An aperiodic hexagonal tile

• Published in 2010
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.

Other information

keywords
Combinatorics,Other Condensed Matter
pages
21

BibTeX entry

@article{Socolar2010,
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.},
author = {Socolar, Joshua E. S. and Taylor, Joan M.},
keywords = {Combinatorics,Other Condensed Matter},
month = {mar},
pages = 21,
title = {An aperiodic hexagonal tile},
url = {http://arxiv.org/abs/1003.4279 http://arxiv.org/pdf/1003.4279v2},
year = 2010,
archivePrefix = {arXiv},
eprint = {1003.4279},
primaryClass = {math.CO},
urldate = {2010-03-26}
}