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
becomes

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
becomes

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

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.

7-Life

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.