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Demipenteract
In Five-dimensional space, five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with Alternation (geometry), alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only Semiregular polytope, semiregular 5-polytope (made of more than one type of regular Facet (geometry), facets), he called it a Semiregular polytope, 5-ic semi-regular. Emanuel Lodewijk Elte, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM5 for a 5-dimensional ''half measure'' polytope. Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol \left\ or . It exists in the semiregular k 21 polytope, k21 polytope family as 121 with the Gosset polytopes: Gosset 2 21 polytope, 221, Gosset 3 21 polytope, 321, and Gosset 4 21 polytope, 421. The graph formed by the vertices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-demicube Verf
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' ( penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM5 for a 5-dimensional ''half measure'' polytope. Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol \left\ or . It exists in the k21 polytope family as 121 with the Gosset polytopes: 221, 321, and 421. The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead. Cartesian coordinates Cartesian coordinates for the vertice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demihypercube
In geometry, demihypercubes (also called ''n-demicubes'', ''n-hemicubes'', and ''half measure polytopes'') are a class of ''n''-polytopes constructed from alternation of an ''n''-hypercube, labeled as ''hγn'' for being ''half'' of the hypercube family, ''γn''. Half of the vertices are deleted and new facets are formed. The 2''n'' facets become 2''n'' (''n''−1)-demicubes, and 2''n'' (''n''−1)-simplex facets are formed in place of the deleted vertices. They have been named with a ''demi-'' prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered ''semiregular'' for having only regular facets. Higher forms do not have all regular facets but are all uniform polytopes. The vertices and edges of a demihypercube form two copies of the halved cube graph. An ''n''-demicube has inversion symmetry if ''n'' is even. Dis ... [...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-polytopes, but can be extended backwards to include the 5- 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)- 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 tessellation of 9-dimensional hyperbolic space. The complete family of 1k2 polytope polytopes are: # 5-cell: 102, (5 tetrahedral cells) # 112 polytope, (16 5-cell, and 10 16-cell facets) # 122 polytope, (54 demipenteract facets) # 132 polytope, (56 122 and 126 demihexeract facets) # ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular K 21 Polytope
In geometry, a uniform ''k''21 polytope is a polytope in ''k'' + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol ''k''21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the ''k''-node sequence. Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the ''5-ic semiregular figure''. Family members The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of hyperbolic 9-space constructed of ∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.) The family starts uniquely as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-polytope
In geometry, a five-dimensional polytope (or 5-polytope or polyteron) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell. Definition A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron, and a 4-face is a 4-polytope. Furthermore, the following requirements must be met: # Each cell must join exactly two 4-faces. # Adjacent 4-faces are not in the same four-dimensional hyperplane. # The figure is not a compound of other figures which meet the requirements. Characteristics The topology of any given 5-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternation (geometry)
In geometry, an alternation or ''partial truncation'', is an operation on a polygon, polyhedron, tessellation, tiling, or higher dimensional polytope that removes alternate vertices.Coxeter, Regular polytopes, pp. 154–156 8.6 Partial truncation, or alternation Coxeter labels an ''alternation'' by a prefixed ''h'', standing for ''hemi'' or ''half''. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be ''alternated''. For example, the alternation of a vertex figure with ''2a.2b.2c'' is ''a.3.b.3.c.3'' where the three is the number of elements in this vertex figure. A special case is square faces whose order divides in half into degenerate digons. So for example, the cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and for his generalization of Descartes' theorem on tangent circles to four and higher dimensions. Biography Thorold Gosset was born in Thames Ditton, the son of John Jackson Gosset, a civil servant and statistical officer for HM Customs,UK Census 1871, RG10-863-89-23 and his wife Eleanor Gosset (formerly Thorold). He was admitted to Pembroke College, Cambridge as a pensioner on 1 October 1888, graduated BA in 1891, was called to the bar of the Inner Temple in June 1895, and graduated LLM in 1896. In 1900 he married Emily Florence Wood, and they subsequently had two children, named Kathleen and John.UK Census 1911, RG14-181-9123-19 Mathematics According to H. S. M. Coxeter, after obtaining his law degree in 1896 and having no client ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular Polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polytopes of the Hyperspaces'' which included a wider definition. Gosset's list In three-dimensional space and below, the terms ''semiregular polytope'' and '' uniform polytope'' have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the ''k''21 polytopes, where the rectified 5-cell is the special case of ''k'' = 0. These were all listed by Gosset, but a proof o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Penteract
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, , around each cubic ridge. Related polytopes It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. Applying an '' alternation'' operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes. The 5-cube can be seen as an ''order-3 tesseractic honeycomb'' on a 4-sphere. It is related to the Euclidean 4-space (order-4) tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic honeycomb. As a configuration This configuration matrix represents the 5-cube. The rows and columns correspond to vertices, edges, faces, cells, and 4- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: * In three-dimensional geometry, some authors call a facet of a polyhedron any polygon whose corners are vertices of the polyhedron, including polygons that are not ''Face (geometry), faces''. To ''facetting, facet'' a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to ''stellation'' and may also be applied to higher-dimensional polytopes. * In polyhedral combinatorics and in the general theory of polytopes, a Face (geometry), face that has dimension ''n'' − 1 (an (''n'' − 1)-face or hyperface) is called a Face (geometry)#Facet, facet. In this terminology, every facet is a face. * A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex.. For (boundary complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
