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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 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 ...
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.
In geometry, an Ammann A1 tiling is a tiling from the 6 piece prototile set shown on the right. They were found in 1977 by Robert Ammann. [1] 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]
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.
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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 ...
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