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Chair Tiling
In geometry, a chair tiling (or L tiling) is a nonperiodic substitution tiling created from L-tromino prototiles. These prototiles are examples of rep-tiles and so an iterative process of decomposing the L tiles into smaller copies and then rescaling them to their original size can be used to cover patches of the plane. Chair tilings do not possess translational symmetry, i.e., they are examples of ''nonperiodic tilings'', but the chair tiles are not aperiodic tiles since they are not forced to tile nonperiodically by themselves. The ''trilobite'' and ''cross'' tiles are aperiodic tiles that enforce the chair tiling substitution structure and these tiles have been modified to a simple aperiodic set of tiles using matching rules enforcing the same structure. Barge et al. have computed the ÄŒech cohomology In mathematics, specifically algebraic topology, ÄŒech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L Substitution Tiling
L, or l, is the twelfth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''el'' (pronounced ), plural ''els''. History Lamedh may have come from a pictogram of an ox goad or cattle prod. Some have suggested a shepherd's staff. Use in writing systems Phonetic and phonemic transcription In phonetic and phonemic transcription, the International Phonetic Alphabet uses to represent the lateral alveolar approximant. English In English orthography, usually represents the phoneme , which can have several sound values, depending on the speaker's accent, and whether it occurs before or after a vowel. The alveolar lateral approximant (the sound represented in IPA by lowercase ) occurs before a vowel, as in ''lip'' or ''blend'', while the velarized alveolar lateral approximant (IPA ) occurs in ''bell'' and ''milk''. This velarization does not occur in many European langu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Tiling
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 the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. Definition and illustration Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution Tiling
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. Introduction A tile substitution is described by a set of prototiles (tile shapes) T_1,T_2,\dots, T_m, an expanding map Q and a dissection rule showing how to dissect the expanded prototiles Q T_i to form copies of some prototiles T_j. Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. 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 til ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tromino
A tromino or triomino is a polyomino of order 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge. Symmetry and enumeration When rotations and reflections are not considered to be distinct shapes, there are only two different ''free'' trominoes: "I" and "L" (the "L" shape is also called "V"). Since both free trominoes have reflection symmetry, they are also the only two ''one-sided'' trominoes (trominoes with reflections considered distinct). When rotations are also considered distinct, there are six ''fixed'' trominoes: two I and four L shapes. They can be obtained by rotating the above forms by 90°, 180° and 270°. Rep-tiling and Golomb's tromino theorem Both types of tromino can be dissected into ''n''2 smaller trominos of the same type, for any integer ''n'' > 1. That is, they are rep-tiles. Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an aperiodic tiling. In this context, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prototile
In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation. Definition A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others. If is the set of tiles in a tessellation, a set of shapes is called a set of prototiles if no two shapes in are congruent to each other, and every tile in is congruent to one of the shapes in . It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same cardinality, so the number of prototiles is well defined. A tessellation is said to be monohedral if it has exactly one prototile. Aperiodicity A set of prototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling. It is unknown whethe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rep-tile
In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of ''Scientific American''. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in ''Mathematics Magazine''. Terminology A rep-tile is labelled rep-''n'' if the dissection uses ''n'' copies. Such a shape necessarily forms the prototile for a tiling of the plane, in many cases an aperiodic tiling. A rep-tile dissection using different sizes of the original shape is called an irregular rep-tile or irreptile. If the dissection uses ''n'' copies, the shape is said to be irrep-''n''. If all these sub-tiles are of different sizes then the tiling is additionally described as perfect. A shape that is rep-''n'' or irrep-''n'' is tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translational Symmetry
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation. Analogously an operator on functions is said to be translationally invariant with respect to a translation operator T_\delta if the result after applying doesn't change if the argument function is translated. More precisely it must hold that \forall \delta \ A f = A (T_\delta f). Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law. Translational symmetry of an object means that a particular translation does not change the object. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Set Of Prototiles
A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non- periodic. The ''aperiodicity'' referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic. A given set of tiles, in the Euclidean plane or some other geometric setting, ''admits a tiling'' if non-overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings that remain invariant after being shifted by a translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings. (For example, randomly arranged tilings using a 2×2 square and 2×1 rectangle are typically non-periodic.) However, an aperiodic set of tiles can ''only'' produce non-periodic tilings. Infinit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ÄŒech Cohomology
In mathematics, specifically algebraic topology, ÄŒech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard ÄŒech. Motivation Let ''X'' be a topological space, and let \mathcal be an open cover of ''X''. Let N(\mathcal) denote the nerve of the covering. The idea of ÄŒech cohomology is that, for an open cover \mathcal consisting of sufficiently small open sets, the resulting simplicial complex N(\mathcal) should be a good combinatorial model for the space ''X''. For such a cover, the ÄŒech cohomology of ''X'' is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of ''X'', ordered by refinement. This is the approach adopted below. Construction Let ''X'' be a topological space, and l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Tiling
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 the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. Definition and illustration Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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