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Regular 5-orthoplex
5-cube t4.svg
Orthogonal projection
inside Petrie polygon
Type Regular 5-polytope
Family orthoplex
Schläfli symbol {3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.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 split1.pngCDel nodes.png
4-faces 32 {33}Cross graph 4.png
Cells 80 {3,3}Cross graph 3.png
Faces 80 {3}Cross graph 2.png
Edges 40
Vertices 10
Vertex figure Pentacross verf.png
Petrie polygon decagon
Coxeter groups BC5, [3,3,3,4]
D5, [32,1,1]
Dual 5-cube
Properties convex

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

It has two constructed forms, the first being regular with Schläfli symbol {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.


Alternate namesEdit

  • pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
  • Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).

As a configurationEdit

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]


Cartesian coordinatesEdit

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

(±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)


There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure(s)
regular 5-orthoplex           {3,3,3,4} [3,3,3,4] 3840        
Quasiregular 5-orthoplex         {3,3,31,1} [3,3,31,1] 1920      
          {3,3,3,4} [4,3,3,3] 3840        
          {3,3,4}+{} [4,3,3,2] 768        
          {3,4}+{4} [4,3,2,4] 384        
          {3,4}+2{} [4,3,2,2] 192        
          2{4}+{} [4,2,4,2] 128        
          {4}+3{} [4,2,2,2] 64        
          5{} [2,2,2,2] 32        

Other imagesEdit

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Dihedral symmetry [4] [4]
The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.

Related polytopes and honeycombsEdit

This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.


  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. "5D uniform polytopes (polytera) x3o3o3o4o - tac".

External linksEdit