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A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. [3] The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are ...
Shahar Mozes has found many alternative constructions of aperiodic sets of tiles, some in more exotic settings; for example in semi-simple Lie groups. [31] Block and Weinberger used homological methods to construct aperiodic sets of tiles for all non-amenable manifolds. [32]
This first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger later reduced his set to 104, and Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles. [9] The set of 13 tiles given in the illustration on the right is an aperiodic set published by Karel Culik, II, in 1996.
Ammann was inspired by the Robinsion tilings, which were found by Robinson in 1971. The A1 tiles are one of five sets of tiles discovered by Ammann and described in Tilings and patterns. [2] The A1 tile set is aperiodic, [2] i.e. they tile the whole Euclidean plane, but only without ever creating a periodic tiling.
See List of aperiodic sets of tiles for examples. Pages in category "Aperiodic tilings" The following 19 pages are in this category, out of 19 total.
A set of prototiles is said to be aperiodic if all of its tilings are non-periodic, and in this case its tilings are also called aperiodic tilings. [5] Penrose tilings are among the simplest known examples of aperiodic tilings of the plane by finite sets of prototiles. [3]
The Socolar–Taylor tile was proposed in 2010 as a solution to the einstein problem, but this tile is not a connected set. In 1996, Petra Gummelt constructed a decorated decagonal tile and showed that when two kinds of overlaps between pairs of tiles are allowed, the tiles can cover the plane, but only non-periodically. [6]
All of this infinite global structure is forced through local matching rules on a pair of tiles, among the very simplest aperiodic sets of tiles ever found, Ammann's A5 set. [1] Various methods to describe the tilings have been proposed: matching rules, substitutions, cut and project schemes [2] and coverings.