---
tag: BAN4VASZF0
kind: concept
name: Substitution tiling
msc: [37B52, 52C23]
sources:
  - url: https://en.wikipedia.org/wiki/Substitution_tiling
same_as:
  wikidata: Q7632143
  wikipedia: Substitution tiling
---

## Summary

A substitution tiling is a tiling of Euclidean space built from finitely many tile shapes.
The rule inflates each tile and cuts the result into tiles of the original shapes.
Repeating the rule produces larger and larger patches, and the tilings whose patches all arise in this way are the substitution tilings.
They give the standard examples of aperiodic order, such as the Penrose tilings, and they model quasicrystals.
The tilings of one substitution form a compact space on which translations act, and this [[dynamical system|BBZP1R872N]] carries much of their structure.

## Explanations

### L1 human

Start with a few shapes of tiles.
Blow each tile up by a fixed factor, then cut the big tile into tiles of the original shapes.
Do it again and again, and the patches grow to cover the whole plane.
A unit square, blown up by $2$ and cut into four unit squares, gives the ordinary square grid.
Cleverer rules, like the one for the Penrose tilings, give tilings that never repeat periodically.
See the [[definition|T4403SAMB2]].