Hexacross

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Regular hexacross 6-cross-polytope
Graph
Type	Regular 6-polytope
Family	orthoplex
Schläfli symbol	{3,3,3,3,4} {33,1,1}
Coxeter-Dynkin diagrams
Hypercells	64 5-simplices
Hypercells	192 5-cells
Cells	240 tetrahedra
Faces	160 triangles
Edges	60
Vertices	12
Vertex figure	Pentacross
Symmetry group	B6, [3,3,3,3,4] C6, [33,1,1]
Dual	Hexeract
Properties	convex

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.

Contents 1 Construction 2 Cartesian coordinates 3 See also 4 External links

[edit] Construction

There's two Coxeter groups associated with the hexacross, one regular, dual of the hexeract with the B₆ or [4,3,3,3] symmetry group, and a lower symmetry with two copies of 5-simplex facets, alternating, with the C₆ or [3^3,1,1] symmetry group.

[edit] 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.

[edit] See also

• Other regular 6-polytopes:
  • 6-simplex - {3,3,3,3,3}
  • Hexeract - {4,3,3,3,3}
• 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}

[edit] External links

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