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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 ...
The Penrose tiles, and shortly thereafter Amman's several different sets of tiles, [21] were the first example based on explicitly forcing a substitution tiling structure to emerge. Joshua Socolar , [ 22 ] [ 23 ] Roger Penrose , [ 24 ] Ludwig Danzer , [ 25 ] and Chaim Goodman-Strauss [ 20 ] have found several subsequent sets.
[9] [23] [41] The substitution rules decompose each tile into smaller tiles of the same shape as those used in the tiling (and thus allow larger tiles to be "composed" from smaller ones). This shows that the Penrose tiling has a scaling self-similarity, and so can be thought of as a fractal , using the same process as the pentaflake .
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 ...
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]
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 ...
A tile substitution with respect to the prototiles P is a pair (,), where : is a linear map, all of whose eigenvalues are larger than one in modulus, together with a substitution rule that maps each to a tiling of .
The elaborate and colourful zellige tessellations of glazed tiles at the Alhambra in Spain that attracted the attention of M. C. Escher. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. These tiles may be polygons ...