Octacross

From Wikipedia, the free encyclopedia

Regular heptacross 8-cross-polytope
Graph
Type	Regular 8-polytope
Family	orthoplex
Schläfli symbol	{3,3,3,3,3,3,4} {3^5,1,1}
Coxeter-Dynkin diagrams
7-faces	256 7-simplexes
6-faces	1024 6-simplexes
5-faces	1792 5-simplexes
4-faces	1792 5-cells
Cells	1120 tetrahedra
Faces	448 triangles
Edges	112
Vertices	16
Vertex figure	Heptacross
Symmetry group	B₈, [3,3,3,3,3,3,4] C₈, [3^5,1,1]
Dual	Octeract
Properties	convex

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.

1 Construction
2 Cartesian coordinates
3 See also
4 External links

[edit] Construction

There are two Coxeter groups associated with the octacross, one regular, dual of the octeract with the B₈ or [4,3,3,3,3,3,3] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the C₈ or [3^5,1,1] symmetry group.

[edit] 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.

[edit] See also

Other regular 8-polytopes:
- 8-simplex - {3,3,3,3,3,3,3}
- Octeract - {4,3,3,3,3,3,3}
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}