 8orthoplex

8orthoplex
Heptacross
Orthogonal projection
inside Petrie polygonType Regular 8polytope Family orthoplex Schläfli symbol {3^{6},4}
{3^{5,1,1}}CoxeterDynkin diagrams
7faces 256 {3^{6}} 6faces 1024 {3^{5}} 5faces 1792 {3^{4}} 4faces 1792 {3^{3}} Cells 1120 {3,3} Faces 448 {3} Edges 112 Vertices 16 Vertex figure 7orthoplex Petrie polygon hexadecagon Coxeter groups C_{8}, [3^{6},4]
D_{8}, [3^{5,1,1}]Dual 8cube Properties convex In geometry, an 8orthoplex, or 8cross polytope is a regular 8polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5cells 4faces, 1792 5faces, 1024 6faces, and 256 7faces.
It has two constructive forms, the first being regular with Schläfli symbol {3^{6},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3^{5,1,1}} or Coxeter symbol 5_{11}.
Contents
Alternate names
 Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
 Diacosipentacontahexazetton as a 256facetted 8polytope (polyzetton)
Construction
There are two Coxeter groups associated with the 8cube, one regular, dual of the octeract with the C_{8} or [4,3,3,3,3,3,3] symmetry group, and a lower symmetry with two copies of 7simplex facets, alternating, with the D_{8} or [3^{5,1,1}] symmetry group.
Images
orthographic projections B_{8} B_{7} [16] [14] B_{6} B_{5} [12] [10] B_{4} B_{3} B_{2} [8] [6] [4] A_{7} A_{5} A_{3} [8] [6] [4] Related tessellations
Related polytopes
It is a part of an infinite family of polytopes, called crosspolytopes or orthoplexes. The dual polytope is an 8hypercube, or octeract.
It is used in its alternated form 5_{11} with the 8simplex to form the 5_{21} honeycomb.
Cartesian coordinates
Cartesian coordinates for the vertices of an 8cube, centered at the origin are
 (±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0),
 (0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (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.
References
 H.S.M. Coxeter:
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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.
 Richard Klitzing, 8D uniform polytopes (polyzetta), x3o3o3o3o3o3o4o  ek
External links
 Olshevsky, George, Cross polytope at Glossary for Hyperspace.
 Polytopes of Various Dimensions
 Multidimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: 8polytopes
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