9-orthoplex: Difference between revisions
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|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3<sup>7</sup>,4}<BR>{3<sup>6,1,1</sup>} |
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {3<sup>7</sup>,4}<BR>{3<sup>6,1,1</sup>} |
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|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|3|node||3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|nodea_1|3a|nodea|3a|nodea||3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea}}<BR>{{CDD|node_h|4|node|3|node|3|node||3|node|3|node|3|node|3|node|3|node}}<BR>{{CDD|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h}} |
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|3|node|3|node||3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|nodea_1|3a|nodea|3a|nodea||3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea}}<BR>{{CDD|node_h|4|node|3|node|3|node||3|node|3|node|3|node|3|node|3|node}}<BR>{{CDD|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h|2|node_h}} |
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|bgcolor=#e7dcc3|8-faces||512 [[8-simplex|{3<sup>7</sup>}]][[Image:8-simplex_t0.svg|25px]] |
|bgcolor=#e7dcc3|8-faces||512 [[8-simplex|{3<sup>7</sup>}]][[Image:8-simplex_t0.svg|25px]] |
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Revision as of 04:00, 30 January 2011
| Regular Enneacross (9-orthoplex) | |
|---|---|
 Orthogonal projection inside Petrie polygon | |
| Type | Regular 9-polytope |
| Family | orthoplex |
| Schläfli symbol | {37,4} {36,1,1} |
| Coxeter-Dynkin diagrams |          
|
| 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 | Enneract |
| 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 {36,1,1} or Coxeter symbol 611.
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)
Related polytopes
It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.
Construction
There are two Coxeter groups associated with the enneacross, 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 an enneacross, 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.
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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