# 6-orthoplex

(Redirected from Hexacross)
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 {3,3,3,31,1} or Coxeter symbol 311.

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 6{} [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:

2D 3D

Icosahedron
H3 Coxeter plane

6-orthoplex
D6 Coxeter plane

Icosahedron

An icosahedrally symmetric projection of the 6-orthoplex down to three dimensions
This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a regular icosahedron. Every pair of vertices of the icosahedra are connected, except opposite ones.
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
A3A1 A5 D6 E7 ${\displaystyle {\tilde {E}}_{7}}$=E7+ ${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [[31,3,1]] [32,3,1] [33,3,1] [34,3,1]
Order 48 720 46,080 2,903,040
Graph - -
Name 31,-1 310 311 321 331 341

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

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