Pentacross

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Regular pentacross 5-cross-polytope
Graph
Type	Regular 5-polytope
Family	orthoplex
Schläfli symbol	{3,3,3,4} {3^2,1,1}
Coxeter-Dynkin diagrams
Hypercells	32 5-cells
Cells	80 tetrahedra
Faces	80 triangles
Edges	40
Vertices	10
Vertex figure	16-cell
Symmetry group	B₅, [3,3,3,4] C₅, [3^2,1,1]
Dual	Penteract
Properties	convex

(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.

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.

1 Construction
2 Cartesian coordinates
3 See also
4 External links

[edit] Construction

There's two Coxeter groups associated with the pentacross, one regular, dual of the penteract with the B₅ or [4,3,3] symmetry group, and a lower symmetry with two copies of 5-cell facets, alternating, with the C₅ or [3^2,1,1] symmetry group.