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Heptacross - Wikipedia, the free encyclopedia

Heptacross

From Wikipedia, the free encyclopedia

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 Image:CDW_ring.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.pngImage:CDW_4.pngImage:CDW_dot.png
Image:CD ring.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD_downbranch-00.pngImage:CD 3b.pngImage:CD dot.png
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

[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 [34,1,1] 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

[edit] External links


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