5-orthoplex

Regular 5-orthoplex (pentacross)
Orthogonal projection inside Petrie polygon
Type	Regular 5-polytope
Family	orthoplex
Schläfli symbol	{3,3,3,4} {3,3,3^1,1}
Coxeter-Dynkin diagrams
4-faces	32 {3³}
Cells	80 {3,3}
Faces	80 {3}
Edges	40
Vertices	10
Vertex figure	16-cell
Petrie polygon	decagon
Coxeter groups	BC₅, [3,3,3,4] D₅, [3^2,1,1]
Dual	5-cube
Properties	convex, Hanner polytope

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

It has two constructed forms, the first being regular with Schläfli symbol {3³,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3^1,1} or Coxeter symbol 2₁₁.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.

Alternate names

pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).

As a configuration

This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[1]^[2]

${\begin{bmatrix}{\begin{matrix}10&8&24&32&16\\2&40&6&12&8\\3&3&80&4&4\\4&6&4&80&2\\5&10&10&5&32\end{matrix}}\end{bmatrix}}$

Cartesian coordinates

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

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

Construction

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C₅ or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D₅ or [3^2,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

Name	Schläfli symbol	Symmetry	Order
regular 5-orthoplex	{3,3,3,4}	[3,3,3,4]	3840
Quasiregular 5-orthoplex	{3,3,3^1,1}	[3,3,3^1,1]	1920
5-fusil
	{3,3,3,4}	[4,3,3,3]	3840
	{3,3,4}+{}	[4,3,3,2]	768
	{3,4}+{4}	[4,3,2,4]	384
	{3,4}+2{}	[4,3,2,2]	192
	2{4}+{}	[4,2,4,2]	128
	{4}+3{}	[4,2,2,2]	64
	5{}	[2,2,2,2]	32

Other images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph
Dihedral symmetry	[4]	[4]

The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.

Related polytopes and honeycombs

2_k1 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E₃=A₂A₁	E₄=A₄	E₅=D₅	E₆	E₇	E₈	E₉ = ${\tilde {E}}_{8}$ = E₈⁺	E₁₀ = ${\bar {T}}_{8}$ = E₈⁺⁺
Coxeter diagram
Symmetry	[3^−1,2,1]	[3^0,2,1]	[[3^1,2,1]]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	384	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	2_−1,1	2₀₁	2₁₁	2₂₁	2₃₁	2₄₁	2₅₁	2₆₁

This polytope is one of 31 uniform 5-polytopes generated from the B₅ Coxeter plane, including the regular 5-cube and 5-orthoplex.

B5 polytopes
β₅	t₁β₅	t₂γ₅	t₁γ₅	γ₅	t_0,1β₅	t_0,2β₅	t_1,2β₅
t_0,3β₅	t_1,3γ₅	t_1,2γ₅	t_0,4γ₅	t_0,3γ₅	t_0,2γ₅	t_0,1γ₅	t_0,1,2β₅
t_0,1,3β₅	t_0,2,3β₅	t_1,2,3γ₅	t_0,1,4β₅	t_0,2,4γ₅	t_0,2,3γ₅	t_0,1,4γ₅	t_0,1,3γ₅
t_0,1,2γ₅	t_0,1,2,3β₅	t_0,1,2,4β₅	t_0,1,3,4γ₅	t_0,1,2,4γ₅	t_0,1,2,3γ₅	t_0,1,2,3,4γ₅

References

^ Coxeter, Regular Polytopes, sec 1.8 Configurations
^ Coxeter, Complex Regular Polytopes, p.117

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. "5D uniform polytopes (polytera) x3o3o3o4o - tac".

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

[1] Coxeter, Regular Polytopes, sec 1.8 Configurations

[2] Coxeter, Complex Regular Polytopes, p.117

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