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6-orthoplex

## Summary

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, [4,34]
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.

## As a configuration

This configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]

${\displaystyle {\begin{bmatrix}{\begin{matrix}12&10&40&80&80&32\\2&60&8&24&32&16\\3&3&160&6&12&8\\4&6&4&240&4&4\\5&10&10&5&192&2\\6&15&20&15&6&64\end{matrix}}\end{bmatrix}}}$

## 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 Schläfli Symmetry Order
Regular 6-orthoplex             {3,3,3,3,4} [4,3,3,3,3] 46080
Quasiregular 6-orthoplex           {3,3,3,31,1} [3,3,3,31,1] 23040
6-fusil             {3,3,3,4}+{} [4,3,3,3,3] 7680
{3,3,4}+{4} [4,3,3,2,4] 3072
2{3,4} [4,3,2,4,3] 2304
{3,3,4}+2{} [4,3,3,2,2] 1536
{3,4}+{4}+{} [4,3,2,4,2] 768
3{4} [4,2,4,2,4] 512
{3,4}+3{} [4,3,2,2,2] 384
2{4}+2{} [4,2,4,2,2] 256
{4}+4{} [4,2,2,2,2] 128
6{} [2,2,2,2,2] 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.[3]

2D 3D

Icosahedron
{3,5} =
H3 Coxeter plane

6-orthoplex
{3,3,3,31,1} =
D6 Coxeter plane

Icosahedron

6-orthoplex
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. This represents a geometric folding of the D6 to H3 Coxeter groups:  :       to      . On the left, seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron.

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]]
= [4,3,3,3,3]
[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.

B6 polytopes

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
• Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o3o3o4o - gee".
Specific
1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
2. ^ Coxeter, Complex Regular Polytopes, p.117
3. ^ Quasicrystals and Geometry, Marjorie Senechal, 1996, Cambridge University Press, p64. 2.7.1 The I6 crystal