| 释义 |
- Alternate names
- Construction
- Cartesian coordinates
- Images
- References
- External links
| Regular 9-orthoplex |
Orthogonal projection
inside Petrie polygon | | Type | Regular 9-polytope | | Family | orthoplex | | Schläfli symbol | {37,4}
{36,31,1} | | Coxeter-Dynkin diagrams | node_1|3|node|3|node|3|node | node|3|node|3|node|3|node|4|node}}
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}} | | 8-faces | 512 {37} | | 7-faces | 2304 {36} | | 6-faces | 4608 {35} | | 5-faces | 5376 {34} | | 4-faces | 4032 {33} | | Cells | 2016 {3,3} | | Faces | 672 {3} | | Edges | 144 | | Vertices | 18 | | Vertex figure | Octacross | | Petrie polygon | Octadecagon | | Coxeter groups | C9, [37,4]
D9, [36,1,1] | | Dual | 9-cube | | Properties | convex |
In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces. It has two constructed forms, the first being regular with Schläfli symbol {37,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {36,31,1} or Coxeter symbol 611. It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract. Alternate names- Enneacross, derived from combining the family name cross polytope with ennea for nine (dimensions) in Greek
- Pentacosidodecayotton as a 512-facetted 9-polytope (polyyotton)
Construction There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C9 or [4,37] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D9 or [36,1,1] symmetry group. Cartesian coordinates Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are (±1,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,±1) Every vertex pair is connected by an edge, except opposites. Images {{B9 Coxeter plane graphs|t8|200|NoA7=true|NoA5=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.
- {{KlitzingPolytopes|polyyotta.htm|9D uniform polytopes (polyyotta)|x3o3o3o3o3o3o3o4o - vee}}
External links - {{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
{{Polytopes}} 1 : 9-polytopes |