Hexicated 8-simplexes

Hexicated 8-simplex
Orthogonal projection on A8 Coxeter plane
Type	uniform 8-polytope
Schläfli symbol	t0,6{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	2268
Vertices	252
Vertex figure
Coxeter groups	A8, [37], order 362880
Properties	convex

In eight-dimensional geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication (6th order truncation) of the regular 8-simplex.

Coordinates

The Cartesian coordinates of the vertices of the hexicated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,1,1,2). This construction is based on facets of the hexicated 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, PhD
• Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3x3o".

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

This page was last edited on 15 January 2023, at 17:24