Gosper's Glider Gun creating "gliders". (Animated GIF)

The evolution and movement of a glider. (Animated GIF)

A sample of a 48-step oscillator from a 2D hexagonal Game of Life (rule 34/2). (Animated GIF)

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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. A variant exists where two players compete.

Rules

The universe of the Game of Life is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, live or dead. Every cell interacts with its eight neighbours, which are the cells that are directly horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions 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 'seed' of the system. The first generation is created by applying the above rules simultaneously to every cell in the seed—births and deaths happen simultaneously, and the discrete moment at which this happens is sometimes called a tick. (In other words, each generation is a pure function of the one before.) 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 rectangular 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:

Ever since its publication, Conway's Game of Life has attracted much interest because of the surprising ways in which 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. The game can also serve as a didactic analogy, used to convey the somewhat counterintuitive notion that \"design\" and \"organization\" can spontaneously emerge in the absence of a designer. For example, philosopher and cognitive scientist Daniel C. Dennett has used the analog of Conway's Life \"universe\" extensively to illustrate the possible evolution of complex philosophical constructs, such as consciousness and free will, from the relatively simple set of deterministic physical laws governing our own universe.