---
tag: 29AR3Z107J
kind: definition
name: Periodic tiling
msc: [52C20, 52C22, 05B45]
defines: WFRGSBFZGV
uses: [GKJK293DXS, 087VJM5Q77]
sources:
  - url: https://en.wikipedia.org/wiki/Tessellation
---

## Definition

Let $\mathcal{T}$ be a [[tiling|GKJK293DXS]] of $\R^n$.
For $v \in \R^n$, let $\mathcal{T} + v = \set{T + v : T \in \mathcal{T}}$ be the image of $\mathcal{T}$ under the [[translation|087VJM5Q77]] $x \mapsto x + v$.
A vector $v$ is a *period* of $\mathcal{T}$ if $\mathcal{T} + v = \mathcal{T}$.
The tiling $\mathcal{T}$ is *periodic* if it has $n$ linearly independent periods.

## Remarks

Example: the tiling of $\R^2$ by the unit squares $[a, a+1] \times [b, b+1]$, $(a, b) \in \Z^2$, is periodic.
Translation by $(1, 0)$ maps $[a, a+1] \times [b, b+1]$ to $[a+1, a+2] \times [b, b+1]$, so $(1, 0)$ is a period, and in the same way $(0, 1)$ is a period; these two vectors are independent.

Non-example: shift the row of squares $[a, a+1] \times [0, 1]$ by $1/2$ to the right and leave all other rows as they are.
The vector $(1, 0)$ is still a period, but no vector with a nonzero second coordinate is one, because the shifted row is the only row with vertices at half-integers.
So the tiling has periods in only one direction and is not periodic.

A tiling with no nonzero period at all is called *non-periodic*.
The [[Penrose tilings|MBE83FTRBG]] are non-periodic.