Cellulare automata in two dimensions

The Game of Life

The classic example of a CA in two dimensions is the so called Game of Life, invented by John Horton Conway in the 1960ies as what he called recreational mathematics. Similar to a CA in one dimension, the Game of Life consists of a grid with cells that in dependence of their neighboring cells can take on one of two states, usually either white or black. In this case however, the set of neighboring cells consists of the eight adjacent cells surrounding a cell in two dimensions. This neighborhood is called Moore-neighborhood (see below). The game starts out with an initial (small) constellation of black cells that comply with the following rule-set:

1. If a white cell has exactly three black cells in its Moore-neighborhood, it becomes black (it becomes “alive”)

According to this rule, the cell-constellation triple becomes square

2. If a black cell has less than two or more than three black cells in its Moore-neighborhood, it becomes white (it “dies”)

According to this rule, the cell-constellation horizontal becomes vertical

When iterated this rule set generates interesting cell constellations, as can be tested with the following interactive model:

f-Pentomino Glider U Glider gun Random

A short way to determine the Game of Life is the formula B3/S23, meaning that a black square in the grid is born (B) when it has 3 neighbors, and it survives (S) when it has not less than 2 and not more than 3 neighbors. The interactive model above allows testing all possibilities of Bij/Skl.


A nice variation of the Game of Life is called 7-Life. It conforms to the rule B37/S23. This game, when started with a particular cell configuration (as depicted in the left-most picture below), has the interesting property of generating a cluster of interacting cell-cultures which live on for a long time without showing particular structure. Then after about 45000 iterations suddenly a highly ordered extension emerges on the right side of the cluster, which keeps growing as long as it remains uninterrupted (right picture below).


Although the process is still absolutely deterministic, one could be inclined to call this an example for order from noise.