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, [4,34] D6, [33,1,1]
Dual	6-cube
Properties	convex, Hanner polytope

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 {3,3,3,3^1,1} 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.

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.

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.

[math]\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} }[/math]

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 [3^3,1,1] 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.

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 3_k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

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". https://bendwavy.org/klitzing/dimensions/polypeta.htm. 

Specific

External links

• Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Cross. 
• Polytopes of Various Dimensions
• Multi-dimensional Glossary