In geometry, a 7orthoplex, or 7cross polytope, is a regular
7polytope
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 128facetted
7polytope
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 (fvectors). 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 7orthoplex, 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
6simplex
Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure regular 7orthoplex 3,3,3,3,3,4 [3,3,3,3,3,4] 645120 Quasiregular 7orthoplex 3,3,3,3,31,1 [3,3,3,3,31,1] 322560 7fusil 7 [26] 128 Cartesian coordinates[edit]
Cartesian coordinates
(±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 7orthoplex Truncated 7orthoplex 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, WileyInterscience Publication, 1995, ISBN 9780471010036 [1] (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591] (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345] 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 Multidimensional 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 pgon Hexagon Pentagon Uniform polyhedron
Tetrahedron
Octahedron
Dodecahedron • Icosahedron Uniform 4polytope
5cell
16cell
Uniform 5polytope
5simplex
5orthoplex
Uniform 6polytope
6simplex
6orthoplex
Uniform 7polytope
7simplex
7orthoplex
Uniform 8polytope
8simplex
8orthoplex
Uniform 9polytope
9simplex
9orthoplex
Uniform 10polytope
10simplex
10orthoplex
Uniform npolytope nsimplex northoplex • ncube ndemicube 1k2 • 2k1 • k21 npentagonal polytope Topics:
Polytope
