Regular 9-orthoplex
Ennecross | |
---|---|
![]() Orthogonal projection inside Petrie polygon | |
Type | Regular 9-polytope |
Family | orthoplex |
Schläfli symbol | {37,4} {36,31,1} |
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 | 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 {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.
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 for the vertices of a 9-orthoplex, centered at the origin, are
Every vertex pair is connected by an edge, except opposites.
B9 | B8 | B7 | |||
---|---|---|---|---|---|
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[18] | [16] | [14] | |||
B6 | B5 | ||||
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[12] | [10] | ||||
B4 | B3 | B2 | |||
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[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
— | — | — | |||
[8] | [6] | [4] |
Regular 9-orthoplex
Ennecross | |
---|---|
![]() Orthogonal projection inside Petrie polygon | |
Type | Regular 9-polytope |
Family | orthoplex |
Schläfli symbol | {37,4} {36,31,1} |
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 | 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 {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.
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 for the vertices of a 9-orthoplex, centered at the origin, are
Every vertex pair is connected by an edge, except opposites.
B9 | B8 | B7 | |||
---|---|---|---|---|---|
![]() |
![]() |
![]() | |||
[18] | [16] | [14] | |||
B6 | B5 | ||||
![]() |
![]() | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
![]() |
![]() |
![]() | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
— | — | — | |||
[8] | [6] | [4] |