9-orthoplex
Regular 9-orthoplex
Ennecross | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 9-polytope |
Family | orthoplex |
Schläfli symbol | {3^{7},4} {3^{6},3^{1,1}} |
Coxeter-Dynkin diagrams | |
8-faces | 512 {3^{7}} |
7-faces | 2304 {3^{6}} |
6-faces | 4608 {3^{5}} |
5-faces | 5376 {3^{4}} |
4-faces | 4032 {3^{3}} |
Cells | 2016 {3,3} |
Faces | 672 {3} |
Edges | 144 |
Vertices | 18 |
Vertex figure | Octacross |
Petrie polygon | Octadecagon |
Coxeter groups | C_{9}, [3^{7},4] D_{9}, [3^{6,1,1}] |
Dual | 9-cube |
Properties | convex, Hanner polytope |
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 {3^{7},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3^{6},3^{1,1}} or Coxeter symbol 6_{11}.
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 C_{9} or [4,3^{7}] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D_{9} or [3^{6,1,1}] 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
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 [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.
- Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o4o - vee". https://bendwavy.org/klitzing/dimensions/polyyotta.htm.
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Cross.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Original source: https://en.wikipedia.org/wiki/9-orthoplex.
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