Pentacross

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 [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

See also

* Other 5-polytopes:
** 5-simplex - {3,3,3,3}
** 5-cube (penteract) - {4,3,3,3}
** 5-demicube (demipenteract) - {3^1,2,1}
* Others in the cross-polytope family
** Octahedron - {3,4}
** Hexadecachoron - {3,3,4}
** Pentacross - {3³,4}
** Hexacross - {3⁴,4}
** Heptacross - {3⁵,4}
** Octacross - {3⁶,4}
** Enneacross - {3⁷,4}
** Decacross - {3⁸,4}
* 1_k2 polytope family

External links

*GlossaryForHyperspace | anchor=Cross | title=Cross polytope
* [http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]