5-orthoplex

Regular 5-orthoplex (pentacross)
Orthogonal projection inside Petrie polygon
Type	Regular 5-polytope
Family	orthoplex
Schläfli symbol	{3,3,3,4} {3,3,31,1}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD
4-faces	32 {33}
Cells	80 {3,3}
Faces	80 {3}
Edges	40
Vertices	10
Vertex figure	16-cell
Petrie polygon	decagon
Coxeter groups	BC5, [3,3,3,4] D5, [32,1,1]
Dual	5-cube
Properties	convex, Hanner polytope

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

It has two constructed forms, the first being regular with Schläfli symbol {3³,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3^1,1} or Coxeter symbol 2₁₁.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.

Alternate names

• pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
• Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).

As a configuration

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

$\left[\begin{array}{c}\begin{array}{ccccc}10& 8& 24& 32& 16\\ 2& 40& 6& 12& 8\\ 3& 3& 80& 4& 4\\ 4& 6& 4& 80& 2\\ 5& 10& 10& 5& 32\end{array}\end{array}\right]$

Cartesian coordinates

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

(±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)

Construction

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C₅ or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D₅ or [3^2,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

Name	Coxeter diagram	Schläfli symbol	Symmetry	Order	Vertex figure(s)
regular 5-orthoplex	Template:CDD	{3,3,3,4}	[3,3,3,4]	3840	Template:CDD
Quasiregular 5-orthoplex	Template:CDD	{3,3,31,1}	[3,3,31,1]	1920	Template:CDD
5-fusil
	Template:CDD	{3,3,3,4}	[4,3,3,3]	3840	Template:CDD
	Template:CDD	{3,3,4}+{}	[4,3,3,2]	768	Template:CDD
	Template:CDD	{3,4}+{4}	[4,3,2,4]	384	Template:CDD Template:CDD
	Template:CDD	{3,4}+2{}	[4,3,2,2]	192	Template:CDD Template:CDD
	Template:CDD	2{4}+{}	[4,2,4,2]	128	Template:CDD
	Template:CDD	{4}+3{}	[4,2,2,2]	64	Template:CDD Template:CDD
	Template:CDD	5{}	[2,2,2,2]	32	Template:CDD

Other images

The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.

Related polytopes and honeycombs

2k1 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\tilde{E}}_8$ = E8+	E10 = ${\overline{T}}_8$ = E8++
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Symmetry	[3−1,2,1]	[30,2,1]	[[31,2,1]]	[32,2,1]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	384	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	2−1,1	201	211	221	231	241	251	261

This polytope is one of 31 uniform 5-polytopes generated from the B₅ Coxeter plane, including the regular 5-cube and 5-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, Template:ISBN [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)
• Template:KlitzingPolytopes

External links

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