# 7-orthoplex

Regular 7-orthoplex
(heptacross)

Orthogonal projection
inside Petrie polygon
Type Regular 7-polytope
Family orthoplex
Schläfli symbol {35,4}
{3,3,3,3,31,1}
Coxeter-Dynkin diagrams
6-faces 128 {35}
5-faces 448 {34}
4-faces 672 {33}
Cells 560 {3,3}
Faces 280 {3}
Edges 84
Vertices 14
Vertex figure 6-orthoplex
Coxeter groups C7, [3,3,3,3,3,4]
D7, [34,1,1]
Dual 7-cube
Properties convex

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-cells 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.
• Hecatonicosoctaexon as a 128-facetted 7-polytope (polyexon).

## As a configuration

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[1][2]

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

## Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## 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.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure
regular 7-orthoplex {3,3,3,3,3,4} [3,3,3,3,3,4] 645120
Quasiregular 7-orthoplex {3,3,3,3,31,1} [3,3,3,3,31,1] 322560
7-fusil 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

1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
2. ^ Coxeter, Complex Regular Polytopes, p.117
• 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, ISBN 978-0-471-01003-6 [1]
• (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)
• Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o4o - zee".