8-orthoplex

8-orthoplex Octacross
Orthogonal projection inside Petrie polygon
Type	Regular 8-polytope
Family	orthoplex
Schläfli symbol	{3⁶,4} {3,3,3,3,3,3^1,1}
Coxeter-Dynkin diagrams
7-faces	256 {3⁶}
6-faces	1024 {3⁵}
5-faces	1792 {3⁴}
4-faces	1792 {3³}
Cells	1120 {3,3}
Faces	448 {3}
Edges	112
Vertices	16
Vertex figure	7-orthoplex
Petrie polygon	hexadecagon
Coxeter groups	C₈, [3⁶,4] D₈, [3^5,1,1]
Dual	8-cube
Properties	convex, Hanner polytope

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.

It has two constructive forms, the first being regular with Schläfli symbol {3⁶,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,3^1,1} or Coxeter symbol 5₁₁.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract.

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Alternate names

Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
Diacosipentacontahexazetton as a 256-facetted 8-polytope (polyzetton)

As a configuration

This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[1]^[2]

${\begin{bmatrix}{\begin{matrix}16&14&84&280&560&672&448&128\\2&112&12&60&160&240&192&64\\3&3&448&10&40&80&80&32\\4&6&4&1120&8&24&32&16\\5&10&10&5&1792&6&12&8\\6&15&20&15&6&1792&4&4\\7&21&35&35&21&7&1024&2\\8&28&56&70&56&28&8&256\end{matrix}}\end{bmatrix}}$

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors. ^[3]

B₈	k-face	f_k	f₀	f₁	f₂	f₃	f₄	f₅	f₆	f₇	k-figure	notes
B₇	( )	f₀	16	14	84	280	560	672	448	128	{3,3,3,3,3,4}	B₈/B₇ = 2^8*8!/2^7/7! = 16
A₁B₆	{ }	f₁	2	112	12	60	160	240	192	64	{3,3,3,3,4}	B₈/A₁B₆ = 2^8*8!/2/2^6/6! = 112
A₂B₅	{3}	f₂	3	3	448	10	40	80	80	32	{3,3,3,4}	B₈/A₂B₅ = 2^8*8!/3!/2^5/5! = 448
A₃B₄	{3,3}	f₃	4	6	4	1120	8	24	32	16	{3,3,4}	B₈/A₃B₄ = 2^8*8!/4!/2^4/4! = 1120
A₄B₃	{3,3,3}	f₄	5	10	10	5	1792	6	12	8	{3,4}	B₈/A₄B₃ = 2^8*8!/5!/8/3! = 1792
A₅B₂	{3,3,3,3}	f₅	6	15	20	15	6	1792	4	4	{4}	B₈/A₅B₂ = 2^8*8!/6!/4/2 = 1792
A₆A₁	{3,3,3,3,3}	f₆	7	21	35	35	21	7	1024	2	{ }	B₈/A₆A₁ = 2^8*8!/7!/2 = 1024
A₇	{3,3,3,3,3,3}	f₇	8	28	56	70	56	28	8	256	( )	B₈/A₇ = 2^8*8!/8! = 256

Construction

There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C₈ or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D₈ or [3^5,1,1] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an 8-fusil.

Name	Schläfli symbol	Symmetry	Order
regular 8-orthoplex	{3,3,3,3,3,3,4}	[3,3,3,3,3,3,4]	10321920
Quasiregular 8-orthoplex	{3,3,3,3,3,3^1,1}	[3,3,3,3,3,3^1,1]	5160960
8-fusil	8{}	[2⁷]	256

Cartesian coordinates

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are

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

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

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

Images

orthographic projections
B₈		B₇

[16]		[14]
B₆		B₅

[12]		[10]
B₄	B₃		B₂

[8]	[6]		[4]
A₇	A₅		A₃

[8]	[6]		[4]

It is used in its alternated form 5₁₁ with the 8-simplex to form the 5₂₁ honeycomb.

References

^ Coxeter, Regular Polytopes, sec 1.8 Configurations
^ Coxeter, Complex Regular Polytopes, p.117
^ Klitzing, Richard. "x3o3o3o3o3o3o4o - ek".

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.
Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o4o - ek".

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

This page was last edited on 17 November 2022, at 00:28

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