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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\R^n$ be [[Euclidean space|0XPR1E85W0]].\nA *tessellation* or *tiling* of $\\R^n$ is a countable family $\\mathcal{T} = \\set{T_1, T_2, \\dots}$ of subsets of $\\R^n$, called *tiles*, such that:\n\n1. each tile $T_i$ is a [[closed set|0ZCK8DQYN1]] with nonempty [[interior|0RZH6W98HK]];\n2. the union of all tiles is $\\R^n$;\n3. the interiors of two distinct tiles are disjoint.\n\n## Remarks\n\nExample: the squares $T_{a,b} = [a, a+1] \\times [b, b+1]$ with $(a, b) \\in \\Z^2$ form a tiling of $\\R^2$.\nEach square is closed with nonempty interior, every point $(x, y)$ lies in $T_{\\floor{x}, \\floor{y}}$, and two distinct open squares $(a, a+1) \\times (b, b+1)$ do not meet.\n\nNon-example: the closed unit disks centred at the points of $\\Z^2$ cover $\\R^2$, but the disks centred at $(0, 0)$ and $(1, 0)$ have overlapping interiors, so clause 3 fails.\n\nMany authors also require each tile to be homeomorphic to a closed ball, so that tiles are \"topological disks\" in the plane.\nA *prototile* set of a tiling is a set of shapes such that each tile is congruent to one of them.\nIn dimension $3$ and higher, tilings by polyhedra are also called [[honeycombs|HTJ03RFKFH]].\n",
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