Game of Life

John Conway invented a game in the late 60's that shows some features which are very life-like. The game is an example of a cellular automaton: A simulation run on a two-dimensional lattice of cells each of which interacts with its eight neighbours, Fig.(6.1), via a few simple rules. (Indeed one of the exercises in the last chapter required you to investigate a cellular automaton model of the BZ chemical reaction.)

The game is run in discrete time-steps, and with the cells of the lattice taking one of two states: alive or dead. At each time-step the state of a cell is determined according to the following two rules:

1. A living cell will remain alive only if it has two or three living neighbours. It dies from exposure or loneliness if it has less than two neighbours, and from overcrowding if it has more than three neighbours.
2. A dead (or vacant) cell can come alive if it is surrounded be exactly three live cells. (One can think of this as reproduction)

Though the above rules are a caricature of ecosystems, they lead to surprising and sometimes complex pattern formation. Indeed this game is a beautiful exemplification of emergent behaviour, as the reader will apprecaite once (s)he plays it (see the exercises). Starting with a given configuration of live cells, many possible outcomes can result. Three simple types of objects are : static, periodic and moving, as shown in the figures below. More interesting objects are those that breed, mimicking real life, as best seen in the simulations.





Note the large variety of patterns that are possible from different initial conditions eventhough the rules for "evolution" are few, local and completely deterministic.