An aperiodic hexagonal tile

• Published in 2010
• Added on 2010-03-26

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.

Links

• http://arxiv.org/abs/1003.4279
• http://arxiv.org/pdf/1003.4279v2

Other information

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.},
	date_added = {2010-03-26},
	date_published = {2010-03-01},
	urls = {http://arxiv.org/abs/1003.4279,http://arxiv.org/pdf/1003.4279v2},
	collections = {Geometry},
	keywords = {Combinatorics,Other Condensed Matter},
	month = {mar},
	pages = 21,
	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}
}