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5-orthoplex

Convex regular 5-polytope in geometry

Convex regular 5-polytope in geometry

Regular 5-orthoplexPentacross
[[Image:5-cube t4.svg281px]]Orthogonal projectioninside Petrie polygon
Type
Family
Schläfli symbol
Coxeter-Dynkin diagrams
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter groups
Dual
Properties

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 names

  • 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). Acronym: tac

As a configuration

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

\begin{bmatrix}\begin{matrix} 10 & 8 & 24 & 32 & 16 \ 2 & 40 & 6 & 12 & 8 \ 3 & 3 & 80 & 4 & 4 \ 4 & 6 & 4 & 80 & 2 \ 5 & 10 & 10 & 5 & 32 \end{matrix}\end{bmatrix}

Cartesian coordinates

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)

Construction

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.

NameCoxeter diagramSchläfli symbolSymmetryOrderVertex figure(s)regular 5-orthoplexQuasiregular 5-orthoplex5-fusil
{3,3,3,4}[3,3,3,4]3840
{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 images

[[Image:Pentacross wire.png220px]]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.

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)
  • x3o3o3o4o - tac

References

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