CellularAutomaton

CellularAutomaton[rule,init,t] generates a list representing the evolution of the cellular automaton with the specified rule from initial condition init for t steps.

CellularAutomaton[rule,init] gives the result of evolving init for one step.

CellularAutomaton[rule,init,{tspec,xspec,…}] gives only those parts of the evolution specified by tspec, xspec, etc.

CellularAutomaton[rule,init,{t,All,…}] includes at each step all cells that could be affected over the course of t steps.

CellularAutomaton[rule] is an operator form of CellularAutomaton that represents one step of evolution.

Examples

Evolve Rule 30 for 50 steps:

ArrayPlot[CellularAutomaton[30, {{1}, 0}, 50]]

One step of Rule 110:

CellularAutomaton[110, {0, 1, 1, 0, 1}]
(* {1, 1, 0, 1, 1} *)

Please visit the official Wolfram Language Reference for more details.