Hexacross

A hexacross, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 octahedron cells, 192 5-cell "4-faces", and 64 "5-faces".

It is a part of an infinite family of polytopes, called cross-polytopes or "orthoplexes". The dual polytope is the 6-hypercube, or hexeract.

The name "hexacross" is derived from combining the family name "cross polytope" with "hex" for six (dimensions) in Greek.

Construction

There's two Coxeter groups associated with the "hexacross", one regular, dual of the hexeract with the C₆ or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-simplex facets, alternating, with the D₆ or [3^3,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a hexacross, centered at the origin are: (±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

See also

* Other 6-polytopes:
** 6-simplex (heptapeton) - {3⁵}
** 6-cube (hexeract) - {4,3⁴}
** 6-demicube (demihexeract) - {3^1,3,1}
** 1₂₂ polytope - {3^1,2,2}
** 2₂₁ polytope - {3^2,2,1}
* 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}

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]