In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
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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...
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...
wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form...
The Voderberg tiling is a mathematical spiral tiling, invented in 1936 by mathematician Heinz Voderberg [de] (1911-1945). Karl August Reinhardt asked the...
if a finite set of Wang tiles can tile the plane, then there also exists a periodic tiling, which, mathematically, is a tiling that is invariant under...
stills are much smaller. ASCII art Impressionist mosaics Micrography Substitutiontiling Wikimedia logo mosaic Cartwright (2007) p.102 quote: Photographic...
of hyperbolic manifolds. Substitutiontilings are a well-studied type of subdivision rule. A subdivision rule takes a tiling of the plane by polygons...
group. Joker substitution A joker may not be retrieved before the initial meld. A joker can be retrieved from a set by replacing it with a tile of the same...
cut-and-project method for constructing aperiodic tilings. Therefore, the vertices of the pinwheel tiling and Penrose tiling are not Danzer sets. A variation of the...
problem asked for a single polyhedron tiling Euclidean 3-space, such that no tiling by it is isohedral (an anisohedral tile). The problem as stated was solved...
the tiles). A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is...
Scrabble in different languages have differing letter distributions of the tiles, because the frequency of each letter of the alphabet is different for every...
(which in turn depends on both the degree of substitution of the methoxy group and the molar substitution). That is, the higher the concentration of the...
The Toynbee tiles, also called Toynbee plaques, are messages of unknown origin found embedded in asphalt of streets in about two dozen major cities in...
Mahjong or mah-jongg (English pronunciation: /mɑːˈdʒɒŋ/ mah-JONG) is a tile-based game that was developed in the 19th century in China and has spread throughout...
affiliate's news and production facilities (including damage to ceiling tiles, piping and HVAC machinery in the newsroom, flooding from broken pipes and...
same authors demonstrated elsewhere that certain one-dimensional substitutiontilings can either be hyperuniform or anti-hyperuniform. Cheney Lin, Steinhardt...
CAN / Ctrl-X10 U+0019 25 031 End of Medium EM / Ctrl-Y U+001A 26 032 Substitute character SUB / Ctrl-Z11 U+001B 27 033 Escape character ESC U+001C 28...
can be used as the prototiles for a form of the Penrose tiling. The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles...
Laves tilings are unique except for the square tiling (1 degree of freedom), barn pentagonal tiling (1 degree of freedom), and hexagonal tiling (2 degrees...
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