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5-orthoplex: Wikis

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Regular pentacross
(5-orthoplex)
5-orthoplex.svg
Orthogonal projection
inside Petrie polygon
Type Regular 5-polytope
Family orthoplex
Schläfli symbol {3,3,3,4}
{32,1,1}
Coxeter-Dynkin diagrams CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
CD ring.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
Hypercells 32 {33}Cross graph 4.png
Cells 80 {3,3}Cross graph 3.png
Faces 80 {3}Cross graph 2.png
Edges 40
Vertices 10
Vertex figure Pentacross verf.png
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.

Contents

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

Pentacross wire.png
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.

External links

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