Rectified 5-cubes

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


5-cube

Rectified 5-cube

Birectified 5-cube
Birectified 5-orthoplex

5-orthoplex

Rectified 5-orthoplex
Orthogonal projections in A5 Coxeter plane

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

Rectified 5-cube

Rectified 5-cube
rectified penteract (rin)
Type uniform 5-polytope
Schläfli symbol r{4,3,3,3}
Coxeter diagram =
4-faces42
Cells200
Faces400
Edges 320
Vertices 80
Vertex figure
tetrahedral prism
Coxeter group B5, [4,33], order 3840
Dual
Base point (0,1,1,1,1,1)√2
Circumradius sqrt(2) = 1.414214
Properties convex, isogonal

Alternate names

  • Rectified penteract (acronym: rin) (Jonathan Bowers)

Construction

The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.

Coordinates

The Cartesian coordinates of the vertices of the rectified 5-cube with edge length is given by all permutations of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Birectified 5-cube

Birectified 5-cube
birectified penteract (nit)
Type uniform 5-polytope
Schläfli symbol 2r{4,3,3,3}
Coxeter diagram =
4-faces4210 {3,4,3}
32 t1{3,3,3}
Cells280
Faces640
Edges 480
Vertices 80
Vertex figure
{3}×{4}
Coxeter group B5, [4,33], order 3840
D5, [32,1,1], order 1920
Dual
Base point (0,0,1,1,1,1)√2
Circumradius sqrt(3/2) = 1.224745
Properties convex, isogonal

E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr52 as a second rectification of a 5-dimensional cross polytope.

Alternate names

  • Birectified 5-cube/penteract
  • Birectified pentacross/5-orthoplex/triacontiditeron
  • Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
  • Rectified 5-demicube/demipenteract

Construction and coordinates

The birectified 5-cube may be constructed by birectifing the vertices of the 5-cube at of the edge length.

The Cartesian coordinates of the vertices of a birectified 5-cube having edge length 2 are all permutations of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8 n
Name t{4} r{4,3} 2t{4,3,3} 2r{4,3,3,3} 3t{4,3,3,3,3} 3r{4,3,3,3,3,3} 4t{4,3,3,3,3,3,3} ...
Coxeter
diagram
Images
Facets {3}
{4}
t{3,3}
t{3,4}
r{3,3,3}
r{3,3,4}
2t{3,3,3,3}
2t{3,3,3,4}
2r{3,3,3,3,3}
2r{3,3,3,3,4}
3t{3,3,3,3,3,3}
3t{3,3,3,3,3,4}
Vertex
figure
( )v( )
{ }×{ }

{ }v{ }

{3}×{4}

{3}v{4}
{3,3}×{3,4} {3,3}v{3,4}

These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

Notes

    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, ISBN 978-0-471-01003-6
        • (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.
    • Klitzing, Richard. "5D uniform polytopes (polytera)". o3x3o3o4o - rin, o3o3x3o4o - nit
    Fundamental convex regular and uniform polytopes in dimensions 2–10
    Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
    Regular polygon Triangle Square p-gon Hexagon Pentagon
    Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
    Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
    Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
    Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
    Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
    Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
    Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
    Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
    Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
    Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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