Runcic 6-cubes

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In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

More information Orthogonal projections in D6 Coxeter plane ...

6-demicube
=

Runcic 6-cube
=

Runcicantic 6-cube
=
Orthogonal projections in D6 Coxeter plane
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Runcic 6-cube

Runcic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,2{3,33,1}
h3{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges3840
Vertices640
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex

Alternate names

  • Cantellated 6-demicube
  • Cantellated demihexeract
  • Small rhombated hemihexeract (Acronym: sirhax) (Jonathan Bowers)[1]

Cartesian coordinates

The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±3)

with an odd number of plus signs.

Images

More information Coxeter plane, B6 ...
orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
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More information n-cubes, n ...
Runcic n-cubes
n45678
[1+,4,3n-2]
= [3,3n-3,1]
[1+,4,32]
= [3,31,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
[1+,4,36]
= [3,35,1]
Runcic
figure
Coxeter
=

=

=

=

=
Schläfli h3{4,32} h3{4,33} h3{4,34} h3{4,35} h3{4,36}
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Runcicantic 6-cube

More information Runcicantic 6-cube ...
Runcicantic 6-cube
Typeuniform 6-polytope
Schläfli symbolt0,1,2{3,33,1}
h2,3{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges5760
Vertices1920
Vertex figure
Coxeter groupsD6, [33,1,1]
Propertiesconvex
Close

Alternate names

  • Cantitruncated 6-demicube
  • Cantitruncated demihexeract
  • Great rhombated hemihexeract (Acronym: girhax) (Jonathan Bowers)[2]

Cartesian coordinates

The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±5,±5)

with an odd number of plus signs.

Images

More information Coxeter plane, B6 ...
orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Close

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

Notes

References

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