10-orthoplex

10-orthoplex Decacross
Orthogonal projection inside Petrie polygon
Type	Regular 10-polytope
Family	Orthoplex
Schläfli symbol	{3⁸,4} {3⁷,3^1,1}
Coxeter-Dynkin diagrams
9-faces	1024 {3⁸}
8-faces	5120 {3⁷}
7-faces	11520 {3⁶}
6-faces	15360 {3⁵}
5-faces	13440 {3⁴}
4-faces	8064 {3³}
Cells	3360 {3,3}
Faces	960 {3}
Edges	180
Vertices	20
Vertex figure	9-orthoplex
Petrie polygon	Icosagon
Coxeter groups	C₁₀, [3⁸,4] D₁₀, [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 {3⁸,4}, and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol {3⁷,3^1,1} 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

Decacross is derived from combining the family name cross polytope with deca for ten (dimensions) in Greek
Chilliaicositetraronnon as a 1024-facetted 10-polytope (polyronnon).

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 [3^7,1,1] 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]
A₉		A₅
—		—
[10]		[6]
A₇		A₃
—		—
[8]		[4]

References

H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  - (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
  - (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
  - (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
Klitzing, Richard. "10D uniform polytopes (polyxenna) x3o3o3o3o3o3o3o3o4o - ka".

External links

Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Polytopes of Various Dimensions
Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds