Rectified 5-cubes

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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-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

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[edit]

Rectified 5-cube
rectified penteract (rin)
Type uniform 5-polytope
Schläfli symbol r{4,3,3,3}
Coxeter diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1-43.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-faces 42
Cells 200
Faces 400
Edges 320
Vertices 80
Vertex figure Rectified 5-cube verf.png
tetrahedral prism
Coxeter group BC5, [4,33], order 3840
Dual
Base point (0,1,1,1,1,1)√2
Circumradius sqrt(2) = 1.414214
Properties convex, isogonal

Alternate names[edit]

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

Construction[edit]

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

Coordinates[edit]

The Cartesian coordinates of the vertices of the rectified 5-cube with edge length \sqrt{2} is given by all permutations of:

(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)

Images[edit]

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t1.svg 5-cube t1 B4.svg 5-cube t1 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t1 B2.svg 5-cube t1 A3.svg
Dihedral symmetry [4] [4]

Birectified 5-cube[edit]

Birectified 5-cube
birectified penteract (nit)
Type uniform 5-polytope
Schläfli symbol 2r{4,3,3,3}
Coxeter diagram CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-faces 42 10 {3,4,3}
32 t1{3,3,3}
Cells 280
Faces 640
Edges 480
Vertices 80
Vertex figure Birectified penteract verf.png
3-4 duoprism
Coxeter group BC5, [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

Alternate names[edit]

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

Construction and coordinates[edit]

The birectified 5-cube may be constructed by birectifing the vertices of the 5-cube at \sqrt{2} of the edge length.

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

\left(0,\ 0,\ \pm1,\ \pm1,\ \pm1\right)

Images[edit]

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t2.svg 5-cube t2 B4.svg 5-cube t2 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t2 B2.svg 5-cube t2 A3.svg
Dihedral symmetry [4] [4]

Related polytopes[edit]

2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8
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
CDel label4.pngCDel branch 11.png CDel node 1.pngCDel split1-43.pngCDel nodes.png CDel branch 11.pngCDel 4a3b.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png
Images Truncated square.png 3-cube t1.svgCuboctahedron.png 4-cube t12.svgSchlegel half-solid bitruncated 8-cell.png 5-cube t2.svg5-cube t2 A3.svg 6-cube t23.svg6-cube t23 A5.svg 7-cube t3.svg7-cube t3 A5.svg 8-cube t34.svg8-cube t34 A7.svg ...
Facets {3} Regular polygon 3 annotated.svg
{4} Regular polygon 4 annotated.svg
t{3,3} Uniform polyhedron-33-t01.png
t{3,4} Uniform polyhedron-43-t12.png
r{3,3,3} Schlegel half-solid rectified 5-cell.png
r{3,3,4} Schlegel wireframe 24-cell.png
2t{3,3,3,3} 5-simplex t12.svg
2t{3,3,3,4} 5-cube t23.svg
2r{3,3,3,3,3} 6-simplex t2.svg
2r{3,3,3,3,4} 6-cube t4.svg
3t{3,3,3,3,3,3} 7-simplex t23.svg
3t{3,3,3,3,3,4} 7-cube t45.svg
Vertex
figure
Cuboctahedron vertfig.png
Rectangle
Bitruncated 8-cell verf.png
Disphenoid
Birectified penteract verf.png
{3}×{4} duoprism
{3,3}×{3,4} duoprism

Related polytopes[edit]

Thes polytopes are a part of 31 uniform polytera 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 - rin, o3o3x3o4o - nit

External links[edit]