---
tag: GKJK293DXS
kind: concept
name: Tessellation
msc: [52C20, 52C22, 05B45]
sources:
  - url: https://en.wikipedia.org/wiki/Tessellation
same_as:
  wikidata: Q214856
  wikipedia: Tessellation
---

## Summary

A tessellation, or tiling, is a covering of the plane or of [[Euclidean space|0XPR1E85W0]] by pieces called tiles.
The tiles do not overlap except along their boundaries.
Tilings are the basic objects of the geometry of space filling.
Their symmetries lead to the [[crystallographic groups|7KTJKFQT2N]].
Their non-periodic forms lead to [[aperiodic sets of prototiles|Z3FCJZF6WK]] and to the mathematics of quasicrystals.

## Explanations

### L1 human

Think of the floor of a bathroom covered by tiles with no gaps and no overlaps.
A tessellation is the same idea carried out over the whole infinite plane.
Tiles may touch along their edges and corners, but no two tiles share any inner point.
The simplest example is the grid of unit squares $[a, a+1] \times [b, b+1]$ with $a$ and $b$ integers.
Hexagons, as in a honeycomb of bees, also tile the plane, but regular pentagons do not.