Cantellated 6-orthoplexes

(Redirected from Cantitruncated 6-orthoplex)
 Orthogonal projections in B6 Coxeter plane 6-orthoplex Cantellated 6-orthoplex Bicantellated 6-orthoplex 6-cube Cantellated 6-cube Bicantellated 6-cube Cantitruncated 6-orthoplex Bicantitruncated 6-orthoplex Bicantitruncated 6-cube Cantitruncated 6-cube

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

There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed from the dual 5-cube

Cantellated 6-orthoplex

Cantellated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t0,2{3,3,3,3,4}
rr{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces 136
4-faces 1656
Cells 5040
Faces 6400
Edges 3360
Vertices 480
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

• Cantellated hexacross
• Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers)[1]

Construction

There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates

Cartesian coordinates for the 480 vertices of a cantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of

(2,1,1,0,0,0)

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]

Bicantellated 6-orthoplex

Bicantellated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t1,3{3,3,3,3,4}
2rr{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges 8640
Vertices 1440
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

• Bicantellated hexacross, bicantellated hexacontatetrapeton
• Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers)[2]

Construction

There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates

Cartesian coordinates for the 1440 vertices of a bicantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of

(2,2,1,1,0,0)

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]

Cantitruncated 6-orthoplex

Cantitruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t0,1,2{3,3,3,3,4}
tr{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges 3840
Vertices 960
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

• Cantitruncated hexacross, cantitruncated hexacontatetrapeton
• Great rhombihexacontatetrapeton (acronym: grog) (Jonathan Bowers)[3]

Construction

There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates

Cartesian coordinates for the 960 vertices of a cantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of

(3,2,1,0,0,0)

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]

Bicantitruncated 6-orthoplex

Bicantitruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t1,2,3{3,3,3,3,4}
2tr{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges 10080
Vertices 2880
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

• Bicantitruncated hexacross, bicantitruncated hexacontatetrapeton
• Great birhombihexacontatetrapeton (acronym: gaborg) (Jonathan Bowers)[4]

Construction

There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates

Cartesian coordinates for the 2880 vertices of a bicantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of

(3,3,2,1,0,0)

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

Notes

1. ^ Klitzing, (x3o3x3o3o4o - srog)
2. ^ Klitzing, (o3x3o3x3o4o - siborg)
3. ^ Klitzing, (x3x3x3o3o4o - grog)
4. ^ Klitzing, (o3x3x3x3o4o - gaborg)

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.
• Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3x3o3o4o - srog, o3x3o3x3o4o - siborg, x3x3x3o3o4o - grog, o3x3x3x3o4o - gaborg