Rectified 8-simplexes

  (Redirected from Birectified 8-simplex)
8-simplex t0.svg
8-simplex
8-simplex t1.svg
Rectified 8-simplex
8-simplex t2.svg
Birectified 8-simplex
8-simplex t3.svg
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-simplexEdit

Rectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 061
Schläfli symbol t1{37}
r{37} = {36,1}
or  
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 06,1 for its branching Coxeter-Dynkin diagram, shown as              .

CoordinatesEdit

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.

ImagesEdit

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-simplexEdit

Birectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 052
Schläfli symbol t2{37}
2r{37} = {35,2} or
 
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 05,2 for its branching Coxeter-Dynkin diagram, shown as            .

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

CoordinatesEdit

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.

ImagesEdit

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-simplexEdit

Trirectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 043
Schläfli symbol t3{37}
3r{37} = {34,3} or
 
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 04,3 for its branching Coxeter-Dynkin diagram, shown as          .

CoordinatesEdit

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.

ImagesEdit

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 polytopesEdit

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.

NotesEdit

ReferencesEdit

  • 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 linksEdit

Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds