Socolar–Taylor tile
The Socolar–Taylor tile is a single tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane, with rotations and reflections of the tile allowed.[1] It is the first known example of a single aperiodic tile, or "einstein".[2] The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed.[3] This rule cannot be geometrically implemented in two dimensions while keeping the tile a connected set.[2][3]
This is, however, possible in three dimensions, and in their original paper Socolar and Taylor suggest a three-dimensional analogue to the monotile.[1] The 3D monotile aperiodically tiles three-dimensional space; however, much as the structure of the 2D tile prevents it from being fitted together just by sliding the tiles together in 2D space, physical copies of the three-dimensional tile could not be fitted together without access to four-dimensional space.[4] This tile is likewise the only known 3D aperiodic tile which can fill three-dimensional space unpredictably.[2]
Gallery
| The monotile implemented geometrically. Black lines are included to show how the structure is enforced. |
| A three-dimensional analogue of the Socolar-Taylor tile, without decoration (all matching rules implemented geometrically). |
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| A three-dimensional analogue of the monotile, with matching rules implemented geometrically. Red lines are included only to illuminate the structure of the tiling. Note that this shape remains a connected set. |
| A partial tiling of three-dimensional space with the 3D monotile. |
| A tiling of 3D space with one tile removed to demonstrate the structure. |
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References
- 1 2 Socolar, Joshua E. S.; Taylor, Joan M. (2011), "An aperiodic hexagonal tile", Journal of Combinatorial Theory, Series A 118 (8): 2207–2231, arXiv:1003.4279, doi:10.1016/j.jcta.2011.05.001, MR 2834173 .
- 1 2 3 Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144
- 1 2 Frettlöh, Dirk. "Hexagonal aperiodic monotile". Tilings Encyclopedia. Retrieved 3 June 2013.
- ↑ Harriss, Edmund. "Socolar and Taylor's Aperiodic Tile". Maxwell's Demon. Retrieved 3 June 2013.
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