The Mandelbrot Set
Fractals - The Colours of Infinity is an excellent documentary on fractal geometry with particular regard to the Mandelbrot Set which is generated by iterating: fc(z) <-> z^2 + c. More can be learned from Wikipedia. The afore linked documentary is narrated by science fiction author Arthur C. Clarke. It features colour cycling of the Mandelbrot Set which result in stunningly beautiful visualisations of the infinite complexity and repetitive order inherent to this particular fractal at any magnification. Indeed, one begins to appreciate that the geometry of the natural world is seemingly a special case of fractal geometry.