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  "body": "\n## Definition\n\nLet $A$ be a finite set of states and let $d \\ge 1$.\nA configuration is an element $x = (x_u)_{u \\in \\Z^d}$ of $A^{\\Z^d}$.\nLet $N \\subseteq \\Z^d$ be a finite set, called the neighbourhood, and let $f \\colon A^N \\to A$ be a map, called the local rule.\nThe *cellular automaton* with neighbourhood $N$ and local rule $f$ is the map $F \\colon A^{\\Z^d} \\to A^{\\Z^d}$ given by\n$$F(x)_u = f\\big((x_{u + n})_{n \\in N}\\big) \\quad \\text{for all } u \\in \\Z^d.$$\n\n## Remarks\n\nExample: rule 90 has $d = 1$, $A = \\set{0, 1}$, $N = \\set{-1, 1}$ and $f(a, b) = a + b \\bmod 2$.\nSo $F(x)_i = x_{i-1} + x_{i+1} \\bmod 2$, and the value at $i$ depends only on the sites $i \\pm 1$, as required.\nNon-example: the map $F(x)_i = x_{2i}$ on $\\set{0,1}^{\\Z}$ is not a cellular automaton.\nThe value at $i$ reads the site $2i$, so no fixed finite set $N$ of offsets works for all $i$.\nEquivalently, it does not commute with the shift, which the [[Curtis–Hedlund–Lyndon theorem|N35QM9G83G]] requires.\nThe dynamics is the iteration of $F$, with $F^t(x)$ the configuration at time $t$.\nIn one dimension with $A = \\set{0,1}$ and $N = \\set{-1, 0, 1}$, the $2^8 = 256$ local rules are the elementary cellular automata, numbered by Wolfram's convention.\n",
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