Octacross

Octacross

An octacross, is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 octahedron cells, 1792 5-cells "4-faces", 1792 "5-faces", 1024 "6-faces", and 256 "7-faces".

It is a part of an infinite family of polytopes, called cross-polytopes or "orthoplexes". The dual polytope is an 8-hypercube, or octeract.

The name "octacross" is derived from combining the family name "cross polytope" with "oct" for eight (dimensions) in Greek.

Construction

There are two Coxeter groups associated with the "octacross", one regular, dual of the octeract with the C8 or [4,3,3,3,3,3,3] symmetry group, and a lower symmetry with two copies of 7-simplex facets, alternating, with the D8 or [35,1,1] symmetry group.

Cartesian coordinates

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

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

See also

* Other 8-polytopes:
** 8-simplex - {37}
** 8-cube (octeract) - {4,36}
** 8-demicube (demiocteract) - {31,5,1}
** 421 polytope - {34,2,1}
** 241 polytope - {32,4,1}
** 142 polytope - {31,4,2}
* 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]


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