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Game of Life

The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is the best-known example of a cellular automaton. more...

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The "game" is actually a zero-player game, meaning that its evolution is determined by its initial state, needing no input from human players. One interacts with the Game of Life by creating an initial configuration and observing how it evolves.

Rules

The universe of the Game of Life is an infinite two-dimensional square grid of square cells, each of which is in exactly one of two possible states, live or dead. Cells interact with their eight neighbours, which are the cells that are directly horizontally, vertically, or diagonally adjacent. At each step in time, the following effects occur:

Any live cell with fewer than two live neighbours dies, as if by loneliness.; Any live cell with more than three live neighbours dies, as if by overcrowding.; Any live cell with two or three live neighbours lives, unchanged, to the next generation.; Any dead cell with exactly three live neighbours comes to life.;

The initial pattern constitutes the first generation of the system. The second generation is created by applying the above rules simultaneously to every cell in the first generation -- births and deaths happen simultaneously, and the discrete moment at which this happens is sometimes called a tick. The rules continue to be applied repeatedly to create further generations.

Origins

Conway was interested in a problem presented in the 1940s by renowned mathematician John von Neumann, who tried to find a hypothetical machine that could build copies of itself and succeeded when he found a mathematical model for such a machine with very complicated rules on a Cartesian grid. Conway tried to simplify von Neumann's ideas and eventually succeeded. By coupling his previous success with Leech's problem in group theory with his interest in von Neumann's ideas concerning self-replicating machines, Conway devised the Game of Life.

It made its first public appearance in the October 1970 issue of Scientific American, in Martin Gardner's "Mathematical Games" column. From a theoretical point of view, it is interesting because it has the power of a universal Turing machine: that is, anything that can be computed algorithmically can be computed within Conway's Game of Life. Gardner wrote:

"The game made Conway instantly famous, but it also opened up a whole new field of mathematical research, the field of cellular automata (...) Because of Life's analogies with the rise, fall and alterations of a society of living organisms, it belongs to a growing class of what are called "simulation games" - games that resemble real-life processes."

Ever since its publication, it has attracted much interest because of the surprising ways the patterns can evolve. Life is an example of emergence and self-organization. It is interesting for physicists, biologists, economists, mathematicians, philosophers, generative scientists and others to observe the way that complex patterns can emerge from the implementation of very simple rules.