9-orthoplex

Regular 9-orthoplex

Orthogonal projection
inside Petrie polygon
TypeRegular 9-polytope
Familyorthoplex
Schläfli symbol {37,4}
{36,31,1}
Coxeter-Dynkin diagrams
8-faces512 {37}
7-faces2304 {36}
6-faces4608 {35}
5-faces5376 {34}
4-faces4032 {33}
Cells2016 {3,3}
Faces672 {3}
Edges144
Vertices18
Vertex figureOctacross
Petrie polygonOctadecagon
Coxeter groupsC9, [37,4]
D9, [36,1,1]
Dual9-cube
Propertiesconvex

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 {36,31,1} or Coxeter symbol 611.

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

Construction

There are two Coxeter groups associated with the 9-orthoplex, 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 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

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

References

External links

This article is issued from Wikipedia - version of the Tuesday, April 21, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.