---
tag: WFRGSBFZGV
kind: concept
name: Periodic tiling
msc: [52C20, 52C22, 05B45]
specializes: [GKJK293DXS]
sources:
  - url: https://en.wikipedia.org/wiki/Tessellation
---

## Summary

A periodic tiling is a [[tessellation|GKJK293DXS]] of $\R^n$ that is carried onto itself by [[translations|087VJM5Q77]] in $n$ independent directions.
Such a tiling is determined by a finite patch repeated along a lattice.
The symmetries of periodic patterns are described by [[crystallographic groups|7KTJKFQT2N]].
Tilings that admit no period lead to the theory of [[aperiodic sets of prototiles|Z3FCJZF6WK]].

## Explanations

### L1 human

A periodic tiling looks like wallpaper: one block of the pattern is copied over and over in two directions.
If you slide the whole tiling by the right amount to the right, or by the right amount upward, every tile lands exactly on another tile.
The square grid is periodic, since a shift by one unit to the right or one unit up maps it onto itself.
A tiling with only a single direction of shifts, such as a strip pattern, is not periodic in this sense.