E6 polytope

In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72 vertices respectively.

Orthographic projections in the E6 Coxeter plane

221
122

They can be visualized as symmetric orthographic projections in Coxeter planes of the E6 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 39 polytopes can be made in the E6, D5, D4, D2, A5, A4, A3 Coxeter planes. Ak has k+1 symmetry, Dk has 2(k-1) symmetry, and E6 has 12 symmetry.

Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E6 symmetry. The vertices and edges drawn with vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane graphs Coxeter diagram
Names
Aut(E6)
[18/2]		 E6
[12]		 D5
[8]		 D4 / A2
[6]		 A5
[6]		 D3 / A3
[4]
1
221
Icosihepta-heptacontidipeton (jak)
2
Rectified 221
Rectified icosihepta-heptacontidipeton (rojak)
3
Trirectified 221
Trirectified icosihepta-heptacontidipeton (harjak)
4
Truncated 221
Truncated icosihepta-heptacontidipeton (tojak)
5
Cantellated 221
Cantellated icosihepta-heptacontidipeton
# Coxeter plane graphs Coxeter diagram
Names
Aut(E6)
[18]		 E6
[12]		 D5
[8]		 D4 / A2
[6]		 A5
[6]		 D6 / A4
[10]		 D3 / A3
[4]
6
122
Pentacontatetrapeton (mo)
7
Rectified 122 / Birectified 221
Rectified pentacontatetrapeton (ram)
8
Birectified 122
Birectified pentacontatetrapeton (barm)
9
Truncated 122
Truncated pentacontatetrapeton (tim)

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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
    • (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. "6D uniform polytopes (polypeta)".
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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