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. A_k has k+1 symmetry, D_k 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 4-polytope	5-cell	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