The Socolar–Taylor tile is a single non-connected

tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or o ...

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.. It is the first known example of a single aperiodic tile, or " einstein". 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. It is currently unknown whether this rule may be geometrically implemented in two dimensions while keeping the tile a

connected set In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties th ...

. 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.

Gallery

References

External links

Previewable digital models of the three-dimensional tile, suitable for 3D printing, at ThingiverseOriginal diagrams and further information on Joan Taylor's personal website
{{DEFAULTSORT:Socolar-Taylor tile Aperiodic tilings