Simple rules can create infinitely complex patterns. The background of ChaosScape.com demonstrates two examples of this. (Click on the triangle at the top right to show or hide this content to see the background better.)
Conway's Game of LifeThe Game of Life is a type of system called cellular automata. Each turn, rules are applied to every cell in a grid. The turns on this page occur about every half a second:
- Any cell with 3 adjacent live neighbors becomes alive.
- Any cell with 2 adjacent live neighbors stay the same.
- A cell with any other number of neighbors dies.
Select a starting pattern from the Seeds drop down list and press the ▶ Play button below to observe its behavior.
- Methuselahs are seeds that continue to change for many generations.
- Oscillators last forever as a repeating sequence.
- Spaceships are oscillators that change position.
You can create your own starting patterns by pausing the turn timer, then clicking on cells. Press the play button to see what your pattern does.
But what are the simple rules of the Mandelbrot set? I believe Jonathon Coulton described it best in his song Mandelbrot Set.
Just take a point called c in the complex plane
Let z1 be z2 + c
And z2 is z12 + c
And z3 is z22 + c and so on…
If the series of zs will always stay
Close to c and never trend away
That point is in the Mandelbrot Set
For example, let's choose our complex point c to be 1+0i.
Now let's iterate zs starting with z=0;
z1 = 02 + 1+0i = 1 z2 = 12 + 1+0i = 2 z3 = 22 + 1+0i = 5 z4 = 52 + 1+0i = 26
It seems that these numbers will just keep getting larger and larger. Clearly not a part of the Mandelbrot set.
Now, let's try c = -1+0i.
z1 = 02 + -1+0i = -1 z2 = -12 + -1+0i = 0 z3 = 02 + -1+0i = -1 z4 = -12 + -1+0i = 0
Instead of escaping to infinity, this point will forever oscillate between -1 and 0. While many values of c are clearly part of the Mandelbrot set or not, the boundaries where they change from one to the other can be chaotic—unpredictable but not uniform.