Socolar–Taylor tile

File:Socolar-Taylor tiling.svg

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.{{citation

| last1 = Socolar | first1 = Joshua E. S.

| last2 = Taylor | first2 = Joan M.

| arxiv = 1003.4279

| doi = 10.1016/j.jcta.2011.05.001

| issue = 8

| journal = Journal of Combinatorial Theory

| mr = 2834173

| pages = 2207–2231

| series = Series A

| title = An aperiodic hexagonal tile

| volume = 118

| year = 2011}}. It is the first known example of a single aperiodic tile, or "einstein".{{citation

| last1 = Socolar | first1 = Joshua E. S.

| last2 = Taylor | first2 = Joan M.

| arxiv = 1009.1419

| doi = 10.1007/s00283-011-9255-y

| issue = 1

| journal = The Mathematical Intelligencer

| mr = 2902144

| pages = 18–28

| title = Forcing nonperiodicity with a single tile

| volume = 34

| year = 2012}} 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.{{cite web |url=http://tilings.math.uni-bielefeld.de/substitution/hexagonal-aperiodic-monotile |title=Hexagonal aperiodic monotile |last1=Frettlöh |first1=Dirk |website=Tilings Encyclopedia |access-date=3 June 2013}} One of their papers 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 simply connected set.

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. 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.{{cite web |url=http://maxwelldemon.com/2010/04/01/socolar_taylor_aperiodic_tile/ |title=Socolar and Taylor's Aperiodic Tile |last1=Harriss |first1=Edmund |author1-link = Edmund Harriss |website=Maxwell's Demon |access-date=3 June 2013}}

Gallery

{{Gallery

|File:Socolar-Taylor tile.svg|The monotile implemented geometrically. Black lines are included to show how the structure is enforced.

|File:ST-tile-connected.svg|A connected version of the monotile (in black) and six of its neighbours.

|File:Socolar-Taylor 3D monotile.stl|A three-dimensional analogue of the Socolar-Taylor tile (all matching rules implemented geometrically)}}

{{Gallery

|File:Socolar-Taylor 3D monotile with decoration.png |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.

|File:Socolar-Taylor 3D tiling example.png |A partial tiling of three-dimensional space with the 3D monotile.

|File:Inside Socolar-Taylor 3D tiling example.png |A tiling of 3D space with one tile removed to demonstrate the structure.}}

References

External links

[http://www.thingiverse.com/thing:2101 Previewable digital models of the three-dimensional tile, suitable for 3D printing, at Thingiverse]
[http://taylortiling.com/ Original diagrams and further information on Joan Taylor's personal website]

Category:Aperiodic tilings