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 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.Contents1 Alternate names 2 As a configuration 3 Images 4 Construction 5 Cartesian coordinates 6 See also 7 References 8 External linksAlternate namesHeptacross, derived from combining the family name cross polytope with hept for seven (dimensions) in Greek. Hecatonicosoctaexon as a 128-facetted 7-polytope 7-polytope (polyexon).As a configuration The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[1][2] [ 14 12 60 160 240 192 64 2 84 10 40 80 80 32 3 3 280 8 24 32 16 4 6 4 560 6 12 8 5 10 10 5 672 4 4 6 15 20 15 6 448 2 7 21 35 35 21 7 128 ] displaystyle begin bmatrix begin matrix 14&12&60&160&240&192&64\2&84&10&40&80&80&32\3&3&280&8&24&32&16\4&6&4&560&6&12&8\5&10&10&5&672&4&4\6&15&20&15&6&448&2\7&21&35&35&21&7&128end matrix end bmatrix Imagesorthographic projectionsCoxeter plane B7 / A6 B6 / D7 B5 / D6 / A4GraphDihedral symmetry [14] [12] [10]Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3GraphDihedral symmetry [8] [6] [4]Coxeter plane A5 A3GraphDihedral symmetry [6] [4]Construction There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figureregular 7-orthoplex 3,3,3,3,3,4 [3,3,3,3,3,4] 645120Quasiregular 7-orthoplex 3,3,3,3,31,1 [3,3,3,3,31,1] 3225607-fusil7 [26] 128Cartesian coordinates Cartesian coordinates Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are(±1,0,0,0,0,0,0), (0,±1,0,0,0,0,0), (0,0,±1,0,0,0,0), (0,0,0,±1,0,0,0), (0,0,0,0,±1,0,0), (0,0,0,0,0,±1,0), (0,0,0,0,0,0,±1)Every vertex pair is connected by an edge, except opposites. See alsoRectified 7-orthoplex Truncated 7-orthoplexReferences^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117H.S.M. Coxeter:H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1](Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]Norman Johnson Uniform Polytopes, Manuscript (1991)N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o4o - zee". External linksOlshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.  Polytopes of Various Dimensions Multi-dimensional Glossaryv t eFundamental convex regular and uniform polytopes in dimensions 2–10Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 HnRegular polygon Triangle Square p-gon Hexagon PentagonUniform polyhedron Tetrahedron Octahedron Octahedron • Cube DemicubeDodecahedron • IcosahedronUniform 4-polytope 5-cell 16-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell 120-cell • 600-cellUniform 5-polytope 5-simplex 5-orthoplex 5-orthoplex • 5-cube 5-demicubeUniform 6-polytope 6-simplex 6-orthoplex 6-orthoplex • 6-cube 6-demicube 122 • 221Uniform 7-polytope 7-simplex 7-orthoplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321Uniform 8-polytope 8-simplex 8-orthoplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421Uniform 9-polytope 9-simplex 9-orthoplex 9-orthoplex • 9-cube 9-demicubeUniform 10-polytope 10-simplex 10-orthoplex 10-orthoplex • 10-cube 10-demicubeUniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytopeTopics: Polytope Polytope families • Regular polytope Regular polytope • List of regular polyt

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