| 释义 |
- Alternate names
- Construction
- Cartesian coordinates
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10-orthoplex
Decacross |
Orthogonal projection
inside Petrie polygon | | Type | Regular 10-polytope | | Family | Orthoplex | | Schläfli symbol | {38,4}
{37,31,1} | | Coxeter-Dynkin diagrams | node_1|3|node|3|node|3|node|3|node | node|3|node|3|node|3|node|4|node}}
{{CDD|node_1|3|node|3|node|3|node | node|3|node|3|node|3|node|split1|nodes} | | 9-faces | 1024 {38} | | 8-faces | 5120 {37} | | 7-faces | 11520 {36} | | 6-faces | 15360 {35} | | 5-faces | 13440 {34} | | 4-faces | 8064 {33} | | Cells | 3360 {3,3} | | Faces | 960 {3} | | Edges | 180 | | Vertices | 20 | | Vertex figure | 9-orthoplex | | Petrie polygon | Icosagon | | Coxeter groups | C10, [38,4]
D10, [37,1,1] | | Dual | 10-cube | | Properties | Convex |
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. Alternate names- 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 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 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 {{B10 Coxeter plane graphs|t9|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}}
- (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)
- {{KlitzingPolytopes|polyxenna.htm|10D uniform polytopes (polyxenna)|x3o3o3o3o3o3o3o3o4o - ka}}
External links - {{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
{{Polytopes}} 1 : 10-polytopes |