# Conway's Game of Life

**Type:** Concept  
**Domain:** Recreational Mathematics  
**Invented:** 1970, John Conway  
**Codex URL:** /mathematics/recreational/conways-game-of-life/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Cellular automaton with 4 simple rules producing Turing-complete complex behaviour. Classic example of emergence.

## The 4 rules
1. Living cell, <2 living neighbours → dies (loneliness)
2. Living cell, 2-3 living neighbours → survives
3. Living cell, >3 living neighbours → dies (overcrowding)
4. Dead cell, exactly 3 living neighbours → becomes alive (reproduction)

## Classic patterns
- Still lifes (block) — never change
- Oscillators (blinker) — cycle
- Spaceships (glider) — translate across grid
- Guns (Gosper glider gun, 1970) — emit gliders forever
- Methuselahs (R-pentomino) — small pattern, long chaotic evolution (1103 generations)

## Turing completeness
Rendell and others built working logic gates and full Turing machines within Life — proven Turing complete. Consequence: general "does this pattern die out?" question is undecidable (Halting Problem analogue).

## Wolfram's classification
4 classes of cellular automaton behavior: uniform, periodic, chaotic, complex/universal. Life is in the 4th class.

## Sources
### Tier 1
- Berlekamp, E.R., Conway, J.H. and Guy, R.K. (2004). *Winning Ways*. Vol. 4, 2nd ed.
- Rendell, P. (2011). A Turing machine in Conway's Game of Life. Springer.
### Tier 2
- Gardner, M. (1970). Mathematical Games. *Scientific American*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*