Parallelohedron

Definition

Let $d \ge 1$ and let $P \subset \R^d$ be a $d$ -dimensional convex polytope. For $t \in \R^d$ write $P + t = \set{x + t : x \in P}$ for the translate of $P$ by $t$ . The polytope $P$ is a parallelohedron if there is a set $T \subset \R^d$ such that:

the translates $P + t$ with $t \in T$ cover $\R^d$ ;
for distinct $s, t \in T$ , the interiors of $P + s$ and $P + t$ are disjoint;
the tiling is face to face: the intersection of any two of the translates is empty or a face of both.

Remarks

Example: the square $P = [0, 1]^2$ with $T = \Z^2$ . The 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. Non-example: an equilateral triangle. All 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 $T$ . The Venkov–McMullen theorem characterizes the convex polytopes that tile $\R^d$ by translations. They 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. Such a polytope then admits a face-to-face tiling in which $T$ is a lattice.

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Convex polytope concept
Translation concept
Parallelohedron definition

Tag	`58MV5BK731`
URI	`m2://58MV5BK731`
Kind	definition
Writing	W0 stub
MSC	52C22, 52B11, 52C07
Defines	Parallelohedron
Uses	Convex polytope, Translation

Parallelohedron

Definition

Remarks

Reading path

Further reading

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