Pentacross

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Regular pentacross 5-cross-polytope
Graph
Type	Regular polyteron
Schläfli symbol	{3,3,3,4}
Coxeter-Dynkin diagrams
Hypercells	32 16-cells
Cells	80 octahedra
Faces	80 triangles
Edges	40
Vertices	10
Vertex figure	5-cells in a 16-cell figure
Symmetry group	B5, [3,3,3,4]
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. The dual polytope can be called a penteract, in the family of hypercubes.

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

[edit] 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)

[edit] See also

• Other regular 5-polytopes:
  • Hexateron - {3,3,3,3}
  • Penteract - {4,3,3,3}
• Others in the cross-polytope family
  • Octahedron - {3,4}
  • Hexadecachoron - {3,3,4}
  • Pentacross - {3,3,3,3}
  • Hexacross - {3,3,3,3,4}
  • Heptacross - {3,3,3,3,3,4}
• Tesseract

[edit] External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

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