Hexicated 8-simplexes

Hexicated 8-simplex
280px 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.

The Cartesian coordinates of the vertices of the hexicated 8-simplex can be most simply positioned in 8-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	120pxpx	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[9]	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	120pxpx	120pxpx	120pxpx
Dihedral symmetry	[5]	[4]	[3]

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, PhD
• Richard Klitzing, 8D uniform polytopes (polyzetta), x3o3o3o3o3o3x3o

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary