# Runcinated 7-simplexes

(Redirected from Runcinated 7-simplex)
 Orthogonal projections in A7 Coxeter plane 7-simplex Runcinated 7-simplex Biruncinated 7-simplex Runcitruncated 7-simplex Biruncitruncated 7-simplex Runcicantellated 7-simplex Biruncicantellated 7-simplex Runcicantitruncated 7-simplex Biruncicantitruncated 7-simplex

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

There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.

## Runcinated 7-simplex

Runcinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 2100
Vertices 280
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Small prismated octaexon (acronym: spo) (Jonathan Bowers)[1]

### Coordinates

The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Biruncinated 7-simplex

Biruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 4200
Vertices 560
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Small biprismated octaexon (sibpo) (Jonathan Bowers)[2]

### Coordinates

The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Runcitruncated 7-simplex

runcitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 4620
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)[3]

### Coordinates

The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Biruncitruncated 7-simplex

Biruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)[4]

### Coordinates

The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Runcicantellated 7-simplex

runcicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 840
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)[5]

### Coordinates

The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Biruncicantellated 7-simplex

biruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)

### Coordinates

The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Runcicantitruncated 7-simplex

runcicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 5880
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Great prismated octaexon (acronym: gapo) (Jonathan Bowers)[6]

### Coordinates

The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Biruncicantitruncated 7-simplex

biruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 11760
Vertices 3360
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)[7]

### Coordinates

The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Related polytopes

These polytopes are among 71 uniform 7-polytopes with A7 symmetry.

## Notes

1. ^ Klitzing, (x3o3o3x3o3o3o - spo)
2. ^ Klitzing, (o3x3o3o3x3o3o - sibpo)
3. ^ Klitzing, (x3x3o3x3o3o3o - patto)
4. ^ Klitzing, (o3x3x3o3x3o3o - bipto)
5. ^ Klitzing, (x3o3x3x3o3o3o - paro)
6. ^ Klitzing, (x3x3x3x3o3o3o - gapo)
7. ^ Klitzing, (o3x3x3x3x3o3o- gibpo)

## 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 [1]
• (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.
• Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo