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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 ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombus Penrose Tiling With Arcs
In plane Euclidean geometry, a rhombus (plural 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 grc, ῥόμβος, rhombos, meaning something that spins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 managing editors are Camillo De Lellis (Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ..., Princeton) and Jean-Benoît Bost ( University of Paris-Sud). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robinson Tiling
Robinson may refer to: People and names * Robinson (name) Fictional characters * Robinson Crusoe, the main character, and title of a novel by Daniel Defoe, published in 1719 Geography * Robinson projection, a map projection used since the 1960s to show the entire world in two dimensions * Robinson (crater), a small lunar impact crater southwest of the large walled plain J. Herschel ;United States * Robinson, Illinois * Robinson, Iowa * Robinson, Kansas * Robinson, Kentucky * Robinson, Minnesota * Robinson, North Dakota * Robinson, Texas * Robinson, Washington * Robinson Township, Pennsylvania (other), two townships in the Pittsburgh Metro Area with the same name ;United Kingdom * Robinson (Lake District), a 737 m hill in England's Lake District * Robinson College, Cambridge, a college in England's University of Cambridge ;France * Robinson (Paris RER), a commuter train station in Paris Ships * USS ''Robinson'', the name of more than one United States Navy ship * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robinson Tiles
Robinson may refer to: People and names * Robinson (name) Fictional characters * Robinson Crusoe, the main character, and title of a novel by Daniel Defoe, published in 1719 Geography * Robinson projection, a map projection used since the 1960s to show the entire world in two dimensions * Robinson (crater), a small lunar impact crater southwest of the large walled plain J. Herschel ;United States * Robinson, Illinois * Robinson, Iowa * Robinson, Kansas * Robinson, Kentucky * Robinson, Minnesota * Robinson, North Dakota * Robinson, Texas * Robinson, Washington * Robinson Township, Pennsylvania (other), two townships in the Pittsburgh Metro Area with the same name ;United Kingdom * Robinson (Lake District), a 737 m hill in England's Lake District * Robinson College, Cambridge, a college in England's University of Cambridge ;France * Robinson (Paris RER), a commuter train station in Paris Ships * USS ''Robinson'', the name of more than one United States Navy ship * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raphael Robinson
Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) was an American mathematician. Born in National City, California, Robinson was the youngest of four children of a lawyer and a teacher. He was awarded from the University of California, Berkeley in mathematics: the BA (1932), MA (1933), and Ph.D. (1935). His Ph.D. thesis, on complex analysis, was titled ''Some results in the theory of Schlicht functions''. In 1941, Robinson married his former student Julia Bowman. She became his Berkeley colleague and the first woman president of the American Mathematical Society. Robinson worked on mathematical logic, set theory, geometry, number theory, and combinatorics. In 1937 he set out a simpler and more conventional version of the John von Neumann 1923 axiomatic set theory. Soon after Alfred Tarski joined Berkeley's mathematics department in 1942, Robinson began to do major work on the foundations of mathematics, building on Tarski's concept of essential undecidabilili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer science. Knuth has been called the "father of the analysis of algorithms". He is the author of the multi-volume work ''The Art of Computer Programming'' and contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process, he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar, Knuth created the WEB and CWEB computer programming systems designed to encou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undecidable Problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run. Background A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns ''yes''. These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language. Using some encoding, such as a Gödel numbering, the strings can be encoded as natural numbers. Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meyer Set
In mathematics, a Meyer set or almost lattice is a set relatively dense ''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] |
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 The Hague 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[...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 ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
