# An aperiodic hexagonal tile

- Published in 2010
- Added on

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.

## Links

## Other information

- key
- Socolar2010
- type
- article
- date_added
- 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.}, 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} }