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The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. They are quasicrystals : implemented as a physical structure a Penrose tiling will produce diffraction patterns with Bragg peaks and five-fold symmetry, revealing the repeated patterns and fixed orientations of its ...
In geometry, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called tiles), without gaps or overlaps (other than the boundaries of the tiles). [1] A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself.
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]
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 ...
One of the main themes of the book is to understand how the mathematical properties of aperiodic tilings such as the Penrose tiling, and in particular the existence of arbitrarily large patches of five-way rotational symmetry throughout these tilings, correspond to the properties of quasicrystals including the five-way symmetry of their Bragg ...
The last five chapters survey a variety of advanced topics in tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings with unusual kinds of tiles. Each chapter open with an introduction to the topic, this is followed by the detailed material of the chapter, much previously unpublished, which is ...
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