Cantellated 5-cubes


5-cube

Cantellated 5-cube

Bicantellated 5-cube

Cantellated 5-orthoplex

5-orthoplex

Cantitruncated 5-cube

Bicantitruncated 5-cube

Cantitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

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

Cantellated 5-cube

Cantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol rr{4,3,3,3} =
Coxeter-Dynkin diagram =
4-faces 122
Cells 680
Faces 1520
Edges 1280
Vertices 320
Vertex figure
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

Coordinates

The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Bicantellated 5-cube

Bicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbols 2rr{4,3,3,3} =
r{32,1,1} =
Coxeter-Dynkin diagrams =
4-faces 122
Cells 840
Faces 2160
Edges 1920
Vertices 480
Vertex figure
Coxeter group B5 [4,3,3,3]
Properties convex

In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

Alternate names

Coordinates

The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:

(0,1,1,2,2)

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Cantitruncated 5-cube

Cantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol tr{4,3,3,3} =
Coxeter-Dynkin
diagram
=
4-faces122
Cells680
Faces1520
Edges1600
Vertices640
Vertex figure
Irr. 5-cell
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

Coordinates

The Cartesian coordinates of the vertices of an cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Bicantitruncated 5-cube

Bicantitruncated 5-cube
Typeuniform 5-polytope
Schläfli symbol 2tr{3,3,3,4} =
t{32,1,1} =
Coxeter-Dynkin diagrams =
4-faces122
Cells840
Faces2160
Edges2400
Vertices960
Vertex figure
Coxeter groupsB5, [3,3,3,4]
D5, [32,1,1]
Propertiesconvex

Alternate names

Coordinates

Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of

(±3,±3,±2,±1,0)

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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