Conway's Game of Life
Conway's Game of Life is a two-dimensional cellular automaton played on a square grid. Time advances in discrete generations. Each cell is either alive or dead. The next state of every cell is determined simultaneously from its eight nearest neighbors using Conway's rules: a dead cell with exactly three live neighbors becomes alive (birth); a live cell survives only if it has two or three live neighbors, otherwise it dies (underpopulation or overcrowding). This rule notation is often written B3/S23. The model is deterministic and requires no randomness once an initial pattern is set, yet it can produce remarkably complex, long-lived, and unpredictable behavior from simple seeds—a standard classroom example of emergence. This simulator uses a toroidal topology: the top edge wraps to the bottom and the left to the right, so gliders and spaceships re-enter the field instead of vanishing at a hard boundary. Users can paint or erase cells with the mouse, run continuous evolution at an adjustable rate, advance one generation at a time, clear the board, randomize with a chosen density, or stamp famous patterns such as the glider, lightweight spaceship (LWSS), Gosper glider gun, pulsar, small oscillators, the acorn methuselah, and the R-pentomino. The implementation is a straightforward neighbor count per cell; it omits infinite grids, alternative rules, and pattern libraries beyond the built-in presets.
Who it's for: Middle school through undergraduate learners exploring discrete models, emergence, and recreational mathematics; anyone comparing finite automata to continuous differential-equation models.
Key terms
- Cellular automaton
- Conway's Game of Life
- B3/S23
- Generation
- Glider
- Spaceship
- Toroidal boundary
- Emergence
How it works
John Conway’s Game of Life: a dead cell with exactly three live neighbors is born; a live cell survives only with two or three neighbors (B3/S23). This page uses a toroidal grid so gliders wrap around. Paint your own seeds or load classic patterns including the Gosper glider gun.
Key equations
Frequently asked questions
- Why are the edges wrapped (torus) instead of dead cells outside?
- A torus keeps compact patterns like gliders and spaceships in view forever, which is convenient on a small finite grid. A fixed dead border is another common choice and changes behavior near the edge; both are legitimate finite approximations of an infinite plane.
- Is the Game of Life "chaos" in the technical sense?
- It is deterministic and governed by simple local rules, but many configurations exhibit sensitive dependence on small changes and extremely long transients—behaviors often discussed alongside chaos in popular explanations. It is not a continuous dynamical system with Lyapunov exponents in the usual ODE sense.
- What is the Gosper glider gun?
- It is a small stable pattern that periodically emits gliders. Its discovery proved that Life can sustain unbounded growth from a finite initial pattern, answering an early open question about whether all patterns eventually die out.
- Why does random fill sometimes settle quickly and sometimes look chaotic?
- Above a critical density, random fields tend to leave fragmented activity; below it, they often decay to still lifes and oscillators. The exact outcome depends on density, grid size, topology, and luck—another illustration of emergent behavior from local rules.
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