Heptellated 8-simplexes

8-simplex	Heptellated 8-simplex	Heptihexipentisteriruncicantitruncated 8-simplex (Omnitruncated 8-simplex)
Orthogonal projections in A8 Coxeter plane (A7 for omnitruncation)

In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

There are 35 unique heptellations for the 8-simplex, including all permutations of truncations, cantellations, runcinations, sterications, pentellations, and hexications. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex. The highest form, the heptihexipentisteriruncicantitruncated 8-simplex is more simply called a omnitruncated 8-simplex with all of the nodes ringed.

Heptellated 8-simplex

Heptellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t0,7{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	504
Vertices	72
Vertex figure	6-simplex antiprism
Coxeter group	A8×2, [[37]], order 725760
Properties	convex

Alternate names

• Expanded 8-simplex
• Small exated enneazetton (soxeb) (Jonathan Bowers)^[1]

Coordinates

The vertices of the heptellated 8-simplex can bepositioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthoplex.

A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0,0,0,0)

Root vectors

Its 72 vertices represent the root vectors of the simple Lie group A₈.

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]

Omnitruncated 8-simplex

Omnitruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t0,1,2,3,4,5,6,7{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	1451520
Vertices	362880
Vertex figure	irr. 7-simplex
Coxeter group	A8, [[37]], order 725760
Properties	convex

The symmetry order of an omnitruncated 8-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the omnitruncation, being 362880 (9 factorial) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual.

Alternate names

• Heptihexipentisteriruncicantitruncated 8-simplex
• Great exated enneazetton (goxeb) (Jonathan Bowers)^[2]

Coordinates

The Cartesian coordinates of the vertices of the omnitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t_{0,1,2,3,4,5,6,7}{3⁷,4}

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]

Permutohedron and related tessellation

The omnitruncated 8-simplex is the permutohedron of order 9. The omnitruncated 8-simplex is a zonotope, the Minkowski sum of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can tessellate space by itself, in this case 8-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .

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)". x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

This page was last edited on 4 April 2023, at 04:46