Magical complexity arising from a simple arithmetic recipe!
Consider an ordinary (x,y) coordinate system:
The "magic circle" is defined as a circle with radius = 2 units (the particular unit is irrelevant -- e.g. it could be inches, or meters, etc.)
We begin at the origin:
By making repeated use of the following four-step arithmetic recipe:
(1) x² = x * x
we compute new values for coordinates x and y .
For example, if a=2 and b=1, then
Notice that if we do step(4) before step(3),
1. A point is computed which lies outside the magic circle.
A Mandelbrot image is constructed as follows.
The pixel at coordinates (a,b) is given a color related
If the computed points never hop outside the magic circle,
The scheme which I use assigns colors which roughly follow
The above example is an extreme magnification
Mandelbrot images constitute a small part of a much more general class of mathematical objects known as fractals. By going from two to three dimensions, the variety of patterns which can be generated becomes truly outstanding. Fractal patterns seem to mirror real life in all its complexity. Here are some examples.