---
tag: MR2JWBCZQZ
kind: concept
name: Parallelohedron
msc: [52C22, 52B11, 52C07]
sources:
  - url: https://en.wikipedia.org/wiki/Parallelohedron
same_as:
  wikidata: Q23823961
  wikipedia: Parallelohedron
---

## Summary

A parallelohedron is a convex polytope whose translated copies fill space face to face, with no rotations allowed.
In the plane the parallelohedra are the parallelograms and the centrally symmetric hexagons.
In space there are five combinatorial types, found by Fedorov: the parallelepiped, the hexagonal prism, the rhombic dodecahedron, the elongated dodecahedron and the truncated octahedron.
The [[Voronoi cell|V1R1R88GV6]] of any lattice is a parallelohedron, and [[Voronoi's conjecture|F03A4MKA5V]] asks whether every parallelohedron arises this way up to an affine map.

## Explanations

### L1 human

Think of tiles that you may slide but never turn.
A parallelohedron is a convex tile of this kind that fills the whole plane or space, with each face of one tile matched exactly with a face of a neighbour.
Bricks in a wall laid in a straight grid are an example, and so are hexagonal floor tiles.
A regular triangle is not one: copies obtained only by sliding all point the same way and leave gaps.