E6 polytope

Orthographic projections in the E₆ Coxeter plane

File:Up 2 21 t0 E6.svg 221 Template:CDD

File:Up 1 22 t0 E6.svg 122 Template:CDD

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

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	File:Complex polyhedron 3-3-3-3-3.png	File:Up 2 21 t0 E6.svg	File:Up 2 21 t0 D5.svg	File:Up 2 21 t0 D4.svg	File:Up 2 21 t0 A5.svg	File:Up 2 21 t0 D3.svg	Template:CDD 221 Icosihepta-heptacontadipeton (jak)
2		File:Up 2 21 t1 E6.svg	File:Up 2 21 t1 D5.svg	File:Up 2 21 t1 D4.svg	File:Up 2 21 t1 A5.svg	File:Up 2 21 t1 D3.svg	Template:CDD Rectified 221 Rectified icosihepta-heptacontadipeton (rojak)
3		File:Up 2 21 t3 E6.svg	File:Up 2 21 t3 D5.svg	File:Up 2 21 t3 D4.svg	File:Up 2 21 t3 A5.svg	File:Up 2 21 t3 D3.svg	Template:CDD Trirectified 221 Trirectified icosihepta-heptacontadipeton (harjak)
4		File:Up 2 21 t01 E6.svg	File:Up 2 21 t01 D5.svg	File:Up 2 21 t01 D4.svg	File:Up 2 21 t01 A5.svg	File:Up 2 21 t01 D3.svg	Template:CDD Truncated 221 Truncated icosihepta-heptacontadipeton (tojak)
5		File:2 21 t02 E6.svg	File:2 21 t02 D5.svg	File:2 21 t02 D4.svg	File:2 21 t02 A5.svg	File:2 21 t02 D3.svg	Template:CDD Cantellated 221 Cantellated icosihepta-heptacontadipeton

#	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	File:Complex polyhedron 3-3-3-4-2.png	File:Up 1 22 t0 E6.svg	File:Up 1 22 t0 D5.svg	File:Up 1 22 t0 D4.svg	File:Up 1 22 t0 A5.svg	File:Up 1 22 t0 A4.svg	File:Up 1 22 t0 D3.svg	Template:CDD 122 Pentacontatetrapeton (mo)
7		File:Up 2 21 t2 E6.svg	File:Up 2 21 t2 D5.svg	File:Up 2 21 t2 D4.svg	File:Up 2 21 t2 A5.svg	File:Up 2 21 t2 A4.svg	File:Up 2 21 t2 D3.svg	Template:CDD Rectified 122 / Birectified 221 Rectified pentacontatetrapeton (ram)
8		File:Up 1 22 t2 E6.svg	File:Up 1 22 t2 D5.svg	File:Up 1 22 t2 D4.svg	File:Up 1 22 t2 A5.svg	File:Up 1 22 t2 A4.svg	File:Up 1 22 t2 D3.svg	Template:CDD Birectified 122 Birectified pentacontatetrapeton (barm)
9		File:Up 1 22 t01 E6.svg	File:Up 1 22 t01 D5.svg	File:Up 1 22 t01 D4.svg	File:Up 1 22 t01 A5.svg	File:Up 1 22 t01 A4.svg	File:Up 1 22 t01 D3.svg	Template:CDD 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, wiley.com, Template:Isbn
  • (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
• Template:KlitzingPolytopes