6-orthoplex

6-orthoplex Hexacross
Orthogonal projection inside Petrie polygon
Type	Regular 6-polytope
Family	orthoplex
Schläfli symbols	{3,3,3,3,4} {3,3,3,31,1}
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, [3,3,3,3,4] D6, [33,1,1]
Dual	6-cube
Properties	convex

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.

It has two constructed forms, the first being regular with Schläfli symbol {3⁴,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3^4,1,1} or Coxeter symbol 4₁₁.

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 (polypeton).

Related polytopes

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.

Construction

There are two 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 lower symmetry with two copies of 5-simplex facets, alternating, with the D₆ or [3^3,1,1] Coxeter group.

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

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

β6	t1β6	t2β6	t2γ6	t1γ6	γ6	t0,1β6	t0,2β6
t1,2β6	t0,3β6	t1,3β6	t2,3γ6	t0,4β6	t1,4γ6	t1,3γ6	t1,2γ6
t0,5γ6	t0,4γ6	t0,3γ6	t0,2γ6	t0,1γ6	t0,1,2β6	t0,1,3β6	t0,2,3β6
t1,2,3β6	t0,1,4β6	t0,2,4β6	t1,2,4β6	t0,3,4β6	t1,2,4γ6	t1,2,3γ6	t0,1,5β6
t0,2,5β6	t0,3,4γ6	t0,2,5γ6	t0,2,4γ6	t0,2,3γ6	t0,1,5γ6	t0,1,4γ6	t0,1,3γ6
t0,1,2γ6	t0,1,2,3β6	t0,1,2,4β6	t0,1,3,4β6	t0,2,3,4β6	t1,2,3,4γ6	t0,1,2,5β6	t0,1,3,5β6
t0,2,3,5γ6	t0,2,3,4γ6	t0,1,4,5γ6	t0,1,3,5γ6	t0,1,3,4γ6	t0,1,2,5γ6	t0,1,2,4γ6	t0,1,2,3γ6
t0,1,2,3,4β6	t0,1,2,3,5β6	t0,1,2,4,5β6	t0,1,2,4,5γ6	t0,1,2,3,5γ6	t0,1,2,3,4γ6	t0,1,2,3,4,5γ6

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. 1966
• Richard Klitzing, 6D uniform polytopes (polypeta), x3o3o3o3o4o - gee

External links

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