"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." - Benoit Mandelbrot, introduction to The Fractal Geometry of Nature
Fractals are infinitely complex patterns that are self-similar across different scales and sizes. Fractals are patterns formed from chaotic equations containing self-similar patterns of complexity increasing with magnification. If we divide a fractal pattern into parts you get a nearly identical reduced–size copy of the whole. The mathematical beauty of fractals is that infinite complexity is formed with relatively simple equations. By iterating or repeating fractal generating equations many times, random outputs create beautiful patterns that are unique, yet recognizable. In a mathematical fractal, such as the famous Mandelbrot set, each pattern is made up of smaller copies of itself, and those smaller copies are made up of smaller copies again. They can be generated by a computer calculating a simple equation over and over.
“If I wanted to construct a fractal, I would take a simple rule, like adding three small triangles on the edge of a big triangle. Then, the rule is repeated again and again.”--- Frederi Viens
Self–similarity may be manifested as exact self-similarity which is identical at all scales; e.g. Koch snowflake or Quasi self–similarity which approximates the same pattern at different scales but may contain small copies of the entire fractal in distorted and degenerate forms; Statistical self–similarity, repeating a pattern stochastically; qualitative self–similarity as in a time series and multi fractal scaling. Fractals have a fractional or fractal dimension greater than its topological dimension and Fractals as mathematical equations are no where differentiable.