5-orthoplex

Regular pentacross (5-orthoplex)
Orthogonal projection inside Petrie polygon
Type	Regular 5-polytope
Family	orthoplex
Schläfli symbol	{3,3,3,4} {3,3,31,1}
Coxeter-Dynkin diagrams
Hypercells	32 {33}
Cells	80 {3,3}
Faces	80 {3}
Edges	40
Vertices	10
Vertex figure	16-cell
Petrie polygon	decagon
Coxeter groups	C5, [3,3,3,4] D5, [32,1,1]
Dual	5-cube
Properties	convex

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 hypercells.

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 {3^2,1,1} or Coxeter symbol 2₁₁.

Alternate names

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

Related polytopes

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

Construction

There are two Coxeter groups associated with the pentacross, 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.

Cartesian coordinates

Cartesian coordinates for the vertices of a pentacross, 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)

Other images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Precisely, the Perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D of Pentacross. 10 sets of 4 edges forms 10 circles in the 4D Schlegel diagram: two of these circles are straight lines because contains the center of projection.

Related polytopes

This polytope is one of 63 uniform polypeta generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

B5 polytopes
β5	t1β5	t2γ5	t1γ5	γ5	t0,1β5	t0,2β5	t1,2β5
t0,3β5	t1,3γ5	t1,2γ5	t0,4γ5	t0,3γ5	t0,2γ5	t0,1γ5	t0,1,2β5
t0,1,3β5	t0,2,3β5	t1,2,3γ5	t0,1,4β5	t0,2,4γ5	t0,2,3γ5	t0,1,4γ5	t0,1,3γ5
t0,1,2γ5	t0,1,2,3β5	t0,1,2,4β5	t0,1,3,4γ5	t0,1,2,4γ5	t0,1,2,3γ5	t0,1,2,3,4γ5

External links

• Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
• Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o3o4o - tac".