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The Info List - 10-orthoplex


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In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-faces. It has two constructed forms, the first being regular with Schläfli symbol 38,4 , and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol 37,31,1 or Coxeter symbol 711. It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube.

Contents

1 Alternate names 2 Construction 3 Cartesian coordinates 4 Images 5 References 6 External links

Alternate names[edit]

Decacross is derived from combining the family name cross polytope with deca for ten (dimensions) in Greek Chilliaicositetraxennon as a 1024-facetted 10-polytope (polyxennon).

Construction[edit] There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or [4,38] symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D10 or [37,1,1] symmetry group. Cartesian coordinates[edit] Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are

(±1,0,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0,0), (0,0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites. Images[edit]

Orthographic projections

B10 B9 B8

[20] [18] [16]

B7 B6 B5

[14] [12] [10]

B4 B3 B2

[8] [6] [4]

A9 A5

— —

[10] [6]

A7 A3

— —

[8] [4]

References[edit]

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. "10D uniform polytopes (polyxenna) x3o3o3o3o3o3o3o3o4o - ka". 

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

.