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7-polytope
In seven-dimensional geometry , a 7-POLYTOPE is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets . A UNIFORM 7-POLYTOPE is one which is vertex-transitive , and constructed from uniform 6-polytope facets. CONTENTS * 1 Regular 7-polytopes * 2 Characteristics * 3 Uniform 7-polytopes 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 7-polytopes * 10 References * 11 External links REGULAR 7-POLYTOPESRegular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with U {p,q,r,s,t} 6-polytopes facets around each 4-face. There are exactly three such convex regular 7-polytopes : * {3,3,3,3,3,3} - 7-simplex * {4,3,3,3,3,3} - 7-cube
7-cube
* {3,3,3,3,3,4} - 7-orthoplex There are no nonconvex regular 7-polytopes. CHARACTERISTICSThe topology of any given 7-polytope 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
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Uniform Polytope
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. 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
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7-simplex
In 7-dimensional geometry , a 7-simplex
7-simplex
is a self-dual regular 7-polytope . It has 8 vertices , 28 edges , 56 triangle faces , 70 tetrahedral cells , 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. 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 OCTA-7-TOPE, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym OCA
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Rectified 7-simplex
In seven-dimensional geometry , a RECTIFIED 7-SIMPLEX is a convex uniform 7-polytope , being a rectification of the regular 7-simplex . There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex
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Truncated 7-simplex
In seven-dimensional geometry , a TRUNCATED 7-SIMPLEX is a convex uniform 7-polytope , being a truncation of the regular 7-simplex . There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex. CONTENTS* 1 Truncated 7-simplex * 1.1 Alternate names * 1.2 Coordinates * 1.3 Images * 2 Bitruncated 7-simplex * 2.1 Alternate names * 2.2 Coordinates * 2.3 Images * 3 Tritruncated 7-simplex * 3.1 Alternate names * 3.2 Coordinates * 3.3 Images * 4 Related polytopes * 5 See also * 6 Notes * 7 References * 8 External links TRUNCATED 7-SIMPLEX TRUNCATED 7-SIMPLEX Type uniform 7-polytope Schläfli symbol
Schläfli symbol
t{3,3,3,3,3,3} Coxeter-Dynkin diagrams 6-faces 16 5-faces 4-faces Cells 350 Faces 336 Edges 196 Vertices 56 Vertex figure Elongated 5-simplex pyramid Coxeter groups A7, Properties convex , Vertex-transitive
Vertex-transitive
In seven-dimensional geometry , a TRUNCATED 7-SIMPLEX is a convex uniform 7-polytope , being a truncation of the regular 7-simplex
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Cantellated 7-simplex
In seven-dimensional geometry , a CANTELLATED 7-SIMPLEX is a convex uniform 7-polytope , being a cantellation of the regular 7-simplex . There are unique 6 degrees of cantellation for the 7-simplex, including truncations
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Runcinated 7-simplex
In seven-dimensional geometry , a RUNCINATED 7-SIMPLEX is a convex uniform 7-polytope with 3rd order truncations (runcination ) of the regular 7-simplex . There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations
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Stericated 7-simplex
In seven-dimensional geometry , a STERICATED 7-SIMPLEX is a convex uniform 7-polytope with 4th order truncations (sterication ) of the regular 7-simplex . There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations, and runcinations
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Pentellated 7-simplex
In seven-dimensional geometry , a PENTELLATED 7-SIMPLEX is a convex uniform 7-polytope with 5th order truncations (pentellation ) of the regular 7-simplex . There are 16 unique pentellations of the 7-simplex with permutations of truncations, cantellations, runcinations, and sterications
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Hexicated 7-simplex
In seven-dimensional geometry , a HEXICATED 7-SIMPLEX is a convex uniform 7-polytope , including 6th-order truncations (hexication) from the regular 7-simplex . There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. The simple HEXICATED 7-SIMPLEX is also called an EXPANDED 7-SIMPLEX, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex . The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed
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7-orthoplex
In geometry , a 7-ORTHOPLEX, or 7-cross polytope , is a regular 7-polytope
7-polytope
with 14 vertices , 84 edges , 280 triangle faces , 560 tetrahedron cells , 672 5-cells 4-faces, 448 5-faces, and 128 6-faces. 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 cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube , or hepteract
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Truncated 7-orthoplex
In seven-dimensional geometry , a TRUNCATED 7-ORTHOPLEX is a convex uniform 7-polytope , being a truncation of the regular 7-orthoplex
7-orthoplex
. There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex
7-orthoplex
are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex
7-orthoplex
are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex
7-orthoplex
are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube
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Rectified 7-orthoplex
In seven-dimensional geometry , a RECTIFIED 7-ORTHOPLEX is a convex uniform 7-polytope , being a rectification of the regular 7-orthoplex . There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex
7-orthoplex
, and the 6th and last being the 7-cube . Vertices of the rectified 7-orthoplex
7-orthoplex
are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex
7-orthoplex
are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex
7-orthoplex
are located in the tetrahedral cell centers of the 7-orthoplex
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Cantellated 7-orthoplex
In seven-dimensional geometry , a CANTELLATED 7-ORTHOPLEX is a convex uniform 7-polytope , being a cantellation of the regular 7-orthoplex
7-orthoplex
. There are ten degrees of cantellation for the 7-orthoplex, including truncations
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Runcinated 7-orthoplex
In seven-dimensional geometry , a RUNCINATED 7-ORTHOPLEX is a convex uniform 7-polytope with 3rd order truncations (runcination ) of the regular 7-orthoplex
7-orthoplex
. There are 16 unique runcinations of the 7-orthoplex
7-orthoplex
with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-cube . These polytopes are among 127 uniform 7-polytopes with B7 symmetry
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Stericated 7-orthoplex
In seven-dimensional geometry , a STERICATED 7-ORTHOPLEX is a convex uniform 7-polytope with 4th order truncations (sterication ) of the regular 7-orthoplex
7-orthoplex
. There are 24 unique sterication for the 7-orthoplex
7-orthoplex
with permutations of truncations, cantellations , and runcinations . 14 are more simply constructed from the 7-cube . This polytope is one of 127 uniform 7-polytopes with B7 symmetry
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