## ℂhaosScape

### Simple solutions to complex problems.

# Chaos

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 Life

The*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.

#### Mandelbrot Set

The Mandelbrot Set is a fractal—a infinitely complex pattern. There are no animated turns, like in *Life*, but you can magnify areas to peer into the depths of its complexity (up to the limits of JavaScript's numeric precision.)

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 calledcin the complex plane

Letz_{1}bez^{2}+c

Andz_{2}isz_{1}^{2}+c

Andz_{3}isz_{2}^{2}+cand so on…

If the series ofzs will always stay

Close tocand 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 *z*s starting with *z*=0;

z_{1}= 0^{2}+ 1+0i = 1z_{2}= 1^{2}+ 1+0i = 2z_{3}= 2^{2}+ 1+0i = 5z_{4}= 5^{2}+ 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`.

z_{1}= 0^{2}+ -1+0i = -1z_{2}= -1^{2}+ -1+0i = 0z_{3}= 0^{2}+ -1+0i = -1z_{4}= -1^{2}+ -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.