6-orthoplex

Last updated November 17, 2022

6-orthoplex Hexacross
Orthogonal projection inside Petrie polygon
Type	Regular 6-polytope
Family	orthoplex
Schläfli symbols	{3,3,3,3,4} {3,3,3,3^1,1}
Coxeter-Dynkin diagrams	=
5-faces	64 {3⁴}
4-faces	192 {3³}
Cells	240 {3,3}
Faces	160 {3}
Edges	60
Vertices	12
Vertex figure	5-orthoplex
Petrie polygon	dodecagon
Coxeter groups	B₆, [4,3⁴] D₆, [3^3,1,1]
Dual	6-cube
Properties	convex, Hanner polytope

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

Alternate names

Hexacross, derived from combining the family name cross polytope with hex for six (dimensions) in Greek.
Hexacontitetrapeton as a 64-facetted 6-polytope.

As a configuration

This configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[1]^[2]

${\begin{bmatrix}{\begin{matrix}12&10&40&80&80&32\\2&60&8&24&32&16\\3&3&160&6&12&8\\4&6&4&240&4&4\\5&10&10&5&192&2\\6&15&20&15&6&64\end{matrix}}\end{bmatrix}}$

Construction

There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C₆ or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D₆ or [3^3,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.

Name	Schläfli	Symmetry	Order
Regular 6-orthoplex	{3,3,3,3,4}	[4,3,3,3,3]	46080
Quasiregular 6-orthoplex	{3,3,3,3^1,1}	[3,3,3,3^1,1]	23040
6-fusil	{3,3,3,4}+{}	[4,3,3,3,3]	7680
	{3,3,4}+{4}	[4,3,3,2,4]	3072
	2{3,4}	[4,3,2,4,3]	2304
	{3,3,4}+2{}	[4,3,3,2,2]	1536
	{3,4}+{4}+{}	[4,3,2,4,2]	768
	3{4}	[4,2,4,2,4]	512
	{3,4}+3{}	[4,3,2,2,2]	384
	2{4}+2{}	[4,2,4,2,2]	256
	{4}+4{}	[4,2,2,2,2]	128
	6{}	[2,2,2,2,2]	64

Cartesian coordinates

Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are

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

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

Images

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.^[3]

2D		3D
Icosahedron {3,5} = H₃ Coxeter plane	6-orthoplex {3,3,3,3^1,1} = D₆ Coxeter plane	Icosahedron	6-orthoplex
This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a regular icosahedron. This represents a geometric folding of the D₆ to H₃ Coxeter groups: : to . On the left, seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron.

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

3_k1 dimensional figures
Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A₃A₁	A₅	D₆	E₇	${\tilde {E}}_{7}$ =E₇⁺	${\bar {T}}_{8}$ =E₇⁺⁺
Coxeter diagram
Symmetry	[3^−1,3,1]	[3^0,3,1]	[[3^1,3,1]] = [4,3,3,3,3]	[3^2,3,1]	[3^3,3,1]	[3^4,3,1]
Order	48	720	46,080	2,903,040	∞
Graph					-	-
Name	3_1,-1	3₁₀	3₁₁	3₂₁	3₃₁	3₄₁

This polytope is one of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

Related Research Articles

In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

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.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos⁻¹(1/5), or approximately 78.46°.

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

<span class="mw-page-title-main">8-cube</span> 8-dimensional hypercube

In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

<span class="mw-page-title-main">9-cube</span> 9-dimensional hypercube

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

<span class="mw-page-title-main">7-orthoplex</span>

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos⁻¹(1/6), or approximately 80.41°.

<span class="mw-page-title-main">8-orthoplex</span>

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

<span class="mw-page-title-main">9-orthoplex</span>

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.

<span class="mw-page-title-main">10-cube</span> 10-dimensional hypercube

In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

<span class="mw-page-title-main">10-orthoplex</span>

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.

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

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
  - (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. "6D uniform polytopes (polypeta) x3o3o3o3o4o - gee".

Specific

↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
↑ Coxeter, Complex Regular Polytopes, p.117
↑ Quasicrystals and Geometry , Marjorie Senechal, 1996, Cambridge University Press, p64. 2.7.1 The I₆ crystal

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

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[1] Coxeter, Regular Polytopes, sec 1.8 Configurations

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

[3] Quasicrystals and Geometry , Marjorie Senechal, 1996, Cambridge University Press, p64. 2.7.1 The I₆ crystal

[1]

[2]

[3]

B6 polytopes
β₆	t₁β₆	t₂β₆	t₂γ₆	t₁γ₆	γ₆	t_0,1β₆	t_0,2β₆
t_1,2β₆	t_0,3β₆	t_1,3β₆	t_2,3γ₆	t_0,4β₆	t_1,4γ₆	t_1,3γ₆	t_1,2γ₆
t_0,5γ₆	t_0,4γ₆	t_0,3γ₆	t_0,2γ₆	t_0,1γ₆	t_0,1,2β₆	t_0,1,3β₆	t_0,2,3β₆
t_1,2,3β₆	t_0,1,4β₆	t_0,2,4β₆	t_1,2,4β₆	t_0,3,4β₆	t_1,2,4γ₆	t_1,2,3γ₆	t_0,1,5β₆
t_0,2,5β₆	t_0,3,4γ₆	t_0,2,5γ₆	t_0,2,4γ₆	t_0,2,3γ₆	t_0,1,5γ₆	t_0,1,4γ₆	t_0,1,3γ₆
t_0,1,2γ₆	t_0,1,2,3β₆	t_0,1,2,4β₆	t_0,1,3,4β₆	t_0,2,3,4β₆	t_1,2,3,4γ₆	t_0,1,2,5β₆	t_0,1,3,5β₆
t_0,2,3,5γ₆	t_0,2,3,4γ₆	t_0,1,4,5γ₆	t_0,1,3,5γ₆	t_0,1,3,4γ₆	t_0,1,2,5γ₆	t_0,1,2,4γ₆	t_0,1,2,3γ₆
t_0,1,2,3,4β₆	t_0,1,2,3,5β₆	t_0,1,2,4,5β₆	t_0,1,2,4,5γ₆	t_0,1,2,3,5γ₆	t_0,1,2,3,4γ₆	t_0,1,2,3,4,5γ₆