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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $S$ be a finite set of subsets of $\\R^n$, called *prototiles*, possibly together with matching rules on how they may meet.\nA tiling of $\\R^n$ *by* $S$ is a [[tiling|GKJK293DXS]] in which every tile is congruent to a member of $S$ and the matching rules hold.\nThe set $S$ is *aperiodic* if:\n\n1. there is a tiling of $\\R^n$ by $S$;\n2. no tiling of $\\R^n$ by $S$ is [[periodic|WFRGSBFZGV]].\n\n## Remarks\n\nExample: the two rhombi of the [[Penrose tiling|MBE83FTRBG]], with their matching rules, form an aperiodic set of prototiles in $\\R^2$.\n\nNon-example: the set $S = \\set{[0,1]^2}$ consisting of the unit square is not aperiodic.\nClause 1 holds, but the standard grid of unit squares is a periodic tiling by $S$, so clause 2 fails.\n\nA tiling with no nonzero period is called *non-periodic*.\nA set of prototiles is aperiodic exactly when it admits tilings and all of them are non-periodic.\nThe first aperiodic set, found by Robert Berger in 1966, consisted of a very large number of square tiles with coloured edges.\n",
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