E6 polytope

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Orthographic projections in the E6 Coxeter plane
Up 2 21 t0 E6.svg
221
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 1 22 t0 E6.svg
122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

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.

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

Graphs[edit]

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
Alt(E6)
[9]
E6
[12]
D5
[8]
D4 / A2
[6]
A5
[6]
D3 / A3
[4]
1 Complex polyhedron 3-3-3-3-3.png Up 2 21 t0 E6.svg Up 2 21 t0 D5.svg Up 2 21 t0 D4.svg Up 2 21 t0 A5.svg Up 2 21 t0 D3.svg CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
221
Icosihepta-heptacontidipeton (jak)
2 Up 2 21 t1 E6.svg Up 2 21 t1 D5.svg Up 2 21 t1 D4.svg Up 2 21 t1 A5.svg Up 2 21 t1 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 221
Rectified icosihepta-heptacontidipeton (rojak)
3 Up 2 21 t3 E6.svg Up 2 21 t3 D5.svg Up 2 21 t3 D4.svg Up 2 21 t3 A5.svg Up 2 21 t3 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Trirectified 221
Trirectified icosihepta-heptacontidipeton (harjak)
4 Up 2 21 t01 E6.svg Up 2 21 t01 D5.svg Up 2 21 t01 D4.svg Up 2 21 t01 A5.svg Up 2 21 t01 D3.svg CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 221
Truncated icosihepta-heptacontidipeton (tojak)
5 2 21 t02 E6.svg 2 21 t02 D5.svg 2 21 t02 D4.svg 2 21 t02 A5.svg 2 21 t02 D3.svg CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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 Complex polyhedron 3-3-3-4-2.png Up 1 22 t0 E6.svg Up 1 22 t0 D5.svg Up 1 22 t0 D4.svg Up 1 22 t0 A5.svg Up 1 22 t0 A4.svg Up 1 22 t0 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
122
Pentacontatetrapeton (mo)
7 Up 2 21 t2 E6.svg Up 2 21 t2 D5.svg Up 2 21 t2 D4.svg Up 2 21 t2 A5.svg Up 2 21 t2 A4.svg Up 2 21 t2 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 122 / Birectified 221
Rectified pentacontatetrapeton (ram)
8 Up 1 22 t2 E6.svg Up 1 22 t2 D5.svg Up 1 22 t2 D4.svg Up 1 22 t2 A5.svg Up 1 22 t2 A4.svg Up 1 22 t2 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Bicantellated 221 / Birectified 122
Birectified pentacontatetrapeton (barm)
9 Up 1 22 t01 E6.svg Up 1 22 t01 D5.svg Up 1 22 t01 D4.svg Up 1 22 t01 A5.svg Up 1 22 t01 A4.svg Up 1 22 t01 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 122
Truncated pentacontatetrapeton (tim)

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 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 / 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