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5-orthoplex

Convex regular 5-polytope in geometry

Summary

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
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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