In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 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 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. Contents 1 Alternate names 2 As a configuration 3 Images 4 Construction 5 Cartesian coordinates 6 See also 7 References 8 External links Alternate names[edit] Heptacross, derived from combining the family name cross polytope with hept for seven (dimensions) in Greek. Hecatonicosoctaexon as a 128-facetted 7-polytope (polyexon). As a configuration[edit] 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 Images[edit] orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph Dihedral symmetry [6] [4] Construction[edit] 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 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 figure regular 7-orthoplex 3,3,3,3,3,4 [3,3,3,3,3,4] 645120 Quasiregular 7-orthoplex 3,3,3,3,31,1 [3,3,3,3,31,1] 322560 7-fusil 7 [26] 128 Cartesian coordinates[edit] 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 also[edit] Rectified 7-orthoplex Truncated 7-orthoplex References[edit] ^ Coxeter, Regular Polytopes, sec 1.8 Configurations ^ Coxeter, Complex Regular Polytopes, p.117 H.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 links[edit] Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polyt |