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  "body": "\n## Definition\n\nLet $\\mathcal{T}$ be a [[tiling|GKJK293DXS]] of $\\R^n$.\nFor $v \\in \\R^n$, let $\\mathcal{T} + v = \\set{T + v : T \\in \\mathcal{T}}$ be the image of $\\mathcal{T}$ under the [[translation|087VJM5Q77]] $x \\mapsto x + v$.\nA vector $v$ is a *period* of $\\mathcal{T}$ if $\\mathcal{T} + v = \\mathcal{T}$.\nThe tiling $\\mathcal{T}$ is *periodic* if it has $n$ linearly independent periods.\n\n## Remarks\n\nExample: the tiling of $\\R^2$ by the unit squares $[a, a+1] \\times [b, b+1]$, $(a, b) \\in \\Z^2$, is periodic.\nTranslation by $(1, 0)$ maps $[a, a+1] \\times [b, b+1]$ to $[a+1, a+2] \\times [b, b+1]$, so $(1, 0)$ is a period, and in the same way $(0, 1)$ is a period; these two vectors are independent.\n\nNon-example: shift the row of squares $[a, a+1] \\times [0, 1]$ by $1/2$ to the right and leave all other rows as they are.\nThe vector $(1, 0)$ is still a period, but no vector with a nonzero second coordinate is one, because the shifted row is the only row with vertices at half-integers.\nSo the tiling has periods in only one direction and is not periodic.\n\nA tiling with no nonzero period at all is called *non-periodic*.\nThe [[Penrose tilings|MBE83FTRBG]] are non-periodic.\n",
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