E8 polytope

Orthographic projections in the E8 Coxeter plane

421

241

142

In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280 vertices respectively.

These polytopes can be visualized as symmetric orthographic projections in Coxeter planes of the E8 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 255 polytopes can be made in the E8, E7, E6, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry, and E6, E7, E8 have [12], [18], [30] symmetry respectively. In addition there are two other degrees of fundamental invariants, order [20] and [24] for the E8 group that represent Coxeter planes.

11 of these 255 polytopes are each shown in 14 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane projections Coxeter-Dynkin diagram
Name
E8
[30]
E7
[18]
E6
[12]
[24] [20] D4-E6
[6]
A3
D3
[4]
A2
D4
[6]
D5
[8]
A4
D6
[10]
D7
[12]
A6
B7
[14]
B8
[16/2]
A5
 
[6]
A7
 
[8]
1
421 (fy)
2
Rectified 421 (riffy)
3
Birectified 421 (borfy)
4
Trirectified 421 (torfy)
5
Rectified 142 (buffy)
6
Rectified 241 (robay)
7
241 (bay)
8
Truncated 241
9
Truncated 421 (tiffy)
10
142 (bif)
11
Truncated 142

References

Notes

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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