device that uses pendulums that move pen and paper simultaneously to create these images. The device was invented by a mathematician in 1844 and apparently was a popular form of entertainment at Victorian soirees.
The images vary depending on how the pendulums swing in relation to one another. Harmonic ratios used in music such as 3:2, 4:3 give the images below.

These beautiful origami pieces were constructed by mathematicians Erik and Martin Demaine (first image) and Thomas Hull (second image) to investigate the surfaces that result from different kinds of pleated folding.

"Map the infinite checkerboard to the plane so that black squares map to squares and white squares map to quadrilaterals. Such a map is a disguised version of a discrete analytic function. This is a discrete approximation to the exponential map f(z)=e^z."
source: http://www.math.brown.edu/~rkenyon/gallery/gallery.html

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)