Rectified 5-orthoplexes

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5-cube t0.svg
5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t1.svg
Rectified 5-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t2.svg
Birectified 5-cube
Birectified 5-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t4.svg
5-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t3.svg
Rectified 5-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Orthogonal projections in A5 Coxeter plane

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.

Rectified 5-orthoplex[edit]

Rectified pentacross
Type uniform 5-polytope
Schläfli symbol t1{3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
Hypercells 42 total:
10 {3,3,4}
32 t1{3,3,3}
Cells 240 total:
80 {3,4}
160 {3,3}
Faces 400 total:
80+320 {3}
Edges 240
Vertices 40
Vertex figure Rectified pentacross verf.png
Octahedral prism
Petrie polygon Decagon
Coxeter groups BC5, [3,3,3,4]
D5, [32,1,1]
Properties convex

Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups.

Alternate names[edit]

  • rectified pentacross
  • rectified triacontiditeron (32-faceted 5-polytope)

Construction[edit]

There are two Coxeter groups associated with the rectified pentacross, one with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D5 or [32,1,1] Coxeter group.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a rectified pentacross, centered at the origin, edge length  \sqrt{2}\ are all permutations of:

(±1,±1,0,0,0)

Images[edit]

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t3.svg 5-cube t3 B4.svg 5-cube t3 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t3 B2.svg 5-cube t3 A3.svg
Dihedral symmetry [4] [4]

Related polytopes[edit]

The rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb:

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png or CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.

5-cube t4.svg
β5
5-cube t3.svg
t1β5
5-cube t2.svg
t2γ5
5-cube t1.svg
t1γ5
5-cube t0.svg
γ5
5-cube t34.svg
t0,1β5
5-cube t24.svg
t0,2β5
5-cube t23.svg
t1,2β5
5-cube t14.svg
t0,3β5
5-cube t13.svg
t1,3γ5
5-cube t12.svg
t1,2γ5
5-cube t04.svg
t0,4γ5
5-cube t03.svg
t0,3γ5
5-cube t02.svg
t0,2γ5
5-cube t01.svg
t0,1γ5
5-cube t234.svg
t0,1,2β5
5-cube t134.svg
t0,1,3β5
5-cube t124.svg
t0,2,3β5
5-cube t123.svg
t1,2,3γ5
5-cube t034.svg
t0,1,4β5
5-cube t024.svg
t0,2,4γ5
5-cube t023.svg
t0,2,3γ5
5-cube t014.svg
t0,1,4γ5
5-cube t013.svg
t0,1,3γ5
5-cube t012.svg
t0,1,2γ5
5-cube t1234.svg
t0,1,2,3β5
5-cube t0234.svg
t0,1,2,4β5
5-cube t0134.svg
t0,1,3,4γ5
5-cube t0124.svg
t0,1,2,4γ5
5-cube t0123.svg
t0,1,2,3γ5
5-cube t01234.svg
t0,1,2,3,4γ5

Notes[edit]

References[edit]

  • 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.
  • Richard Klitzing, 5D, uniform polytopes (polytera) o3x3o3o4o - rat

External links[edit]