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	Template:CDD Template:CDD = Template:CDD
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.
• Hexacontatetrapeton as a 64-facetted 6-polytope.
• Acronym: gee (Jonathan Bowers)Template:Sfn

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.Template:Sfnp

$\left[\begin{array}{c}\begin{array}{cccccc}12& 10& 40& 80& 80& 32\\ 2& 60& 8& 24& 32& 16\\ 3& 3& 160& 6& 12& 8\\ 4& 6& 4& 240& 4& 4\\ 5& 10& 10& 5& 192& 2\\ 6& 15& 20& 15& 6& 64\end{array}\end{array}\right]$

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	Template:CDD	{3,3,3,3,4}	[4,3,3,3,3]	46080
Quasiregular 6-orthoplex	Template:CDD	{3,3,3,31,1}	[3,3,3,31,1]	23040
6-fusil	Template:CDD	{3,3,3,4}+{}	[4,3,3,3,3]	7680
	Template:CDD	{3,3,4}+{4}	[4,3,3,2,4]	3072
	Template:CDD	2{3,4}	[4,3,2,4,3]	2304
	Template:CDD	{3,3,4}+2{}	[4,3,3,2,2]	1536
	Template:CDD	{3,4}+{4}+{}	[4,3,2,4,2]	768
	Template:CDD	3{4}	[4,2,4,2,4]	512
	Template:CDD	{3,4}+3{}	[4,3,2,2,2]	384
	Template:CDD	2{4}+2{}	[4,2,4,2,2]	256
	Template:CDD	{4}+4{}	[4,2,2,2,2]	128
	Template:CDD	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.

Related polytopes

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.

2D		3D
Icosahedron {3,5} = Template:CDD H3 Coxeter plane	6-orthoplex {3,3,3,31,1} = Template:CDD 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: : Template:CDD to Template:CDD. 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.)

3_k1 dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\tilde{E}}_7$=E7+	${\overline{T}}_8$=E7++
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
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 also the regular 6-cube. Template:Hexeract family

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973
  • Template:Cite book
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:Isbn
    • (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
• Template:KlitzingPolytopes x3o3o3o3o4o - gee Template:Sfn whitelist

External links

• Template:GlossaryForHyperspace
• Polytopes of Various Dimensions
• Multi-dimensional Glossary