Regular 7-orthoplex
Orthogonal projection
inside Petrie polygon
Type Regular 7-polytope
Family orthoplex
Schläfli symbol {35,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
6-faces 128 {35}6-simplex t0.svg
5-faces 448 {34}5-simplex t0.svg
4-faces 672 {33}4-simplex t0.svg
Cells 560 {3,3}3-simplex t0.svg
Faces 280 {3}2-simplex t0.svg
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

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 namesEdit

  • 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 configurationEdit

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.[1][2]



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


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 coordinatesEdit

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 alsoEdit


  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".

External linksEdit

Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds