Rectified 8-simplexes

8-simplex	Rectified 8-simplex
Birectified 8-simplex	Trirectified 8-simplex
Orthogonal projections in A8 Coxeter plane

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

There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.

Rectified 8-simplex

Rectified 8-simplex
Type	uniform 8-polytope
Coxeter symbol	061
Schläfli symbol	t1{37} r{37} = {36,1} or $\left\{{\begin{array}{l}3,3,3,3,3,3\\3\end{array}}\right\}$
Coxeter-Dynkin diagrams	or
7-faces	18
6-faces	108
5-faces	336
4-faces	630
Cells	756
Faces	588
Edges	252
Vertices	36
Vertex figure	7-simplex prism, {}×{3,3,3,3,3}
Petrie polygon	enneagon
Coxeter group	A8, [37], order 362880
Properties	convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S¹
₈. It is also called 0_6,1 for its branching Coxeter-Dynkin diagram, shown as .

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

Birectified 8-simplex

Birectified 8-simplex
Type	uniform 8-polytope
Coxeter symbol	052
Schläfli symbol	t2{37} 2r{37} = {35,2} or $\left\{{\begin{array}{l}3,3,3,3,3\\3,3\end{array}}\right\}$
Coxeter-Dynkin diagrams	or
7-faces	18
6-faces	144
5-faces	588
4-faces	1386
Cells	2016
Faces	1764
Edges	756
Vertices	84
Vertex figure	{3}×{3,3,3,3}
Coxeter group	A8, [37], order 362880
Properties	convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S²
₈. It is also called 0_5,2 for its branching Coxeter-Dynkin diagram, shown as .

The birectified 8-simplex is the vertex figure of the 1₅₂ honeycomb.

Coordinates

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

Trirectified 8-simplex

Trirectified 8-simplex
Type	uniform 8-polytope
Coxeter symbol	043
Schläfli symbol	t3{37} 3r{37} = {34,3} or $\left\{{\begin{array}{l}3,3,3,3\\3,3,3\end{array}}\right\}$
Coxeter-Dynkin diagrams	or
7-faces	9 + 9
6-faces	36 + 72 + 36
5-faces	84 + 252 + 252 + 84
4-faces	126 + 504 + 756 + 504
Cells	630 + 1260 + 1260
Faces	1260 + 1680
Edges	1260
Vertices	126
Vertex figure	{3,3}×{3,3,3}
Petrie polygon	enneagon
Coxeter group	A7, [37], order 362880
Properties	convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S³
₈. It is also called 0_4,3 for its branching Coxeter-Dynkin diagram, shown as .

Coordinates

The Cartesian coordinates of the vertices of the trirectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 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 the vertex figure of the 9-demicube, and the edge figure of the uniform 2₆₁ honeycomb.

It is also 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)". o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary