# 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 {34,1,1} or Coxeter symbol 411.

## 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 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 [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure
Alternate 6-orthoplex {3,3,3,3,4} [3,3,3,3,4] 46080
regular 6-orthoplex {3,3,3,31,1} [3,3,3,31,1] 23040
6-fusil {}+{}+{}+{}+{}+{} [25] 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, as seen in this 2D projection:

 Icosahedron H3 Coxeter plane 6-orthoplex D6 Coxeter plane This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions. This represents a geometric folding of the D6 to H3 Coxeter groups: Seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping.

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

3k1 dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3×A1 A5 D6 E7 ${\tilde{E}}_{7}$=E7+ E7++
Coxeter
diagram
Symmetry
(order)
[3-1,3,1]
(48)
[30,3,1]
(720)
[[31,3,1]]
(46,080)
[32,3,1]
(2,903,040)
[33,3,1]
(∞)
[34,3,1]
(∞)
Graph
Name 31,-1 310 311 321 331 341

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