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  "body": "\n## Example\n\nA Wang tile is a unit square whose four edges carry colours from a finite set.\nFix a finite set $W$ of Wang tiles.\nA tiling by $W$ is a map $\\tau \\colon \\Z^2 \\to W$ such that adjacent edges match.\nThat is, for each $(i, j)$, the right edge of $\\tau(i, j)$ has the colour of the left edge of $\\tau(i+1, j)$.\nAlso the top edge of $\\tau(i, j)$ has the colour of the bottom edge of $\\tau(i, j+1)$.\nTiles may not be rotated or reflected.\n\nThe set of all tilings by $W$ is a two-dimensional [[subshift of finite type|624XDXX8FN]] in $W^{\\Z^2}$.\nIndeed, its forbidden patterns are the pairs of adjacent tiles whose common edge has two different colours.\nThere are finitely many of them.\n\nCheck of a small case: let $W$ consist of one tile whose four edges all have the colour $a$.\nAll matching conditions hold, so the constant map is a tiling, and it is periodic.\nNow let $W$ consist of one tile with left edge $a$ and right edge $b \\ne a$.\nTwo such tiles can never sit side by side, so $W$ admits no tiling.\n\n## Remarks\n\nThe domino problem asks for an algorithm that decides, given $W$, whether $W$ tiles the plane.\nBerger proved that no such algorithm exists, by simulating a [[Turing machine|W6ESDYBF8Z]] with Wang tiles and reducing to the [[halting problem|Z7TN2B35B0]].\nA key step is the existence of an aperiodic set of Wang tiles: a set that tiles the plane, but only in non-periodic ways.\nJeandel and Rao found an aperiodic set of $11$ Wang tiles and showed that no set with fewer tiles is aperiodic.\n\n## Explanations\n\n### L1 human\n\nWang tiles are square dominoes with coloured edges, and neighbouring edges must have the same colour.\nSome sets of such tiles cover the plane in a repeating pattern, some cannot cover it at all.\nSurprisingly, some sets cover the plane but never in a repeating way.\nBecause of this, no computer program can decide for every set of tiles whether it covers the plane.\n",
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