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Quarter Hypercubic Honeycomb
In geometry, the quarter hypercubic honeycomb (or quarter n-cubic honeycomb) is a dimensional infinite series of Honeycomb (geometry), honeycombs, based on the hypercube honeycomb. It is given a Schläfli symbol q or Coxeter symbol qδ4 representing the regular form with three quarters of the vertices removed and containing the symmetry of Coxeter diagram#Infinite Coxeter groups, Coxeter group _ for n ≥ 5, with _4 = _4 and for quarter n-cubic honeycombs _5 = _5.Coxeter, Regular and semi-regular honeycoms, 1988, p.318-319 See also * Hypercubic honeycomb * Alternated hypercubic honeycomb * Simplectic honeycomb * Truncated simplectic honeycomb * Omnitruncated simplectic honeycomb References * Coxeter, Coxeter, H.S.M. ''Regular Polytopes (book), Regular Polytopes'', (3rd edition, 1973), Dover edition, *# pp. 122–123, 1973. (The lattice of hypercubes γn form the ''cubic honeycombs'', δn+1) *# pp. 154–156: Partial truncation or alternation, represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid .Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68 It is a part of an infinite family of polytopes, called cross-polytopes or ''orthoplexes'', and is analogous to the octahedron in three dimensions. It is Coxeter's \beta_4 polytope. Conway's name for a cross-polytope is orthoplex, for ''orthant complex''. The dual polytope is the tesseract (4-cube), which it can be combined with to form a compound figure. The 16-cell has 16 cells as the tesseract has 16 vertices. Geometry The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). Each of its 4 successor convex regular 4-polytopes can be constructed as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demihexeract Ortho Petrie
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' (hexeract) 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 HM6 for a 6-dimensional ''half measure'' polytope. Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol \left\ or . Cartesian coordinates Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract: : (±1,±1,±1,±1,±1,±1) with an odd number of plus signs. As a configuration This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quarter 6-cubic Honeycomb
In Six-dimensional space, six-dimensional Euclidean geometry, the quarter 6-cubic honeycomb is a uniform space-filling tessellation (or honeycomb (geometry), honeycomb). It has half the vertices of the 6-demicubic honeycomb, and a quarter of the vertices of a 6-cube honeycomb.Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318 Its facets are 6-demicubes, stericated 6-demicubes, and × duoprisms. Related honeycombs See also Regular and uniform honeycombs in 5-space: *6-cube honeycomb *6-demicube honeycomb * 6-simplex honeycomb * Truncated 6-simplex honeycomb * Omnitruncated 6-simplex honeycomb Notes References * Kaleidoscopes: Selected Writings of Harold Scott MacDonald Coxeter, H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] See p31 * {{Honeycombs Honeycombs (geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism. Topologically, under its highest symmetry, ,3,3 there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedron-octahedron segmentochoron. The vertex figure of the ''rectified 5-cell'' is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends. Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5-cell is not self-dual becau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quarter 5-cubic Honeycomb Verf
A quarter is one-fourth, , 25% or 0.25. Quarter or quarters may refer to: Places * Quarter (urban subdivision), a section or area, usually of a town Placenames * Quarter, South Lanarkshire, a settlement in Scotland * Le Quartier, a settlement in France * The Quarter, Anguilla * Quartier, Sud, Haiti Arts, entertainment, and media * Quarters (children's game) or bloody knuckles, a schoolyard game involving quarters or other coins * Quarters (game), a drinking game * ''Quarters!'', a 2015 album by the psychedelic rock group King Gizzard and the Lizard Wizard * Quarter note, in music one quarter of a whole note * "Quarters" (Wilco song) * "Quarter" (song) Coins * Quarter (Canadian coin), valued at one-fourth of a Canadian dollar * Quarter (United States coin), valued at one-fourth of a U.S. dollar ** Washington quarter, the current design of this coin * Quarter farthing, a British monetary unit * Quarter dollar, unit of currencies that are named dollar * Quarter guinea, a Britis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 5-demicube
In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes. Steric 5-cube Alternate names * Steric penteract, runcinated demipenteract * Small prismated hemipenteract (siphin) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of : (±1,±1,±1,±1,±3) with an odd number of plus signs. Images Related polytopes Stericantic 5-cube Alternate names * Prismatotruncated hemipenteract (pithin) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±5) with an odd number of plus signs. Images Steriruncic 5-cube Alternate names * Prismatorhombated hemipenteract (pirhin) (Jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-demicube T03 D5
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 vertices of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-demicube
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 vertices of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demipenteract Graph Ortho
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 vertices of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quarter 5-cubic Honeycomb
In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb.Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318 Its facets are 5-demicubes and runcinated 5-demicubes. Related honeycombs See also Regular and uniform honeycombs in 5-space: * 5-cube honeycomb * 5-demicube honeycomb * 5-simplex honeycomb * Truncated 5-simplex honeycomb * Omnitruncated 5-simplex honeycomb Notes References * Kaleidoscopes: Selected Writings of H. S. M. Coxeter 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 t ..., edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Prism
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms. Alternative names *Octahedral dyadic prism ( Norman W. Johnson) *Ope (Jonathan Bowers, for octahedral prism) *Triangular antiprismatic prism *Triangular antiprismatic hyperprism Coordinates It is a Hanner polytope with vertex coordinates, permuting first 3 coordinates: :( ±1,0,0 ±1) Structure The octahedral prism consists of two octahedra connected to each other via 8 triangular prisms. The triangular prisms are joined to each other via their square faces. Projections The octahedron-first orthographic projection of the octahedral prism into 3D space has an octahedral envelope. The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces. The triangular-prism-first orthographic projection of the octahedral prism into 3D space has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
