"The mathematics of fractals is amazingly simple because you need only one equation, using only simple multiplication and addition. the same equation is then repeated ad infinitum. For example the "Mandelbrot set" is based on the simple formula of taking a number, multiplying it by itself and then adding the original number. The result of that equation is then use as the input fro the next equation and so on. The challenge is that even though each equation follows the same formula, these equations must be repeated millions of times to actually visualise a fractal patter. The manual labor and time needed to complete millions of equations prevented early mathematicians from recognizing the value of fractal geometry. With the advent of powerful computers Mandelbrot was able to define this new math.
Inherent in the geometry of fractals is the creation of ever repeating, "self-similar" patterns nested within one another...Fractal geometry emphasises the relationship between the patterns in a whole structure and the patterns seen in parts of a structure. For example, the pattern of twigs on a branch resembles the pattern of limbs branching off the trunk."