Hexicated 7-cubes

Orthogonal projections in B4 Coxeter plane

7-cube

Hexicated 7-cube

Hexitruncated 7-cube

Hexicantellated 7-cube

Hexiruncinated 7-cube

Hexicantitruncated 7-cube

Hexiruncitruncated 7-cube

Hexiruncicantellated 7-cube

Hexisteritruncated 7-cube

Hexistericantellated 7-cube

Hexipentitruncated 7-cube

Hexiruncicantitruncated 7-cube

Hexistericantitruncated 7-cube

Hexisteriruncitruncated 7-cube

Hexisteriruncicantellated 7-cube

Hexipenticantitruncated 7-cube

Hexipentiruncitruncated 7-cube

Hexisteriruncicantitruncated 7-cube

Hexipentiruncicantitruncated 7-cube

Hexipentistericantitruncated 7-cube

Hexipentisteriruncicantitruncated 7-cube
(Omnitruncated 7-cube)

In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-cube.

There are 32 hexications for the 7-cube, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 20 are represented here, while 12 are more easily constructed from the 7-orthoplex.

The simple hexicated 7-cube is also called an expanded 7-cube, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-cube. The highest form, the hexipentisteriruncicantitruncated 7-cube is more simply called a omnitruncated 7-cube with all of the nodes ringed.

These polytope are among a family of 127 uniform 7-polytopes with B7 symmetry.

Hexicated 7-cube

Hexicated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-cube, or alternately can be seen as an expansion operation.

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexitruncated 7-cube

hexitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexicantellated 7-cube

Hexicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,2,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexiruncinated 7-cube

Hexiruncinated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,3,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexicantitruncated 7-cube

Hexicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexiruncitruncated 7-cube

Hexiruncitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,3,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexiruncicantellated 7-cube

Hexiruncicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,2,3,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

In seven-dimensional geometry, a hexiruncicantellated 7-cube is a uniform 7-polytope.

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexisteritruncated 7-cube

hexisteritruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexistericantellated 7-cube

hexistericantellated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,2,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexipentitruncated 7-cube

Hexipentitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,5,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexiruncicantitruncated 7-cube

Hexiruncicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,3,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph too complex too complex
Dihedral symmetry [6] [4]

Hexistericantitruncated 7-cube

Hexistericantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexisteriruncitruncated 7-cube

Hexisteriruncitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexisteriruncicantellated 7-cube

Hexisteriruncitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,2,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexipenticantitruncated 7-cube

hexipenticantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,5,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexipentiruncitruncated 7-cube

Hexisteriruncicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexisteriruncicantitruncated 7-cube

Hexisteriruncicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexipentiruncicantitruncated 7-cube

Hexipentiruncicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,3,5,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Hexipentistericantitruncated 7-cube

Hexipentistericantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,4,5,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Omnitruncated 7-cube

Omnitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbol t0,1,2,3,4,5,6{36}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

The omnitruncated 7-cube is the largest uniform 7-polytope in the B7 symmetry of the regular 7-cube. It can also be called the hexipentisteriruncicantitruncated 7-cube which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

Alternate names

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Notes

    References

    External links

    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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