# Runcinated 7-orthoplexes

(Redirected from Biruncitruncated 7-orthoplex)
Orthogonal projections in B6 Coxeter plane

7-orthoplex

Runcinated 7-orthoplex

Biruncinated 7-orthoplex

Runcitruncated 7-orthoplex

Biruncitruncated 7-orthoplex

Runcicantellated 7-orthoplex

Biruncicantellated 7-orthoplex

Runcicantitruncated 7-orthoplex

Biruncicantitruncated 7-orthoplex

In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex.

There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-cube.

These polytopes are among 127 uniform 7-polytopes with B7 symmetry.

## Runcinated 7-orthoplex

Runcinated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 23520
Vertices 2240
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Small prismated hecatonicosoctaexon (acronym: spaz) (Jonathan Bowers)[1]

### 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]

## Biruncinated 7-orthoplex

Biruncinated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 60480
Vertices 6720
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Small biprismated hecatonicosoctaexon (Acronym sibpaz) (Jonathan Bowers)[2]

### 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]

## Runcitruncated 7-orthoplex

Runcitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 50400
Vertices 6720
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Prismatotruncated hecatonicosoctaexon (acronym: potaz) (Jonathan Bowers)[3]

### 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]

## Biruncitruncated 7-orthoplex

Biruncitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,2,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 120960
Vertices 20160
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Biprismatotruncated hecatonicosoctaexon (acronym: baptize) (Jonathan Bowers)[4]

### 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]

## Runcicantellated 7-orthoplex

Runcicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 33600
Vertices 6720
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Prismatorhombated hecatonicosoctaexon (acronym: parz) (Jonathan Bowers)[5]

### 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]

## Biruncicantellated 7-orthoplex

biruncicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 100800
Vertices 20160
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Biprismatorhombated hecatonicosoctaexon (acronym: boparz) (Jonathan Bowers)[6]

### 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]

## Runcicantitruncated 7-orthoplex

Runcicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 60480
Vertices 13440
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Great prismated hecatonicosoctaexon (acronym: gopaz) (Jonathan Bowers)[7]

### 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]

## Biruncicantitruncated 7-orthoplex

biruncicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 161280
Vertices 40320
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

### Alternate names

• Great biprismated hecatonicosoctaexon (acronym: gibpaz) (Jonathan Bowers)[8]

### 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]

## Notes

1. ^ Klitzing, (o3o3o3x3o3o4x - spaz)
2. ^ Klitzing, (o3x3o3o3x3o4o - sibpaz)
3. ^ Klitzing, (o3o3o3x3x3o4x - potaz)
4. ^ Klitzing, (o3o3x3o3x3x4o - baptize)
5. ^ Klitzing, (o3o3o3x3x3o4x - parz)
6. ^ Klitzing, (o3x3o3x3x3o4o - boparz)
7. ^ Klitzing, (o3o3o3x3x3x4x - gopaz)
8. ^ Klitzing, (o3o3x3x3x3x3o - gibpaz)

## 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 Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 Wiley: Kaleidoscopes: Selected Writings of H.S.M. Coxeter
• (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. (1966)
• Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3o3x3o3o4x - spaz, o3x3o3o3x3o4o - sibpaz, o3o3o3x3x3o4x - potaz, o3o3x3o3x3x4o - baptize, o3o3o3x3x3o4x - parz, o3x3o3x3x3o4o - boparz, o3o3o3x3x3x4x - gopaz, o3o3x3x3x3x3o - gibpaz