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10-polytope
In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets. Regular 10-polytopes Regular 10-polytopes can be represented by the Schläfli symbol , with x 9-polytope facets around each peak. There are exactly three such convex regular 10-polytopes: # - 10-simplex # - 10-cube # - 10-orthoplex There are no nonconvex regular 10-polytopes. Euler characteristic The topology of any given 10-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 Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10-polytopes, whatever their underlying topology. This inadequacy of the E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified Decacross
In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex. These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry. Rectified 10-orthoplex In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex. Rectified 10-orthoplex The ''rectified 10-orthoplex'' is the vertex figure for the demidekeractic honeycomb. : or Alternate names * rectified decacross (Acronym rake) (Jonathan Bowers) Construction There are two Coxeter groups associated with the ''rectifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 10-orthoplex
In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex. These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry. Rectified 10-orthoplex In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex. Rectified 10-orthoplex The ''rectified 10-orthoplex'' is the vertex figure for the demidekeractic honeycomb. : or Alternate names * rectified decacross (Acronym rake) (Jonathan Bowers) Construction There are two Coxeter groups associated with the ''r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 10-cube
In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube. There are 10 rectifications of the ''10-cube'', with the zeroth being the 10-cube itself. Vertices of the ''rectified 10-cube'' are located at the edge-centers of the ''10-cube''. Vertices of the ''birectified 10-cube'' are located in the square face centers of the ''10-cube''. Vertices of the ''trirectified 10-cube'' are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the 10-orthoplex. These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry. Rectified 10-cube Alternate names * Rectified dekeract (Acronym rade) (Jonathan Bowers) Cartesian coordinates Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length \sqrt are all permutations of: : (±1,±1,±1,±1,±1,±1,±1,±1,±1,0) Images Birectified ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10-orthoplex
In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells ''4-faces'', 13440 ''5-faces'', 15360 ''6-faces'', 11520 ''7-faces'', 5120 ''8-faces'', and 1024 ''9-faces''. It has two constructed forms, the first being regular with Schläfli symbol , and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol or Coxeter symbol 711. It is one of an infinite family of polytopes, called cross-polytopes or ''orthoplexes''. The dual polytope is the 10-hypercube or 10-cube. Alternate names *Decacross is derived from combining the family name ''cross polytope'' with ''deca'' for ten (dimensions) in Greek * Chilliaicositetraxennon as a 1024- facetted 10-polytope (polyxennon). Construction There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or ,38symmetry group, and a lower symmetry with two copies of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 10-simplex
In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex. These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry. There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex. Rectified 10-simplex The ''rectified 10-simplex'' is the vertex figure of the 11-demicube. Alternate names * Rectified hendecaxennon (Acronym ru) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''rectified 10-simplex'' can be most simply positioned in 11-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10-demicube
In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube 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 HM10 for a ten-dimensional ''half measure'' polytope. Coxeter named this polytope as 171 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 demidekeract centered at the origin are alternate halves of the dekeract: : (±1,±1,±1,±1,±1,±1,±1,±1,±1,±1) with an odd number of plus signs. Images References * H.S.M. Coxeter: ** Coxeter, ''Regular Polytopes'', (3rd edition, 1973), Dover edition, , p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10-cube
In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces. It can be named by its Schläfli symbol , being composed of 3 9-cubes around each 8-face. It is sometimes called a dekeract, a portmanteau of tesseract (the ''4-cube'') and ''deka-'' for ten (dimensions) in Greek, It can also be called an icosaxennon or icosa-10-tope as a 10 dimensional polytope, constructed from 20 regular facets. It is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes. Cartesian coordinates Cartesian coordinates for the vertices of a dekeract centered at the origin and edge length 2 are : (±1,±1,±1,±1,±1,±1,±1,±1,±1,±1) while the interior of the same consis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10-simplex
In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos−1(1/10), or approximately 84.26°. It can also be called a hendecaronnon, or hendeca-10-tope, as an 11- facetted polytope in 10-dimensions. The name ''hendecaronnon'' is derived from ''hendeca'' for 11 facets in Greek and -ronn (variation of ennea for nine), having 9-dimensional facets, and ''-on''. Coordinates The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are: :\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \pm1\right) :\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0\right) :\left(\sqrt,\ \sqrt,\ 1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0\right) :\ ... [...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] |
Truncated 10-cube
Truncation is the term used for limiting the number of digits right of the decimal point by discarding the least significant ones. Truncation may also refer to: Mathematics * Truncation (statistics) refers to measurements which have been cut off at some value * Truncation (numerical analysis) refers to truncating an infinite sum by a finite one * Truncation (geometry) is the removal of one or more parts, as for example in truncated cube * Propositional truncation, a type former which truncates a type down to a mere proposition Computer science * Data truncation, an event that occurs when a file or other data is stored in a location too small to accommodate its entire length * Truncate (SQL), a command in the SQL data manipulation language to quickly remove all data from a table Biology * Truncate, a leaf shape * Truncated protein, a protein shortened by a mutation which specifically induces premature termination of messenger RNA translation Other uses * Cheque truncation, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10-simplex T07
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 10-orthoplex
Truncation is the term used for limiting the number of digits right of the decimal point by discarding the least significant ones. Truncation may also refer to: Mathematics * Truncation (statistics) refers to measurements which have been cut off at some value * Truncation (numerical analysis) refers to truncating an infinite sum by a finite one * Truncation (geometry) is the removal of one or more parts, as for example in truncated cube * Propositional truncation, a type former which truncates a type down to a mere proposition Computer science * Data truncation, an event that occurs when a file or other data is stored in a location too small to accommodate its entire length * Truncate (SQL), a command in the SQL data manipulation language to quickly remove all data from a table Biology * Truncate, a leaf shape * Truncated protein, a protein shortened by a mutation which specifically induces premature termination of messenger RNA translation Other uses * Cheque truncation, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
