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Penrose Tiling
A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and a tiling is ''aperiodic'' if it does not contain arbitrarily large Periodic tiling, periodic regions or patches. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s. There are several variants of Penrose tilings with different tile shapes. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombus, rhombi, or two different quadrilaterals called kite (geometry), kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together in a way that avoids periodic tiling. This ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penrose Tiling (Rhombi)
Penrose may refer to: Places United States * Penrose, Arlington, Virginia, a neighborhood * Penrose, Colorado, a town * Penrose, St. Louis, Missouri, a neighborhood * Penrose, Philadelphia, Pennsylvania, a neighborhood * Penrose, North Carolina, an unincorporated community * Penrose, Utah, an unincorporated community * Penrose, Virginia, an historic district in Arlington County Elsewhere * Penrose, New South Wales (Wingecarribee), Australia * Penrose, New South Wales (Wollongong), New South Wales, Australia * Penrose, Cornwall, a country house and National Trust estate in England * Penrose, New Zealand * Penrose Peak (other) * Penrose railway station (other) People * Penrose Stout (1887–1934), American architect * Penrose Hallowell (c. 1928–2021), Pennsylvania secretary of agriculture * Penrose (surname), including a list of people with the name Other uses * Penrose (brand), a List of Conagra brands, brand name owned by ConAgra Foods, Inc. * ''The Penrose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bragg Peak
The Bragg peak is a pronounced peak on the Bragg curve which plots the energy loss of ionizing radiation during its travel through matter. For protons, α-rays, and other ion rays, the peak occurs immediately before the particles come to rest. It is named after William Henry Bragg, who discovered it in 1903 using alpha particles from radium, and wrote the first empirical formula for ionization energy loss per distance with Richard Kleeman. When a fast charged particle moves through matter, it ionizes atoms of the material and deposits a dose along its path. A peak occurs because the interaction cross section increases as the charged particle's energy decreases. Energy lost by charged particles is inversely proportional to the square of their velocity, which explains the peak occurring just before the particle comes to a complete stop. In the upper figure, it is the peak for alpha particles of 5.49 MeV moving through air. In the lower figure, it is the narrow peak of the "nat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wang Domino
Wang tiles (or Wang dominoes), first proposed by mathematician, logician, and philosopher Hao Wang in 1961, is a class of formal systems. They are modeled visually by square tiles with a color on each side. A set of such tiles is selected, and copies of the tiles are arranged side by side with matching colors, ''without'' rotating or reflecting them. The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern. Domino problem In 1961, Wang conjectured that 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 translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern, where the overall pattern is a repetition of some smaller pattern. He also observed that this conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hao Wang (academic)
Hao Wang (; 20 May 1921 – 13 May 1995) was a Chinese-American logician, philosopher, mathematician, and commentator on Kurt Gödel. Biography Born in Jinan, Shandong, in the Republic of China (today in the People's Republic of China), Wang received his early education in China. He obtained a BSc degree in mathematics from the National Southwestern Associated University in 1943 and an M.A. in philosophy from Tsinghua University in 1945, where his teachers included Feng Youlan and Jin Yuelin, after which he moved to the United States for further graduate studies. He studied logic under W. V. O. Quine at Harvard University, culminating in a Ph.D. in 1948. He was appointed to an assistant professorship at Harvard the same year. During the early 1950s, Wang studied with Paul Bernays in Zürich. In 1956, he was appointed Reader in the Philosophy of Mathematics at the University of Oxford. In 1959, Wang wrote on an IBM 704 computer a program that in only 9 minutes mechanica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wang Tiles
Wang tiles (or Wang dominoes), first proposed by mathematician, logician, and philosopher Hao Wang in 1961, is a class of formal systems. They are modeled visually by square tiles with a color on each side. A set of such tiles is selected, and copies of the tiles are arranged side by side with matching colors, ''without'' rotating or reflecting them. The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern. Domino problem In 1961, Wang conjectured that 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 translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern, where the overall pattern is a repetition of some smaller pattern. He also observed that this conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prototile
In mathematics, 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. In March 2023, four researchers, Chaim Goodm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Tiling
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional spaces, higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include ''regular tilings'' with regular polygonal tiles all of the same shape, and ''semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 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 a repeating pattern (an aperiodic set of prototiles). A ''tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
