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1 42 Polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421. These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . The projection of 142 to the E8 Coxeter plane (aka. the Petrie projection) with polytope radius 4\sqrt is shown below with 483,840 edges of length 2\sqrt culled 53% on the interior to only 226,444: 142 polytope The 142 is composed of 2400 facets: 240 132 polytopes, and 2160 7-demicubes (141). Its vertex figure is a birectified 7-simplex. This polytope, along ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4 21 T0 E6
4 (four) is a number, numeral (linguistics), numeral and numerical digit, digit. It is the natural number following 3 and preceding 5. It is the smallest semiprime and composite number, and is tetraphobia, considered unlucky in many East Asian cultures. In mathematics Four is the smallest composite number, its proper divisors being and . Four is the sum and product of two with itself: 2 + 2 = 4 = 2 x 2, the only number b such that a + a = b = a x a, which also makes four the smallest squared prime number p^. In Knuth's up-arrow notation, , and so forth, for any number of up arrows. By consequence, four is the only square one more than a prime number, specifically 3, three. The sum of the first four prime numbers 2, two + 3, three + 5, five + 7, seven is the only sum of four consecutive prime numbers that yields an Parity (mathematics), odd prime number, 17 (number), seventeen, which is the fourth super-prime. Four lies between the first proper pair of twin primes, 3, three and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 8-polytope
In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), facets. A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes Regular 8-polytopes can be represented by the Schläfli symbol , with v 7-polytope Facet (mathematics), facets around each Peak (geometry), peak. There are exactly three such List of regular polytopes#Convex 4, convex regular 8-polytopes: # - 8-simplex # - 8-cube # - 8-orthoplex There are no nonconvex regular 8-polytopes. Characteristics The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficient (topology), torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-demicube
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM7 for a 7-dimensional ''half measure'' polytope. Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol \left\ or . Cartesian coordinates Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract: : (±1,±1,±1,±1,±1,±1,±1) with an odd number of plus signs. Images As a configuration This configuration matrix represents the 7-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-demicube. The nondiagonal n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosset 1 32 Petrie
Gosset, founded in 1584, is the oldest wine house in Champagne. In 1584, Pierre Gosset, alderman of Aÿ and wine-grower, made still, mostly red, wines from the grapes he harvested from his own vines. In those days, two wines vied for pride of place at the table to the Kings of France: the wine of Aÿ and, from some hundreds of leagues further South, the wine of Beaune. Then, in the 18th century the wine made in around Aÿ began to bubble and the Gosset family turned naturally to the production of champagne. History In 1994, the house was sold to the Renaud-Cointreau group (family owned and also the owners of Cognac Frapin). Under the new management, Gosset succeeded in increasing its production to one million bottles in 2005. The house is now successfully headed by Jean-Pierre Cointreau and exports 65% of its production to more than 70 countries. In 2009, as he prepared to celebrate Gosset’s 425th birthday, Jean-Pierre Cointreau took the decision to buy new premises in Eperna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 32 Polytope
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences. The rectified 132 is constructed by points at the mid-edges of the 132. These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . 1_32 polytope This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E7* lattice. Alternate names * Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes. * Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch. * ''Pentacontihexa-hecatonicosihexa-exon'' (Acronym lin) - 56-126 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. Definition The Schläfli symbol is a recursive description, starting with for a ''p''-sided regular polygon that is convex. For example, is an equilateral triangle, is a square, a convex regular pentagon, etc. Regular star polygons are not convex, and their Schläfli symbols contain irreducible fractions ''p''/''q'', where ''p'' is the number of vertices, and ''q'' is their turning number. Equivalently, is created from the vertices of , connected every ''q''. For example, is a pentagram; is a pentagon. A regular polyhedron that has ''q'' regular ''p''-sided Face (geometry), polygon faces around each Verte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 1 K2 Polytope
In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol . Family members The family starts uniquely as 6-polytope In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. Definition A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. A ...s, but can be extended backwards to include the 5-demihypercube, demicube (demipenteract) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions. Each polytope is constructed from 1k-1,2 and (n-1)-Demihypercube, demicube facets. Each has a vertex figure of a ' polytope is a birectified n-simplex, ''t2''. The sequence ends with k=6 (n=10), as an infinite tessellatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 8-polytope
In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), facets. A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes Regular 8-polytopes can be represented by the Schläfli symbol , with v 7-polytope Facet (mathematics), facets around each Peak (geometry), peak. There are exactly three such List of regular polytopes#Convex 4, convex regular 8-polytopes: # - 8-simplex # - 8-cube # - 8-orthoplex There are no nonconvex regular 8-polytopes. Characteristics The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficient (topology), torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E8 142 Petrie Projection
E8 may refer to: Mathematics * E8, an exceptional simple Lie group with root lattice of rank 8 * E8 lattice, special lattice in R8 * E8 manifold, mathematical object with no smooth structure or topological triangulation * E8 polytope, alternate name for the 421 semiregular (uniform) polytope * Elementary abelian group of order 8 Physics * E8 Theory, term sometimes loosely used to refer to ''An Exceptionally Simple Theory of Everything'' Transport * E-8 Joint STARS, USAF command and control aircraft * EMD E8, 1949 diesel passenger train locomotive * European route E8, part of the international E-road network, running between Tromsø, Norway and Turku, Finland * European walking route E8, a walking route from Ireland to Turkey * HMS E8, 1912 British E class submarine * London Buses route E8, runs between Ealing Broadway station and Brentford * Mikoyan-Gurevich Ye-8, 1962 supersonic jet fighter developed in the Soviet Union * E8, IATA code for the former Alpi Eagles air ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of ''semiregular'' polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures ( star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well. Operations Nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the great dirhombic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
