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Matching Rules
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. In March 2023, four researchers, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, announced the proof that the tile discovered by David Smith is an aperiodic monotile, i.e., a solution to the einstein problem, a problem that seeks the existence of any single shape aperiodic tile. In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rhombus Penrose Tiling With Arcs
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the Diamonds (suit), diamonds suit in playing cards which resembles the projection of an Octahedron#Orthogonal projections, octahedral diamond, or a lozenge (shape), lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after calisson, the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple polygon, simple (non-self-intersecting), and is a special case of a parallelogram and a Kite (geometry), kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , roman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Uniformly Dense
Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences ... See also * * Homogeneous distribution {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Socolar–Taylor Tile
The Socolar–Taylor tile is a single non-connected tessellation, tile which is aperiodic on the Euclidean plane, meaning that it admits only aperiodic tiling, non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and reflections of the tile allowed.. It is the first known example of a single aperiodic tile, or "einstein problem, einstein". The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed. One of their papers shows a realization of the tile as a connected set. It is currently unknown whether this rule may be geometrically implemented in two dimensions while keeping the tile a simply connected space, simply connected set. This is, however, confirmed to be possible in three dimensions, and, in their original paper, Socolar and Taylor suggest a three-dimensional analogue to the monotile. Taylor and Socolar remark that the 3D monotile ape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Meyer Set
In mathematics, a Meyer set or almost lattice is a relatively dense set ''X'' of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after Yves Meyer, who introduced and studied them in the context of diophantine approximation. Nowadays Meyer sets are best known as mathematical model for quasicrystals. However, Meyer's work precedes the discovery of quasicrystals by more than a decade and was entirely motivated by number theoretic questions.. Definition and characterizations A subset ''X'' of a metric space is relatively dense if there exists a number ''r'' such that all points of ''X'' are within distance ''r'' of ''X'', and it is uniformly discrete if there exists a number ''ε'' such that no two points of ''X'' are within distance ''ε'' of each other. A set that is both relatively dense and uniformly discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Nicolaas Govert De Bruijn
Nicolaas Govert "Dick" de Bruijn (; 9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many contributions in the fields of analysis, number theory, combinatorics and logic.Nicolaas Govert de Bruijn's obituary 2012 Biography De Bruijn was born in where he attended elementary school between 1924 and 1930 and secondary school until 1934. He started studies in mathematics at in 1936 but his studies were interrupted by the outbreak of |
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Cut-and-project Method
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. In March 2023, four researchers, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, announced the proof that the tile discovered by David Smith is an aperiodic monotile, i.e., a solution to the einstein problem, a problem that seeks the existence of any single shape aperiodic tile. In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Substitution Tiling
In geometry, a tile substitution is a method for constructing highly ordered Tessellation, 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 (mathematics), 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 function, periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be "enforced by matchin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Robert Ammann
Robert Ammann (October 1, 1946 – May, 1994) was an List of amateur mathematicians, amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. Ammann attended Brandeis University, but generally did not go to classes, and left after three years. He worked as a programmer for Honeywell. After twelve years, his position was eliminated as part of a routine cutback, and Ammann ended up working as a mail sorter for a post office. In 1975, Ammann read an announcement by Martin Gardner of new work by Roger Penrose. Penrose had discovered two simple sets of aperiodic tiles, each consisting of just two quadrilaterals. Since Penrose was taking out a patent, he wasn't ready to publish them, and Gardner's description was rather vague. Ammann wrote a letter to Gardner, describing his own work, which duplicated one of Penrose's sets, plus a foursome of "Golden rhombus, golden rhombohedra" that formed aperiodic tiling ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fellow of Wadham College, Oxford, and an honorary fellow of St John's College, Cambridge, and University College London. Penrose has contributed to the mathematical physics of general relativity and physical cosmology, cosmology. He has received several prizes and awards, including the 1988 Wolf Prize in Physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems, and the 2020 Nobel Prize in Physics "for the discovery that black hole formation is a robust prediction of the general theory of relativity". He won the Royal Society Prizes for Science Books, Royal Society Science Books Prize for ''The Emperor's New Mind'' (1989), which outlines his views on physics and con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Raphael M
Raffaello Sanzio da Urbino (; March 28 or April 6, 1483April 6, 1520), now generally known in English as Raphael ( , ), was an Italian painter and architect of the High Renaissance. His work is admired for its clarity of form, ease of composition, and visual achievement of the Neoplatonic ideal of human grandeur. Together with Leonardo da Vinci and Michelangelo, he forms the traditional trinity of great masters of that period. His father Giovanni Santi was court painter to the ruler of the small but highly cultured city of Urbino. He died when Raphael was eleven, and Raphael seems to have played a role in managing the family workshop from this point. He probably trained in the workshop of Pietro Perugino, and was described as a fully trained "master" by 1500. He worked in or for several cities in north Italy until in 1508 he moved to Rome at the invitation of Pope Julius II, to work on the Apostolic Palace at the Vatican. He was given a series of important commissions there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hans Läuchli
Hans may refer to: __NOTOC__ People * Hans (name), a masculine given name * Hans Raj Hans, Indian singer and politician ** Navraj Hans, Indian singer, actor, entrepreneur, cricket player and performer, son of Hans Raj Hans ** Yuvraj Hans, Punjabi actor and singer, son of Hans Raj Hans * Hans clan, a tribal clan in Punjab, Pakistan Places * Hans, Marne, a commune in France * Hans Island, administrated by Greenland and Canada Arts and entertainment * ''Hans'' (film) a 2006 Italian film directed by Louis Nero * Hans (Frozen), the main antagonist of the 2013 Disney animated film ''Frozen'' * ''Hans'' (magazine), an Indian Hindi literary monthly * ''Hans'', a comic book drawn by Grzegorz Rosiński and later by Zbigniew Kasprzak Other uses * Clever Hans, the "wonder horse" * ''The Hans India'', an English language newspaper in India * HANS device, a racing car safety device * Hans, the ISO 15924 code for Simplified Chinese characters See also *Han (other) Han may refer t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |