# Runcinated 8-simplexes

(Redirected from Triruncinated 8-simplex)
 Orthogonal projections in A8 Coxeter plane 8-simplex Runcinated 8-simplex Biruncinated 8-simplex Triruncinated 8-simplex Runcitruncated 8-simplex Biruncitruncated 8-simplex Triruncitruncated 8-simplex Runcicantellated 8-simplex Biruncicantellated 8-simplex Runcicantitruncated 8-simplex Biruncicantitruncated 8-simplex Triruncicantitruncated 8-simplex

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicantitruncated 8-simplex have a doubled symmetry, showing [18] order reflectional symmetry in the A8 Coxeter plane.

## Runcinated 8-simplex

Runcinated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,3{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 4536
Vertices 504
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Runcinated enneazetton
• Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)[1]

### Coordinates

The Cartesian coordinates of the vertices of the runcinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Biruncinated 8-simplex

Biruncinated 8-simplex
Type uniform 8-polytope
Schläfli symbol t1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 11340
Vertices 1260
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Biruncinated enneazetton
• Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)[2]

### Coordinates

The Cartesian coordinates of the vertices of the biruncinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Triruncinated 8-simplex

Triruncinated 8-simplex
Type uniform 8-polytope
Schläfli symbol t2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 15120
Vertices 1680
Vertex figure
Coxeter group A8×2, [[37]], order 725760
Properties convex

### Alternate names

• Triruncinated enneazetton
• Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)[3]

### Coordinates

The Cartesian coordinates of the vertices of the triruncinated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2,2). This construction is based on facets of the triruncinated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Runcitruncated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Biruncitruncated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Triruncitruncated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Runcicantellated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Biruncicantellated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Runcicantitruncated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Biruncicantitruncated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Triruncicantitruncated 8-simplex

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [[9]] = [18] [8] [[7]] = [14] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] = [10] [4] [[3]] = [6]

## Related polytopes

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.

## Notes

1. ^ Klitzing (x3o3o3x3o3o3o3o - spene)
2. ^ Klitzing (o3x3o3o3x3o3o3o - sabpene)
3. ^ Klitzing (o3o3x3o3o3x3o3o - satpeb)

## 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. "8D uniform polytopes (polyzetta)". x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb