Rectified 6-orthoplexes

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


6-orthoplex

Rectified 6-orthoplex

Birectified 6-orthoplex

Birectified 6-cube

Rectified 6-cube

6-cube
Orthogonal projections in B6 Coxeter plane

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

Rectified hexacross
Typeuniform 6-polytope
Schläfli symbolst1{34,4} or r{34,4}

r{3,3,3,31,1}
Coxeter-Dynkin diagrams =
=
5-faces76 total:
64 rectified 5-simplex
12 5-orthoplex
4-faces576 total:
192 rectified 5-cell
384 5-cell
Cells1200 total:
240 octahedron
960 tetrahedron
Faces1120 total:
160 and 960 triangles
Edges480
Vertices60
Vertex figure16-cell prism
Petrie polygonDodecagon
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex

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

or

Alternate names

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

Construction

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

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)

Images

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

Root vectors

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-simplices 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.

The 60 roots of D6 can be geometrically folded into H3 (Icosahedral symmetry), as to , creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:[1]

Rectified 6-orthoplex 2 icosidodecahedra
3D (H3 projection) A4/B5/D6 Coxeter plane H2 Coxeter plane

Birectified 6-orthoplex

Birectified 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolst2{34,4} or 2r{34,4}

t2{3,3,3,31,1}
Coxeter-Dynkin diagrams =
=
5-faces76
4-faces636
Cells2160
Faces2880
Edges1440
Vertices160
Vertex figure{3}×{3,4} duoprism
Petrie polygonDodecagon
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex

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

Alternate names

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

Cartesian coordinates

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

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

It can also be projected into 3D-dimensions as --> , a dodecahedron envelope.

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.

B6 polytopes

β6

t1β6

t2β6

t2γ6

t1γ6

γ6

t0,1β6

t0,2β6

t1,2β6

t0,3β6

t1,3β6

t2,3γ6

t0,4β6

t1,4γ6

t1,3γ6

t1,2γ6

t0,5γ6

t0,4γ6

t0,3γ6

t0,2γ6

t0,1γ6

t0,1,2β6

t0,1,3β6

t0,2,3β6

t1,2,3β6

t0,1,4β6

t0,2,4β6

t1,2,4β6

t0,3,4β6

t1,2,4γ6

t1,2,3γ6

t0,1,5β6

t0,2,5β6

t0,3,4γ6

t0,2,5γ6

t0,2,4γ6

t0,2,3γ6

t0,1,5γ6

t0,1,4γ6

t0,1,3γ6

t0,1,2γ6

t0,1,2,3β6

t0,1,2,4β6

t0,1,3,4β6

t0,2,3,4β6

t1,2,3,4γ6

t0,1,2,5β6

t0,1,3,5β6

t0,2,3,5γ6

t0,2,3,4γ6

t0,1,4,5γ6

t0,1,3,5γ6

t0,1,3,4γ6

t0,1,2,5γ6

t0,1,2,4γ6

t0,1,2,3γ6

t0,1,2,3,4β6

t0,1,2,3,5β6

t0,1,2,4,5β6

t0,1,2,4,5γ6

t0,1,2,3,5γ6

t0,1,2,3,4γ6

t0,1,2,3,4,5γ6

Notes

  1. Icosidodecahedron from D6 John Baez, January 1, 2015

References

  • 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
      • (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, o3o3x3o3o4o - brag
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 polychoron Pentachoron 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
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