10-orthoplex Decacross | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 10-polytope |
Family | Orthoplex |
Schläfli symbol | {3^{8},4} {3^{7},3^{1,1}} |
Coxeter-Dynkin diagrams | |
9-faces | 1024 {3^{8}} |
8-faces | 5120 {3^{7}} |
7-faces | 11520 {3^{6}} |
6-faces | 15360 {3^{5}} |
5-faces | 13440 {3^{4}} |
4-faces | 8064 {3^{3}} |
Cells | 3360 {3,3} |
Faces | 960 {3} |
Edges | 180 |
Vertices | 20 |
Vertex figure | 9-orthoplex |
Petrie polygon | Icosagon |
Coxeter groups | C_{10}, [3^{8},4] D_{10}, [3^{7,1,1}] |
Dual | 10-cube |
Properties | Convex, Hanner polytope |
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 {3^{8},4}, and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol {3^{7},3^{1,1}} or Coxeter symbol 7_{11}.
It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube.
There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C_{10} or [4,3^{8}] symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D_{10} or [3^{7,1,1}] symmetry group.
Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are
Every vertex pair is connected by an edge, except opposites.
B_{10} | B_{9} | B_{8} |
---|---|---|
[20] | [18] | [16] |
B_{7} | B_{6} | B_{5} |
[14] | [12] | [10] |
B_{4} | B_{3} | B_{2} |
[8] | [6] | [4] |
A_{9} | A_{5} | |
— | — | |
[10] | [6] | |
A_{7} | A_{3} | |
— | — | |
[8] | [4] |