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The cellular automaton (CA) rule that has gained the most popularity by far is Conway's Game of Life. In 1970, English mathematician John Horton Conway, after experimenting with various CA rules, settled on Life, which incorporated analogies to real living organisms—with births, deaths and survival in the natural world.

Live cells (cells in state 1) with too few live neighbors die from isolation, whereas those with too many live neighbors die from suffocation.

The rules are explicitly as follows:

1. Being surrounded by exactly three live neighbors turns a dead cell (state 0) into a live cell.

2. Live cells with exactly two or three live neighbors survive.

3. Otherwise, cells die or remain dead.

Life has inspired an entire category of CA rules referred to as Life-like rules—rules that behave similarly to Life. The convenient naming convention is used to refer to various rules for two-dimensional outer-totalistic CAs. The stands for birth and the stands for survival. This refers to the analogy of cells being dead (0 or white) or alive (1 or black). The and are subsets of digits from 0 to 8 that represent the number of live neighboring cells in a Moore neighborhood that would either birth a cell from being dead or allow a live cell to survive, respectively. For example, Conway's Game of Life is written as in this notation. Any outer-totalistic, two-state, two-dimensional Life-like rule may be simulated with this Demonstration.

References

[1] E. Peña and H. Sayama, "Life Worth Mentioning: Complexity in Life-Like Cellular Automata," Artificial Life, 27(2), 2021 pp. 105–112. doi:10.1162/artl_a_00348.

[2] LifeWiki. "List of Life-Like Cellular Automata." (Apr 19, 2022) conwaylife.com/wiki/List_of_Life-like_cellular_automata.