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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 ...
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".
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.
A patch of 25 monotiles, showing the triangular hierarchical structure. The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and reflections of the tile allowed. [1]
Download as PDF; Printable version; In other projects ... This category is for articles about aperiodic sets of tiles. See List of aperiodic sets of tiles for examples.
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]
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]
As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1974 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry.