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 6-polytope
Schläfli symbol t1{34,4} or r{34,4}
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 1.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.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 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 6-polytope
Schläfli symbol t2{34,4} or 2r{34,4}
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 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a.pngCDel nodea.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 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 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 - rag". 

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