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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $P \\subset \\R^d$ be a $d$-dimensional [[convex polytope|300K6CB1KH]].\nFor $t \\in \\R^d$ write $P + t = \\set{x + t : x \\in P}$ for the [[translate|087VJM5Q77]] of $P$ by $t$.\nThe polytope $P$ is a *parallelohedron* if there is a set $T \\subset \\R^d$ such that:\n\n1. the translates $P + t$ with $t \\in T$ cover $\\R^d$;\n2. for distinct $s, t \\in T$, the interiors of $P + s$ and $P + t$ are disjoint;\n3. the tiling is face to face: the intersection of any two of the translates is empty or a face of both.\n\n## Remarks\n\nExample: the square $P = [0, 1]^2$ with $T = \\Z^2$.\nThe squares $[a, a + 1] \\times [b, b + 1]$ cover the plane, two distinct ones have disjoint interiors, and two of them meet in a common edge, a common vertex, or not at all.\nNon-example: an equilateral triangle.\nAll its translates point the same way, and an upward triangle cannot be surrounded by upward triangles without gaps, so clause 1 or clause 2 fails for every $T$.\nThe Venkov–McMullen theorem characterizes the convex polytopes that tile $\\R^d$ by translations.\nThey are those that are centrally symmetric, with centrally symmetric facets, and whose belts of facets parallel to a $(d-2)$-face have $4$ or $6$ facets.\nSuch a polytope then admits a face-to-face tiling in which $T$ is a lattice.\n",
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