Rectified 6-orthoplexes

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6-cube t5.svg
6-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t4.svg
Rectified 6-orthoplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t3.svg
Birectified 6-orthoplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t2.svg
Birectified 6-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t1.svg
Rectified 6-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
6-cube t0.svg
6-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Orthogonal projections in B6 Coxeter plane

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

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

Rectified 6-orthoplex[edit]

Rectified hexacross
Type uniform polypeton
Schläfli symbol r{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.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 3.pngCDel node.pngCDel split1.pngCDel nodes.png
5-faces 76 total:
64 rectified 5-simplex
12 5-orthoplex
4-faces 576 total:
192 rectified 5-cell
384 5-cell
Cells 1200 total:
240 octahedron
960 tetrahedron
Faces 1120 total:
160 and 960 triangles
Edges 480
Vertices 60
Vertex figure 16-cell prism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.

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

Alternate names[edit]

  • rectified hexacross
  • rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)

Construction[edit]

There are two Coxeter groups associated with the rectified hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or [33,1,1] Coxeter group.

Cartesian coordinates[edit]

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

(±1,±1,0,0,0,0)

Root vectors[edit]

The 60 vertices represent the root vectors of the simple Lie group D6. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplexs cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B6 and C6 simple Lie groups.

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t5.svg 6-cube t5 B5.svg 6-cube t5 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t5 B3.svg 6-cube t5 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t5 A5.svg 6-cube t5 A3.svg
Dihedral symmetry [6] [4]

Birectified 6-orthoplex[edit]

Birectified 6-orthoplex
Type uniform polypeton
Schläfli symbol 2r{3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel 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 3.pngCDel node.pngCDel split1.pngCDel nodes.png
5-faces 76
4-faces 636
Cells 2160
Faces 2880
Edges 1440
Vertices 160
Vertex figure {3}×{3,4} duoprism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.

Alternate names[edit]

  • birectified hexacross
  • birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)

Cartesian coordinates[edit]

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

(±1,±1,±1,0,0,0)

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t4.svg 6-cube t4 B5.svg 6-cube t4 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t4 B3.svg 6-cube t4 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t4 A5.svg 6-cube t4 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

These polytopes are a part a family of 63 uniform polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

6-cube t5.svg
β6
6-cube t4.svg
t1β6
6-cube t3.svg
t2β6
6-cube t2.svg
t2γ6
6-cube t1.svg
t1γ6
6-cube t0.svg
γ6
6-cube t45.svg
t0,1β6
6-cube t35.svg
t0,2β6
6-cube t34.svg
t1,2β6
6-cube t25.svg
t0,3β6
6-cube t24.svg
t1,3β6
6-cube t23.svg
t2,3γ6
6-cube t15.svg
t0,4β6
6-cube t14.svg
t1,4γ6
6-cube t13.svg
t1,3γ6
6-cube t12.svg
t1,2γ6
6-cube t05.svg
t0,5γ6
6-cube t04.svg
t0,4γ6
6-cube t03.svg
t0,3γ6
6-cube t02.svg
t0,2γ6
6-cube t01.svg
t0,1γ6
6-cube t345.svg
t0,1,2β6
6-cube t245.svg
t0,1,3β6
6-cube t235.svg
t0,2,3β6
6-cube t234.svg
t1,2,3β6
6-cube t145.svg
t0,1,4β6
6-cube t135.svg
t0,2,4β6
6-cube t134.svg
t1,2,4β6
6-cube t125.svg
t0,3,4β6
6-cube t124.svg
t1,2,4γ6
6-cube t123.svg
t1,2,3γ6
6-cube t045.svg
t0,1,5β6
6-cube t035.svg
t0,2,5β6
6-cube t034.svg
t0,3,4γ6
6-cube t025.svg
t0,2,5γ6
6-cube t024.svg
t0,2,4γ6
6-cube t023.svg
t0,2,3γ6
6-cube t015.svg
t0,1,5γ6
6-cube t014.svg
t0,1,4γ6
6-cube t013.svg
t0,1,3γ6
6-cube t012.svg
t0,1,2γ6
6-cube t2345.svg
t0,1,2,3β6
6-cube t1345.svg
t0,1,2,4β6
6-cube t1245.svg
t0,1,3,4β6
6-cube t1235.svg
t0,2,3,4β6
6-cube t1234.svg
t1,2,3,4γ6
6-cube t0345.svg
t0,1,2,5β6
6-cube t0245.svg
t0,1,3,5β6
6-cube t0235.svg
t0,2,3,5γ6
6-cube t0234.svg
t0,2,3,4γ6
6-cube t0145.svg
t0,1,4,5γ6
6-cube t0135.svg
t0,1,3,5γ6
6-cube t0134.svg
t0,1,3,4γ6
6-cube t0125.svg
t0,1,2,5γ6
6-cube t0124.svg
t0,1,2,4γ6
6-cube t0123.svg
t0,1,2,3γ6
6-cube t12345.svg
t0,1,2,3,4β6
6-cube t02345.svg
t0,1,2,3,5β6
6-cube t01345.svg
t0,1,2,4,5β6
6-cube t01245.svg
t0,1,2,4,5γ6
6-cube t01235.svg
t0,1,2,3,5γ6
6-cube t01234.svg
t0,1,2,3,4γ6
6-cube t012345.svg
t0,1,2,3,4,5γ6

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, 6D uniform polytopes (polypeta), o3x3o3o3o4o - rag

External links[edit]