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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]
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 ...
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-periodic tilings. The Penrose tilings are a well-known example of aperiodic tilings. [1] [2]
Amman's A and B tiles in his pair A5 are a 45-135-degree silver rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic, hierarchical, and quasiperiodic structures of each of the infinite number of individual Ammann–Beenker tilings.
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 tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A tessellation of space, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions.
However, an aperiodic set of tiles can only produce non-periodic tilings. [1] [2] Infinitely many distinct tilings may be obtained from a single aperiodic set of tiles. [3] The best-known examples of an aperiodic set of tiles are the various Penrose tiles. [4] [5] The known aperiodic sets of prototiles are seen on the list of aperiodic sets of ...
These shapes are called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only these shapes. That is, each tile in the tiling must be congruent to one of these prototiles. [4] A tiling that has no periods is non-periodic.