I have always been fascinated by cellular automata, specifically the brand governed by the rules devised by the mathematician John Horton Conway, which is known as the Game of Life.
What I find particularly fascinating is the fact that such simple rules give rise to such complex behaviors. The Game is played this way:
There exists a grid of squares, each of which can be in one of two states: ON or OFF. The rules for changing states are simple and depend only on the states of the square’s 8 nearest neighbors.
- An ON cell with fewer than two ON neighbours turns OFF.
- An ON cell with more than three ON neighbours turns OFF.
- An OFF cell with exactly three ON neighbours turns ON.
One can think of the first rule as stating that the cell dies from loneliness. In the second rule, the cell dies from overcrowding. To come to life, you need exactly three neighbors. Here is a link to the original Scientific American article from 1970.
Since each cell has eight possible neighbors, but can only stay on if it has one or two ON neighbors, the playing field will be rather sparse. The situation is rather unstable since three ON neighbors can turn ON a cell, but this increases the chances that another ON cell will now have more than three ON neighbors, which would make it turn OFF. In that event, life would lead to death. In a generic sparse environment, the tendency is to grow, but this leads to a crowded environment where the tendency is to diminish. The general result is that the playing field will be mostly empty with pockets of ON cells evolving dynamically.
This game is an excellent example of how simple rules can lead to extremely complex behavior.
To play, you can check out an online applet, or download your own superfast game from Golly Game of Life. The latter is very nice as you can choose from a host of fascinating patterns to load and play with.
One can go to Stephen Silver’s Life Lexicon to see a library (or bestiary) of most of the known creatures.
Above is an emblem I created in Mathematica that is a period-29 oscillator. That is, the pattern repeats every 29 steps. I had previously used it on a website for a course on complex systems that I taught at CUNY in 1998.
Kevin Knuth
Albany NY
Posted under Complexity, Fun
This post was written by drknuth on April 13, 2008
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