# 5-orthoplex: Wikis

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# Encyclopedia

(Redirected to Pentacross article)

Regular pentacross
(5-orthoplex)

Orthogonal projection
inside Petrie polygon
Type Regular 5-polytope
Family orthoplex
Schläfli symbol {3,3,3,4}
{32,1,1}
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 Penteract
Properties convex

In five-dimensional geometry, a pentacross, also called a triacontakaiditeron, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 octahedron cells, 32 5-cell hypercells.

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

The name pentacross is derived from combining the family name cross polytope with pente for five (dimensions) in Greek.

## 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 [32,1,1] 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

 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.