Cantellated 8-simplexes

Cantellated 8-simplex	Bicantellated 8-simplex	Tricantellated 8-simplex
Cantitruncated 8-simplex	Bicantitruncated 8-simplex	Tricantitruncated 8-simplex
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.

There are six unique cantellations for the 8-simplex, including permutations of truncation.

Cantellated 8-simplex

Cantellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	rr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	1764
Vertices	252
Vertex figure	6-simplex prism
Coxeter group	A8, [37], order 362880
Properties	convex

Alternate names

• Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)^[1]

Coordinates

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

Bicantellated 8-simplex

Bicantellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	r2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	5292
Vertices	756
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

Alternate names

• Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)^[2]

Coordinates

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

Tricantellated 8-simplex

tricantellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	r3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	8820
Vertices	1260
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

Alternate names

• Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)^[3]

Coordinates

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

Cantitruncated 8-simplex

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

Alternate names

• Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)^[4]

Coordinates

The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 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]

Bicantitruncated 8-simplex

Bicantitruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

Alternate names

• Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)^[5]

Coordinates

The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 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]

Tricantitruncated 8-simplex

Tricantitruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A8, [37], order 362880
Properties	convex

• Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)^[6]

Coordinates

The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 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]

Related polytopes

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

A8 polytopes
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
t<sub>015</sub>	t<sub>025</sub>	t125	t<sub>035</sub>	t135	t<sub>235</sub>	t<sub>045</sub>	t145	t<sub>016</sub>	t<sub>026</sub>	t<sub>126</sub>	t<sub>036</sub>	t<sub>136</sub>	t<sub>046</sub>	t<sub>056</sub>
t<sub>017</sub>	t<sub>027</sub>	t<sub>037</sub>	t0123	t0124	t0134	t0234	t1234	t<sub>0125</sub>	t<sub>0135</sub>	t<sub>0235</sub>	t1235	t<sub>0145</sub>	t<sub>0245</sub>	t1245
t<sub>0345</sub>	t1345	t2345	t<sub>0126</sub>	t<sub>0136</sub>	t<sub>0236</sub>	t<sub>1236</sub>	t<sub>0146</sub>	t<sub>0246</sub>	t<sub>1246</sub>	t<sub>0346</sub>	t<sub>1346</sub>	t<sub>0156</sub>	t<sub>0256</sub>	t<sub>1256</sub>
t<sub>0356</sub>	t<sub>0456</sub>	t<sub>0127</sub>	t<sub>0137</sub>	t<sub>0237</sub>	t<sub>0147</sub>	t<sub>0247</sub>	t<sub>0347</sub>	t<sub>0157</sub>	t<sub>0257</sub>	t<sub>0167</sub>	t01234	t<sub>01235</sub>	t<sub>01245</sub>	t<sub>01345</sub>
t<sub>02345</sub>	t12345	t<sub>01236</sub>	t<sub>01246</sub>	t<sub>01346</sub>	t<sub>02346</sub>	t<sub>12346</sub>	t<sub>01256</sub>	t<sub>01356</sub>	t<sub>02356</sub>	t<sub>12356</sub>	t<sub>01456</sub>	t<sub>02456</sub>	t<sub>03456</sub>	t<sub>01237</sub>
t<sub>01247</sub>	t<sub>01347</sub>	t<sub>02347</sub>	t<sub>01257</sub>	t<sub>01357</sub>	t<sub>02357</sub>	t<sub>01457</sub>	t<sub>01267</sub>	t<sub>01367</sub>	t<sub>012345</sub>	t<sub>012346</sub>	t<sub>012356</sub>	t<sub>012456</sub>	t<sub>013456</sub>	t<sub>023456</sub>
t<sub>123456</sub>	t<sub>012347</sub>	t<sub>012357</sub>	t<sub>012457</sub>	t<sub>013457</sub>	t<sub>023457</sub>	t<sub>012367</sub>	t<sub>012467</sub>	t<sub>013467</sub>	t<sub>012567</sub>	t<sub>0123456</sub>	t<sub>0123457</sub>	t<sub>0123467</sub>	t<sub>0123567</sub>	t01234567

Notes

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)". x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

This page was last edited on 4 April 2023, at 04:42