Fractal's. Mandelbrot set in particular.

Mandelbrot set of fractals. They are quasi fractals, but very famous due to extended graphical technology.

Picture 1
A fractal is basically a geometric shape which can be divided(at least approximately) to a many parts which have the same geometric shape. Fractals, thus exhibit a property of self-similarity, and are bound by a mathematical equation, largely recursive in nature. This means that it is bound by a set of iterations. A fractal shows the following characteristics,

• Its geometric. Its made up of irregular straight lines(or points).
• It is too irregular to be easily described in traditional geometric language.
• It is self-similar (at least approximately or randomly).
• It has a simple and recursive definition.

Many naturally occurring objects are actually fractals(at least in part) like clouds and lightning bolts.

This is a Koch curve, supposed to be a complete fractal. Its actually made from an equilateral triangle.

 To Mandelbrot then. Mandelbrot set is a quasi-fractal set(i.e, shows many fractal properties, but is not exactly self-similar.)  It has a formula, z_n+1 = z_n² + c, where n is substituted by the number. The condition is, a complex number, c, is in the Mandelbrot set if, when starting with z₀ = 0 and applying the iteration repeatedly, the absolute value of z_n never exceeds a certain number (that number depends on c) however large n gets. i is one of the numbers in this set.
Its famous for the aesthetic value, and for the fact that its a relatively simple equation.
http://www.dailymotion.com/video/x3su4b_mandelbrot-fractal-zoom_creation