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7-orthoplex

Regular 7- polytope

Regular 7-orthoplexHeptacross
[[Image:7-orthoplex.svg	280px]]Orthogonal projectioninside Petrie polygon
Type
Family
Schläfli symbol
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter groups
Dual
Properties

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cell 4-faces, 448 5-faces, and 128 6-faces.

It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411.

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

Alternate names

Heptacross, derived from combining the family name cross polytope with hept for seven (dimensions) in Greek.
Hecatonicosaoctaexon as a 128-facetted 7-polytope (polyexon). Acronym: zee

As a configuration

This configuration matrix represents the 7-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

\begin{bmatrix}\begin{matrix} 14 & 12 & 60 & 160 & 240 & 192 & 64 \ 2 & 84 & 10 & 40 & 80 & 80 & 32 \ 3 & 3 & 280 & 8 & 24 & 32 & 16 \ 4 & 6 & 4 & 560 & 6 & 12 & 8 \ 5 & 10 & 10 & 5 & 672 & 4 & 4 \ 6 & 15 & 20 & 15 & 6 & 448 & 2 \ 7 & 21 & 35 & 35 & 21 & 7 & 128 \end{matrix}\end{bmatrix}

Images

Construction

There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.

Coxeter diagram	Schläfli symbol	Symmetry
{3,3,3,3,3,4}	[3,3,3,3,3,4]	645120
{3,3,3,3,31,1}	[3,3,3,3,31,1]	322560
7{}	[26]	128

Cartesian coordinates

Cartesian coordinates for the vertices of a 7-orthoplex, 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.

References

H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com,
  - (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
  - (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
  - (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
x3o3o3o3o3o4o - zee

References

Coxeter, Regular Polytopes, sec 1.8 Configurations
Coxeter, Complex Regular Polytopes, p.117

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