5-orthoplex explained

bgcolor=#e7dcc3 colspan=2	Regular 5-orthoplex (pentacross)
bgcolor=#ffffff align=center colspan=2	Orthogonal projection inside Petrie polygon
Type	Regular 5-polytope
Family	orthoplex
Schläfli symbol
Coxeter-Dynkin diagrams
4-faces	32
Cells	80
Faces	80
Edges	40
Vertices	10
Vertex figure	16-cell
Petrie polygon	decagon
Coxeter groups	BC₅, [3,3,3,4] D₅, [3<sup>2,1,1</sup>]
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, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol 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.^[1] ^[2]

\begin{bmatrix}\begin{matrix} 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{matrix}\end{bmatrix}

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<sup>2,1,1</sup>] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

Name	Schläfli symbol	Symmetry	Order
regular 5-orthoplex		[3,3,3,4]	3840
Quasiregular 5-orthoplex		[3,3,3<sup>1,1</sup>]	1920
5-fusil
		[4,3,3,3]	3840
	+	[4,3,3,2]	768
	+	[4,3,2,4]	384
	+2	[4,3,2,2]	192
	2+	[4,2,4,2]	128
	+3	[4,2,2,2]	64
	5	[2,2,2,2]	32

Related polytopes and honeycombs

This polytope is one of 31 uniform 5-polytopes generated from the B₅ Coxeter plane, including the regular 5-cube and 5-orthoplex.

References

Coxeter, Regular Polytopes, sec 1.8 Configurations
Coxeter, Complex Regular Polytopes, p.117

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  - (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)