# Rectified 6-orthoplexes

(Redirected from Birectified 6-orthoplex)
 Orthogonal projections in B6 Coxeter plane 6-orthoplex Rectified 6-orthoplex Birectified 6-orthoplex Birectified 6-cube Rectified 6-cube 6-cube

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

Rectified hexacross
Type uniform 6-polytope
Schläfli symbol r{3,3,3,3,4}
Coxeter-Dynkin diagrams
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.

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 $\sqrt{2}\$ are all permutations of:

(±1,±1,0,0,0,0)

#### 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-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

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]

## Birectified 6-orthoplex

Birectified 6-orthoplex
Type uniform 6-polytope
Schläfli symbol 2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
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

• 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 $\sqrt{2}\$ 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]

## Related polytopes

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.

 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

## 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 [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