Aperiodic set of prototiles

Click "show" for description.
A periodic tiling with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. To do this, the basic triangle must be rotated 60 degrees to fit edge-to-edge to a neighboring triangle. Thus a triangular tiling of fundamental units is generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be translated to form an infinite tiling of the plane. It is not necessary to rotate this patch to achieve this.
The Penrose tiles are an aperiodic set of tiles, since they admit only non-periodic tilings of the plane (see next image).
All of the infinitely many tilings by the Penrose tiles are aperiodic. That is, the Penrose tiles are an aperiodic set of prototiles.

A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.

A given set of tiles, in the Euclidean plane or some other geometric setting, admits a tiling if non-overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings that remain invariant after being shifted by a translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings. (For example, randomly arranged tilings using a 2×2 square and 2×1 rectangle are typically non-periodic.)

However, an aperiodic set of tiles can only produce non-periodic tilings.[1][2] Infinitely many distinct tilings may be obtained from a single aperiodic set of tiles.[3]

The best-known examples of an aperiodic set of tiles are the various Penrose tiles.[4][5] The known aperiodic sets of prototiles are seen on the list of aperiodic sets of tiles. The underlying undecidability of the domino problem implies that there exists no systematic procedure for deciding whether a given set of tiles can tile the plane.

  1. ^ Senechal, Marjorie (1996) [1995]. Quasicrystals and geometry (corrected paperback ed.). Cambridge University Press. ISBN 978-0-521-57541-6.
  2. ^ Grünbaum, Branko; Geoffrey C. Shephard (1986). Tilings and Patterns. W.H. Freeman & Company. ISBN 978-0-7167-1194-0.
  3. ^ A set of aperiodic prototiles can always form uncountably many different tilings, even up to isometry, as proven by Nikolaï Dolbilin in his 1995 paper The Countability of a Tiling Family and the Periodicity of a Tiling
  4. ^ Gardner, Martin (January 1977). "Mathematical Games". Scientific American. 236 (5): 111–119. Bibcode:1977SciAm.236e.128G. doi:10.1038/scientificamerican0577-128.
  5. ^ Gardner, Martin (1988). Penrose Tiles to Trapdoor Ciphers. W H Freeman & Co. ISBN 978-0-7167-1987-8.