Runcinated 8-simplexes

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
Orthogonal projections in A8 Coxeter plane

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 triruncicanti​truncated 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)

The Cartesian coordinates of the vertices of the runcinated 8-simplex can be most simply positioned in 9-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)

Coordinates

The Cartesian coordinates of the vertices of the biruncinated 8-simplex can be most simply positioned in 9-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)

Coordinates

The Cartesian coordinates of the vertices of the triruncinated 8-simplex can be most simply positioned in 9-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.

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

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary