Rectified 6-cubes

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6-cube t0.svg
6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t1.svg
Rectified 6-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t2.svg
Birectified 6-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t3.svg
Birectified 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t4.svg
Rectified 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t5.svg
6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.

There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-ocube are located in the square face centers of the 6-cube.

Rectified 6-cube[edit]

Rectified 6-cube
Type uniform 6-polytope
Schläfli symbol t1{4,34} or r{4,34}
Coxeter-Dynkin diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.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.pngCDel 3b.pngCDel nodeb.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-faces 76
4-faces 444
Cells 1120
Faces 1520
Edges 960
Vertices 192
Vertex figure 5-cell prism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names[edit]

  • Rectified hexeract (acronym: rax) (Jonathan Bowers)

Construction[edit]

The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

Coordinates[edit]

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t1.svg 6-cube t1 B5.svg 6-cube t1 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t1 B3.svg 6-cube t1 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t1 A5.svg 6-cube t1 A3.svg
Dihedral symmetry [6] [4]

Birectified 6-cube[edit]

Birectified 6-cube
Type uniform 6-polytope
Coxeter symbol 0311
Schläfli symbol t2{4,34} or 2r{4,34}
Coxeter-Dynkin diagrams CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-faces 76
4-faces 636
Cells 2080
Faces 3200
Edges 1920
Vertices 240
Vertex figure {4}x{3,3} duoprism
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names[edit]

  • Birectified hexeract (acronym: brox) (Jonathan Bowers)

Construction[edit]

The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

Coordinates[edit]

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t2.svg 6-cube t2 B5.svg 6-cube t2 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t2 B3.svg 6-cube t2 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t2 A5.svg 6-cube t2 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-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. "6D uniform polytopes (polypeta)".  o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,

External links[edit]

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