A Socolar tiling is an example of an

aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- peri ...

, developed in 1989 by Joshua Socolar in the exploration of

quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical ...

s. There are 3 tiles a 30° rhombus, square, and regular hexagon. The 12-fold symmmetry set exist similar to the 10-fold Penrose rhombic tilings, and 8-fold

Ammann–Beenker tiling In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. Th ...

s. The 12-fold tiles easily tile periodically, so special rules are defined to limit their connections and force nonperiodic tilings. Each tile disallowed from touching another of itself, while the hexagon can connect to both and itself, but only in alternate edges.

Dodecagonal rhomb tiling

The ''dodecagonal rhomb tiling'' include three tiles, a 30° rhombus, a 60° rhombus, and a square. And expanded set can also include an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

, half of the 60° rhombus.60° rhombus.A Quasiperiodic Tiling With 12-Fold Rotational Symmetry and Inflation Factor 1 + √3
Theo P. Schaad and Peter Stampfli, 10 Feb 2021

References

{{geometry-stub Aperiodic tilings

Dodecagonal rhomb tiling

See Also

References