# 8-orthoplex

8-orthoplex
Heptacross

Orthogonal projection
inside Petrie polygon
Type Regular 8-polytope
Family orthoplex
Schläfli symbol {36,4}
{3,3,3,3,3,31,1}
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
Coxeter groups C8, [36,4]
D8, [35,1,1]
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 {35,1,1} or Coxeter symbol 511.

## 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 [35,1,1] symmetry group.A lowest symmetry construction is based on a dual of a 8-orthotope, called a 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 {3,3,3,3,3,31,1} [3,3,3,3,3,31,1] 5160960
8-fusil {}+{}+{}+{}+{}+{}+{}+{} [27] 256

## Images

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

## Related polytopes

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

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

## 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.

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