E8 polytope

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Orthographic projections in the E8 Coxeter plane
E8 graph.svg
421
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
2 41 t0 E8.svg
241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Gosset 1 42 polytope petrie.svg
142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

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[edit]

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 4 21 t0 E8.svg 4 21 t0 E7.svg 4 21 t0 E6.svg 4 21 t0 p20.svg 4 21 t0 p24.svg 4 21 t0 mox.svg 4 21 t0 B2.svg 4 21 t0 B3.svg 4 21 t0 B4.svg 4 21 t0 B5.svg 4 21 t0 B6.svg 4 21 t0 B7.svg 4 21 t0 B8.svg 4 21 t0 A5.svg 4 21 t0 A7.svg CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
421 (fy)
2 4 21 t1 E8.svg 4 21 t1 E7.svg 4 21 t1 E6.svg 4 21 t1 p20.svg 4 21 t1 p24.svg 4 21 t1 mox.svg 4 21 t1 B2.svg 4 21 t1 B3.svg 4 21 t1 B4.svg 4 21 t1 B5.svg 4 21 t1 B6.svg 4 21 t1 B7.svg 4 21 t1 B8.svg 4 21 t1 A5.svg 4 21 t1 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 421 (riffy)
3 4 21 t2 E8.svg 4 21 t2 E7.svg 4 21 t2 E6.svg 4 21 t2 p20.svg 4 21 t2 p24.svg 4 21 t2 mox.svg 4 21 t2 B2.svg 4 21 t2 B3.svg 4 21 t2 B4.svg 4 21 t2 B5.svg 4 21 t2 B6.svg 4 21 t2 B7.svg 4 21 t2 B8.svg 4 21 t2 A5.svg 4 21 t2 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Birectified 421 (borfy)
4 4 21 t3 E7.svg 4 21 t3 E6.svg 4 21 t3 mox.svg 4 21 t3 B2.svg 4 21 t3 B3.svg 4 21 t3 B4.svg 4 21 t3 B5.svg 4 21 t3 B6.svg 4 21 t3 B7.svg 4 21 t3 A5.svg 4 21 t3 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Trirectified 421 (torfy)
5 4 21 t4 E7.svg 4 21 t4 E6.svg 4 21 t4 mox.svg 4 21 t4 B2.svg 4 21 t4 B3.svg 4 21 t4 B4.svg 4 21 t4 B5.svg 4 21 t4 B6.svg 4 21 t4 A5.svg 4 21 t4 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 142 (buffy)
6 2 41 t1 E8.svg 2 41 t1 E7.svg 2 41 t1 E6.svg 2 41 t1 p20.svg 2 41 t1 p24.svg 2 41 t1 mox.svg 2 41 t1 B2.svg 2 41 t1 B3.svg 2 41 t1 B4.svg 2 41 t1 B5.svg 2 41 t1 B6.svg 2 41 t1 B7.svg 2 41 t1 B8.svg 2 41 t1 A5.svg 2 41 t1 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Rectified 241 (robay)
7 2 41 t0 E8.svg 2 41 t0 E7.svg 2 41 t0 E6.svg 2 41 t0 p20.svg 2 41 t0 p24.svg 2 41 t0 mox.svg 2 41 t0 B2.svg 2 41 t0 B3.svg 2 41 t0 B4.svg 2 41 t0 B5.svg 2 41 t0 B6.svg 2 41 t0 B7.svg 2 41 t0 B8.svg 2 41 t0 A5.svg 2 41 t0 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
241 (bay)
8 2 41 t01 E7.svg 2 41 t01 E6.svg 2 41 t01 B2.svg 2 41 t01 B3.svg 2 41 t01 B4.svg 2 41 t01 B5.svg 2 41 t01 B6.svg 2 41 t01 B7.svg 2 41 t01 A5.svg 2 41 t01 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea 1.png
Truncated 241
9 4 21 t01 E8.svg 4 21 t01 E7.svg 4 21 t01 E6.svg 4 21 t01 p20.svg 4 21 t01 p24.svg 4 21 t01 B2.svg 4 21 t01 B3.svg 4 21 t01 B4.svg 4 21 t01 B5.svg 4 21 t01 B6.svg 4 21 t01 B7.svg 4 21 t01 B8.svg 4 21 t01 A5.svg 4 21 t01 A7.svg CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 421 (tiffy)
10 Gosset 1 42 polytope petrie.svg 1 42 t0 e7.svg 1 42 polytope E6 Coxeter plane.svg 1 42 t0 p20.svg 1 42 t0 p24.svg 1 42 t0 mox.svg 1 42 t0 B2.svg 1 42 t0 B3.svg 1 42 t0 B4.svg 1 42 t0 B5.svg 1 42 t0 B6.svg 1 42 t0 B7.svg 1 42 t0 B8.svg 1 42 t0 A5.svg 1 42 t0 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
142 (bif)
11 1 42 t01 E6.svg 1 42 t01 B2.svg 1 42 t01 B3.svg 1 42 t01 B4.svg 1 42 t01 B5.svg 1 42 t01 B6.svg 1 42 t01 A5.svg 1 42 t01 A7.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 142

References[edit]

  • 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]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Klitzing, Richard. "8D uniform polytopes (polyzetta)". 

Notes[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / 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