Heptacross

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Regular heptacross 7-cross-polytope
Graph
Type	Regular 7-polytope
Family	orthoplex
Schläfli symbol	{3,3,3,3,3,4} {34,1,1}
Coxeter-Dynkin diagrams
6-faces	128 6-simplices
5-faces	448 5-simplices
4-faces	672 5-cells
Cells	560 tetrahedra
Faces	280 triangles
Edges	84
Vertices	14
Vertex figure	Hexacross
Symmetry group	B7, [3,3,3,3,3,4] C7, [34,1,1]
Dual	Hepteract
Properties	convex

A heptacross, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 octahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.

The name heptacross is derived from combining the family name cross polytope with hept for seven (dimensions) in Greek.

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

[edit] Construction

There are two Coxeter groups associated with the heptacross, one regular, dual of the hepteract with the B₇ or [4,3,3,3,3] symmetry group, and a lower symmetry with two copies of 7-simplex facets, alternating, with the C₇ or [3^4,1,1] symmetry group.

[edit] Cartesian coordinates

Cartesian coordinates for the vertices of a heptacross, centered at the origin are

(±1,0,0,0,0,0,0), (0,±1,0,0,0,0,0), (0,0,±1,0,0,0,0), (0,0,0,±1,0,0,0), (0,0,0,0,±1,0,0), (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 7-polytopes:
  • 7-simplex - {3,3,3,3,3,3}
  • Hepteract - {4,3,3,3,3,3}
• 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}

[edit] External links

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