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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]
A portion of tiling by Ammann's aperiodic A5 set of tiles, decorated with finite, local matching rules which force infinite, global structure, that of Amman–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 ...
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]
More letters followed, and Ammann became a correspondent with many of the professional researchers. He discovered several new aperiodic tilings, each among the simplest known examples of aperiodic sets of tiles. He also showed how to generate tilings using lines in the plane as guides for lines marked on the tiles, now called "Ammann bars".
Smith also found a second tile, dubbed the "turtle", which seemed to have the same properties. In March 2023, the team of four announced their proof that the hat and turtle tiles, as well as an infinite family of other tiles interpolating the two, are aperiodic monotiles. [13] [3] [14] Both the hat and turtle tiles require some reflected copies ...