Cantellated 6-orthoplexes

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 6-cube

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Cantellated 6-orthoplex

Alternate names

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

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 [3^3,1,1] 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)

Bicantellated 6-orthoplex

Alternate names

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

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 [3^3,1,1] 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)

Cantitruncated 6-orthoplex

Alternate names

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

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 [3^3,1,1] 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)

Bicantitruncated 6-orthoplex

Alternate names

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

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 [3^3,1,1] 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)

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 and 6-orthoplex.

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com,
    • (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.

• x3o3x3o3o4o - srog, o3x3o3x3o4o - siborg, x3x3x3o3o4o - grog, o3x3x3x3o4o - gaborg

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary