Socolar–Taylor tile

A patch of 25 monotiles, showing the triangular hierarchical structure

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. 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. This rule cannot be geometrically implemented in two dimensions while keeping the tile a connected set.

This is, however, possible in three dimensions, and in their original paper Socolar and Taylor suggest a three-dimensional analogue to the monotile. 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. This tile is likewise the only known 3D aperiodic tile which can fill three-dimensional space unpredictably.

Gallery

• 
  The monotile implemented geometrically. Black lines are included to show how the structure is enforced.
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  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.
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  A partial tiling of three-dimensional space with the 3D monotile.
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  A tiling of 3D space with one tile removed to demonstrate the structure.

External links

• Previewable digital models of the three-dimensional tile, suitable for 3D printing, at Thingiverse
• Original diagrams and further information on Joan Taylor's personal website