10-orthoplex

10-orthoplex
Decacross

Orthogonal projection
inside Petrie polygon
TypeRegular 10-polytope
FamilyOrthoplex
Schläfli symbol {38,4}
{37,31,1}
Coxeter-Dynkin diagrams
9-faces1024 {38}
8-faces5120 {37}
7-faces11520 {36}
6-faces15360 {35}
5-faces13440 {34}
4-faces8064 {33}
Cells3360 {3,3}
Faces960 {3}
Edges180
Vertices20
Vertex figure9-orthoplex
Petrie polygonIcosagon
Coxeter groupsC10, [38,4]
D10, [37,1,1]
Dual10-cube
PropertiesConvex

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 {38,4}, and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol {37,31,1} or Coxeter symbol 711.

It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube.

Alternate names

Construction

There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or [4,38] symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D10 or [37,1,1] symmetry group.

Cartesian coordinates

Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are

(±1,0,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0,0), (0,0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

Images

Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]

References

External links

This article is issued from Wikipedia - version of the Tuesday, January 27, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.