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7polytope In sevendimensional geometry , a 7POLYTOPE is a polytope contained by 6polytope facets. Each 5polytope ridge being shared by exactly two 6polytope facets . A UNIFORM 7POLYTOPE is one which is vertextransitive , and constructed from uniform 6polytope facets. CONTENTS * 1 Regular 7polytopes * 2 Characteristics * 3 Uniform 7polytopes by fundamental Coxeter groups * 4 The A7 family * 5 The B7 family * 6 The D7 family * 7 The E7 family * 8 Regular and uniform honeycombs * 8.1 Regular and uniform hyperbolic honeycombs * 9 Notes on the Wythoff construction Wythoff construction for the uniform 7polytopes * 10 References * 11 External links REGULAR 7POLYTOPESRegular 7polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with U {p,q,r,s,t} 6polytopes facets around each 4face. There are exactly three such convex regular 7polytopes : * {3,3,3,3,3,3}  7simplex * {4,3,3,3,3,3}  7cube 7cube * {3,3,3,3,3,4}  7orthoplex There are no nonconvex regular 7polytopes. CHARACTERISTICSThe topology of any given 7polytope is defined by its Betti numbers and torsion coefficients . The value of the Euler characteristic Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology [...More...]  "7polytope" on: Wikipedia Yahoo 

Uniform Polytope A UNIFORM POLYTOPE of dimension three or higher is a vertextransitive polytope bounded by uniform facets . The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertextransitive evensided 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 (2dimensional tilings and higher dimensional honeycombs ) of Euclidean and hyperbolic space to be considered polytopes as well. CONTENTS* 1 Operations * 1.1 Rectification operators * 1.2 Truncation operators * 1.3 Alternation * 2 Vertex figure * 3 Circumradius * 4 Uniform polytopes by dimension * 4.1 One dimension * 4.2 Two dimensions * 4.3 Three dimensions * 4.3.1 Constructions * 4.4 Four dimensions * 4.4.1 Constructive summary * 4.4.2 Truncated forms * 4.5 Five and higher dimensions * 5 Uniform honeycombs * 6 Scaliform polytope * 7 See also * 8 References * 9 Sources * 10 External links OPERATIONSNearly every uniform polytope can be generated by a Wythoff construction , and represented by a Coxeter diagram [...More...]  "Uniform Polytope" on: Wikipedia Yahoo 

7simplex In 7dimensional geometry , a 7simplex 7simplex is a selfdual regular 7polytope . It has 8 vertices , 28 edges , 56 triangle faces , 70 tetrahedral cells , 56 5cell 5faces, 28 5simplex 6faces, and 8 6simplex 7faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°. CONTENTS * 1 Alternate names * 2 Coordinates * 3 Images * 4 Related polytopes * 5 Notes * 6 External links ALTERNATE NAMESIt can also be called an OCTAEXON, or OCTA7TOPE, as an 8facetted polytope in 7dimensions. The name octaexon is derived from octa for eight facets in Greek and ex for having sixdimensional facets, and on. Jonathan Bowers gives an octaexon the acronym OCA [...More...]  "7simplex" on: Wikipedia Yahoo 

Rectified 7simplex In sevendimensional geometry , a RECTIFIED 7SIMPLEX is a convex uniform 7polytope , being a rectification of the regular 7simplex . There are four unique degrees of rectifications, including the zeroth, the 7simplex itself. Vertices of the rectified 7simplex are located at the edgecenters of the 7simplex. Vertices of the birectified 7simplex are located in the triangular face centers of the 7simplex. Vertices of the trirectified 7simplex are located in the tetrahedral cell centers of the 7simplex [...More...]  "Rectified 7simplex" on: Wikipedia Yahoo 

Truncated 7simplex In sevendimensional geometry , a TRUNCATED 7SIMPLEX is a convex uniform 7polytope , being a truncation of the regular 7simplex . There are unique 3 degrees of truncation. Vertices of the truncation 7simplex are located as pairs on the edge of the 7simplex. Vertices of the bitruncated 7simplex are located on the triangular faces of the 7simplex. Vertices of the tritruncated 7simplex are located inside the tetrahedral cells of the 7simplex. CONTENTS* 1 Truncated 7simplex * 1.1 Alternate names * 1.2 Coordinates * 1.3 Images * 2 Bitruncated 7simplex * 2.1 Alternate names * 2.2 Coordinates * 2.3 Images * 3 Tritruncated 7simplex * 3.1 Alternate names * 3.2 Coordinates * 3.3 Images * 4 Related polytopes * 5 See also * 6 Notes * 7 References * 8 External links TRUNCATED 7SIMPLEX TRUNCATED 7SIMPLEX Type uniform 7polytope Schläfli symbol Schläfli symbol t{3,3,3,3,3,3} CoxeterDynkin diagrams 6faces 16 5faces 4faces Cells 350 Faces 336 Edges 196 Vertices 56 Vertex figure Elongated 5simplex pyramid Coxeter groups A7, Properties convex , Vertextransitive Vertextransitive In sevendimensional geometry , a TRUNCATED 7SIMPLEX is a convex uniform 7polytope , being a truncation of the regular 7simplex [...More...]  "Truncated 7simplex" on: Wikipedia Yahoo 

Cantellated 7simplex In sevendimensional geometry , a CANTELLATED 7SIMPLEX is a convex uniform 7polytope , being a cantellation of the regular 7simplex . There are unique 6 degrees of cantellation for the 7simplex, including truncations [...More...]  "Cantellated 7simplex" on: Wikipedia Yahoo 

Runcinated 7simplex In sevendimensional geometry , a RUNCINATED 7SIMPLEX is a convex uniform 7polytope with 3rd order truncations (runcination ) of the regular 7simplex . There are 8 unique runcinations of the 7simplex with permutations of truncations, and cantellations [...More...]  "Runcinated 7simplex" on: Wikipedia Yahoo 

Stericated 7simplex In sevendimensional geometry , a STERICATED 7SIMPLEX is a convex uniform 7polytope with 4th order truncations (sterication ) of the regular 7simplex . There are 14 unique sterication for the 7simplex with permutations of truncations, cantellations, and runcinations [...More...]  "Stericated 7simplex" on: Wikipedia Yahoo 

Pentellated 7simplex In sevendimensional geometry , a PENTELLATED 7SIMPLEX is a convex uniform 7polytope with 5th order truncations (pentellation ) of the regular 7simplex . There are 16 unique pentellations of the 7simplex with permutations of truncations, cantellations, runcinations, and sterications [...More...]  "Pentellated 7simplex" on: Wikipedia Yahoo 

Hexicated 7simplex In sevendimensional geometry , a HEXICATED 7SIMPLEX is a convex uniform 7polytope , including 6thorder truncations (hexication) from the regular 7simplex . There are 20 unique hexications for the 7simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. The simple HEXICATED 7SIMPLEX is also called an EXPANDED 7SIMPLEX, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7simplex . The highest form, the hexipentisteriruncicantitruncated 7simplex is more simply called a omnitruncated 7simplex with all of the nodes ringed [...More...]  "Hexicated 7simplex" on: Wikipedia Yahoo 

7orthoplex In geometry , a 7ORTHOPLEX, or 7cross polytope , is a regular 7polytope 7polytope with 14 vertices , 84 edges , 280 triangle faces , 560 tetrahedron cells , 672 5cells 4faces, 448 5faces, and 128 6faces. It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411. It is a part of an infinite family of polytopes, called crosspolytopes or orthoplexes. The dual polytope is the 7hypercube , or hepteract [...More...]  "7orthoplex" on: Wikipedia Yahoo 

Truncated 7orthoplex In sevendimensional geometry , a TRUNCATED 7ORTHOPLEX is a convex uniform 7polytope , being a truncation of the regular 7orthoplex 7orthoplex . There are 6 truncations of the 7orthoplex. Vertices of the truncation 7orthoplex 7orthoplex are located as pairs on the edge of the 7orthoplex. Vertices of the bitruncated 7orthoplex 7orthoplex are located on the triangular faces of the 7orthoplex. Vertices of the tritruncated 7orthoplex 7orthoplex are located inside the tetrahedral cells of the 7orthoplex. The final three truncations are best expressed relative to the 7cube [...More...]  "Truncated 7orthoplex" on: Wikipedia Yahoo 

Rectified 7orthoplex In sevendimensional geometry , a RECTIFIED 7ORTHOPLEX is a convex uniform 7polytope , being a rectification of the regular 7orthoplex . There are unique 7 degrees of rectifications, the zeroth being the 7orthoplex 7orthoplex , and the 6th and last being the 7cube . Vertices of the rectified 7orthoplex 7orthoplex are located at the edgecenters of the 7orthoplex. Vertices of the birectified 7orthoplex 7orthoplex are located in the triangular face centers of the 7orthoplex. Vertices of the trirectified 7orthoplex 7orthoplex are located in the tetrahedral cell centers of the 7orthoplex [...More...]  "Rectified 7orthoplex" on: Wikipedia Yahoo 

Cantellated 7orthoplex In sevendimensional geometry , a CANTELLATED 7ORTHOPLEX is a convex uniform 7polytope , being a cantellation of the regular 7orthoplex 7orthoplex . There are ten degrees of cantellation for the 7orthoplex, including truncations [...More...]  "Cantellated 7orthoplex" on: Wikipedia Yahoo 

Runcinated 7orthoplex In sevendimensional geometry , a RUNCINATED 7ORTHOPLEX is a convex uniform 7polytope with 3rd order truncations (runcination ) of the regular 7orthoplex 7orthoplex . There are 16 unique runcinations of the 7orthoplex 7orthoplex with permutations of truncations, and cantellations. 8 are more simply constructed from the 7cube . These polytopes are among 127 uniform 7polytopes with B7 symmetry [...More...]  "Runcinated 7orthoplex" on: Wikipedia Yahoo 

Stericated 7orthoplex In sevendimensional geometry , a STERICATED 7ORTHOPLEX is a convex uniform 7polytope with 4th order truncations (sterication ) of the regular 7orthoplex 7orthoplex . There are 24 unique sterication for the 7orthoplex 7orthoplex with permutations of truncations, cantellations , and runcinations . 14 are more simply constructed from the 7cube . This polytope is one of 127 uniform 7polytopes with B7 symmetry [...More...]  "Stericated 7orthoplex" on: Wikipedia Yahoo 