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 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[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 is given by all permutations of:

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}
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[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 of the edge length.

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

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]

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

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.
  • Klitzing, Richard. "5D uniform polytopes (polytera)".  o3x3o3o4o - rin, o3o3x3o4o - nit

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / 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