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
Petrie polygon	tetradecagon
Coxeter groups	C7, [3,3,3,3,3,4] D7, [34,1,1]
Dual	7-cube
Properties	convex, Hanner polytope

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 {3,3,3,3,3^1,1} 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

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.

[math]\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} }[/math]

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 [3^4,1,1] 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.

See also

• Rectified 7-orthoplex
• Truncated 7-orthoplex

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, 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". https://bendwavy.org/klitzing/dimensions/polyexa.htm. 

External links

• Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Cross. 
• Polytopes of Various Dimensions
• Multi-dimensional Glossary