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.
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 for the vertices of a 9-orthoplex, centered at the origin, are
Every vertex pair is connected by an edge, except opposites.
B_{9} | B_{8} | B_{7} | |||
---|---|---|---|---|---|
[18] | [16] | [14] | |||
B_{6} | B_{5} | ||||
[12] | [10] | ||||
B_{4} | B_{3} | B_{2} | |||
[8] | [6] | [4] | |||
A_{7} | A_{5} | A_{3} | |||
— | — | — | |||
[8] | [6] | [4] |