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A Penrose tiling with rhombi exhibiting fivefold symmetry. A Penrose tiling is an example of an aperiodic tiling.Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches.
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 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]
§10.1 Similarity, §10.2 aperiodic tiling, Raphael M. Robinson, list of aperiodic sets of tiles, Ammann A1 tilings, §10.3 Penrose tiling, golden ratio, §10.4 Ammann–Beenker tiling, aperiodic set of prototiles, §10.7 Roger Penrose, Robert Ammann, John H. Conway, Alan Lindsay Mackay, Dan Shechtman, Einstein problem: 11: Wang tiles
Some substitution tilings are periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution tilings generated by the system. The tilings by these marked tiles are necessarily aperiodic.
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]
This form of the aperiodic Penrose tiling has two prototiles, a thick rhombus (shown blue in the figure) and a thin rhombus (green). In mathematics , a prototile is one of the shapes of a tile in a tessellation .