8-orthoplex

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

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.

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)

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	Coxeter diagram	Schläfli symbol	Symmetry	Order	Vertex figure
regular 8-orthoplex		{3,3,3,3,3,3,4}	[3,3,3,3,3,3,4]	10321920
Alternate 8-orthoplex		{3,3,3,3,3,3^1,1}	[3,3,3,3,3,3^1,1]	5160960
8-fusil		8{}	[2^7]	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

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

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary