# An aperiodic hexagonal tile

• Published in 2010
In the collection
We show that a single tile can fill space uniformly but not admit a periodic tiling. The space--filling tiling that can be built from copies of the tile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. We present the tile, prove that the tilings it admits are not periodic, and discuss some of their remarkable properties, including their relation to a previously known tiling. We also clarify some subtleties in the definitions of the terms "nonperiodic tiling" and "aperiodic tile". For a reasonable interpretation of these terms, the tile presented here is the only known example of an aperiodic tile.

## Comment

A single shape which tiles the plane, but not periodically. The catch is that it's not a single connected component.

## Other information

key
Socolar2010
type
article
2010-03-26
date_published
2010-03-01
keywords
Combinatorics,Other Condensed Matter
pages
21

### BibTeX entry

@article{Socolar2010,
key = {Socolar2010},
type = {article},
title = {An aperiodic hexagonal tile},
author = {Socolar, Joshua E. S. and Taylor, Joan M.},
abstract = {We show that a single tile can fill space uniformly but not admit a periodic tiling. The space--filling tiling that can be built from copies of the tile has the structure of a union of honeycombs with lattice constants of {\$}2^n a{\$}, where {\$}a{\$} sets the scale of the most dense lattice and {\$}n{\$} takes all positive integer values. We present the tile, prove that the tilings it admits are not periodic, and discuss some of their remarkable properties, including their relation to a previously known tiling. We also clarify some subtleties in the definitions of the terms "nonperiodic tiling" and "aperiodic tile". For a reasonable interpretation of these terms, the tile presented here is the only known example of an aperiodic tile.},
comment = {A single shape which tiles the plane, but not periodically. The catch is that it's not a single connected component.},
}