Octacross
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Regular heptacross (8-orthoplex)
Orthogonal projection inside Petrie polygon
Type	Regular 8-polytope
Family	orthoplex
Schläfli symbol	{36,4} {35,1,1}
Coxeter-Dynkin diagrams
7-faces	256 {36}
6-faces	1024 {35}
5-faces	1792 {34}
4-faces	1792 {33}
Cells	1120 {3,3}
Faces	448 {3}
Edges	112
Vertices	16
Vertex figure	Heptacross
Petrie polygon	hexadecagon
Coxeter groups	C8, [36,4] D8, [35,1,1
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.

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 [3^5,1,1 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) - {3^1,5,1}
  • 4₂₁ polytope - {3^4,2,1}
  • 2₄₁ polytope - {3^2,4,1}
  • 1₄₂ polytope - {3^1,4,2}
• 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

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

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