# Runcinated 8-simplex

 Orthogonal projections in A8Coxeter 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 A8Coxeter plane.

## Runcinated 8-simplex

Runcinated 8-simplex
Type uniform polyzetton
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
AkCoxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
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## Biruncinated 8-simplex

Biruncinated 8-simplex
Type uniform polyzetton
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: spene) (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
AkCoxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
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## Triruncinated 8-simplex

Triruncinated 8-simplex
Type uniform polyzetton
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: sabpene) (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
AkCoxeter 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]
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## Runcitruncated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Biruncitruncated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Triruncitruncated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Runcicantellated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Biruncicantellated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Runcicantitruncated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Biruncicantitruncated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Triruncicantitruncated 8-simplex

### Images

orthographic projections
AkCoxeter 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]
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## Related polytopes

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

 t0 t1 t2 t3 t01 t02 t12 t03 t13 t23 t04 t14 t24 t34 t05 t15 t25 t06 t16 t07 t012 t013 t023 t123 t014 t024 t124 t034 t134 t234 t015 t025 t125 t035 t135 t235 t045 t145 t016 t026 t126 t036 t136 t046 t056 t017 t027 t037 t0123 t0124 t0134 t0234 t1234 t0125 t0135 t0235 t1235 t0145 t0245 t1245 t0345 t1345 t2345 t0126 t0136 t0236 t1236 t0146 t0246 t1246 t0346 t1346 t0156 t0256 t1256 t0356 t0456 t0127 t0137 t0237 t0147 t0247 t0347 t0157 t0257 t0167 t01234 t01235 t01245 t01345 t02345 t12345 t01236 t01246 t01346 t02346 t12346 t01256 t01356 t02356 t12356 t01456 t02456 t03456 t01237 t01247 t01347 t02347 t01257 t01357 t02357 t01457 t01267 t01367 t012345 t012346 t012356 t012456 t013456 t023456 t123456 t012347 t012357 t012457 t013457 t023457 t012367 t012467 t013467 t012567 t0123456 t0123457 t0123467 t0123567 t01234567
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## Notes

1. ^ Klitzing (x3o3o3x3o3o3o3o - spene)
2. ^ Klitzing (o3x3o3o3x3o3o3o - sabpene)
3. ^ Klitzing (o3x3o3o3x3o3o3o - sabpene)
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## References

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## 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, editied 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.
• Richard Klitzing, 8D, uniform polytopes (polyzetta) x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb
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