---
tag: 58MV5BK731
kind: definition
name: Parallelohedron
msc: [52C22, 52B11, 52C07]
defines: MR2JWBCZQZ
uses: [300K6CB1KH, 087VJM5Q77]
sources:
  - url: https://en.wikipedia.org/wiki/Parallelohedron
---

## Definition

Let $d \ge 1$ and let $P \subset \R^d$ be a $d$-dimensional [[convex polytope|300K6CB1KH]].
For $t \in \R^d$ write $P + t = \set{x + t : x \in P}$ for the [[translate|087VJM5Q77]] of $P$ by $t$.
The polytope $P$ is a *parallelohedron* if there is a set $T \subset \R^d$ such that:

1. the translates $P + t$ with $t \in T$ cover $\R^d$;
2. for distinct $s, t \in T$, the interiors of $P + s$ and $P + t$ are disjoint;
3. 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.