---
tag: 7KTJKFQT2N
kind: concept
name: Crystallographic group
msc: [20H15, 52C22]
specializes: [0H3YY8CPKH]
sources:
  - url: https://en.wikipedia.org/wiki/Space_group
---

## Summary

A crystallographic group is a discrete group of isometries of [[Euclidean space|0XPR1E85W0]] $\R^n$ whose quotient space is compact.
These groups describe the symmetries of periodic patterns, such as crystals in dimension $3$ and wallpapers in dimension $2$.
By the [[Bieberbach theorems|YE87R5CDJY]], each one contains a lattice of translations of finite index.
In each dimension there are only finitely many of them up to isomorphism.
In dimension $3$ they are the space groups of crystallography, and in dimension $2$ the [[wallpaper groups|MY3CBHARYQ]].

## Explanations

### L1 human

Imagine a pattern that fills all of space and repeats in every direction, such as the atoms of a salt crystal.
The rigid motions that carry the pattern onto itself form a crystallographic group.
The translations that shift the pattern by one period are always among them.
For the square grid of integer points in the plane, the group contains the shifts by integer vectors and the quarter turns about the grid points.