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Cantellated 6 orthoplexes

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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.

Contents

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

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 [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)

    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 [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)

    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 [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)

    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 [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)

    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.

    References

    Cantellated 6-orthoplexes Wikipedia
    (Text) CC BY-SA


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