Socolar–Taylor tile

A patch of 25 monotiles, showing the triangular hierarchical structure
A patch of 25 monotiles, showing the triangular hierarchical structure

The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), 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] One of their papers[2] shows a realization of the tile as a connected set. It is currently unknown whether this rule may be geometrically implemented in two dimensions while keeping the tile a simply connected set.[2][3]

This is, however, confirmed to be possible in three dimensions, and, in their original paper, Socolar and Taylor suggest a three-dimensional analogue to the monotile.[1] Taylor and Socolar remark that the 3D monotile aperiodically tiles three-dimensional space. However the tile does allow tilings with a period, shifting one (non-periodic) two dimensional layer to the next, and so the tile is only "weakly aperiodic".

Physical copies of the three-dimensional tile could not be fitted together without allowing reflections, which would require access to four-dimensional space.[2][4]

  • The monotile implemented geometrically. Black lines are included to show how the structure is enforced.
    The monotile implemented geometrically. Black lines are included to show how the structure is enforced.
  • A connected version of the monotile (in black) and six of its neighbours.
    A connected version of the monotile (in black) and six of its neighbours.
  • A three-dimensional analogue of the Socolar-Taylor tile (all matching rules implemented geometrically)
    A three-dimensional analogue of the Socolar-Taylor tile (all matching rules implemented geometrically)
  • 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 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 partial tiling of three-dimensional space with the 3D monotile.
  • A tiling of 3D space with one tile removed to demonstrate the structure.
    A tiling of 3D space with one tile removed to demonstrate the structure.

References

  1. ^ a b 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.
  2. ^ a b c d 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
  3. ^ a b Frettlöh, Dirk. "Hexagonal aperiodic monotile". Tilings Encyclopedia. Retrieved 3 June 2013.
  4. ^ Harriss, Edmund. "Socolar and Taylor's Aperiodic Tile". Maxwell's Demon. Retrieved 3 June 2013.

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