E6 polytope

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|+ Orthographic projections in the E₆ Coxeter plane

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2₂₁
{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}

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1₂₂
{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}

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.

rowspan=2\|# !colspan=6\|Coxeter plane graphs !rowspan=2\|Coxeter diagram Names
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Aut(E₆) [18/2] !E₆ [12] !D₅ [8] !D₄ / A₂ [6] !A₅ [6] ! D₃ / A₃ [4]
align=center \|1	80px	80px	80px	80px	80px	80px	{{CDD\|nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea}} 2₂₁ Icosihepta-heptacontidipeton (jak)
align=center \|2		80px	80px	80px	80px	80px	{{CDD\|nodea\|3a\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea}} Rectified 2₂₁ Rectified icosihepta-heptacontidipeton (rojak)
style="text-align:center;" \|3		80px	80px	80px	80px	80px	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea_1\|3a\|nodea}} Trirectified 2₂₁ Trirectified icosihepta-heptacontidipeton (harjak)
align=center \|4		80px	80px	80px	80px	80px	{{CDD\|nodea_1\|3a\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea}} Truncated 2₂₁ Truncated icosihepta-heptacontidipeton (tojak)
align=center \|5		80px	80px	80px	80px	80px	{{CDD\|nodea_1\|3a\|nodea\|3a\|branch_10\|3a\|nodea\|3a\|nodea}} Cantellated 2₂₁ Cantellated icosihepta-heptacontidipeton

rowspan=2\|# !colspan=7\|Coxeter plane graphs !rowspan=2\|Coxeter diagram Names
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Aut(E₆) [18] !E₆ [12] !D₅ [8] !D₄ / A₂ [6] !A₅ [6] !D₆ / A₄ [10] ! D₃ / A₃ [4]
style="text-align:center; background:#e0f0e0;" \|6	80px	80px	80px	80px	80px	80px	80px	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea\|3a\|nodea}} 1₂₂ Pentacontatetrapeton (mo)
style="text-align:center; background:#e0f0e0;" \|7		80px	80px	80px	80px	80px	80px	{{CDD\|nodea\|3a\|nodea\|3a\|branch_10\|3a\|nodea\|3a\|nodea}} Rectified 1₂₂ / Birectified 2₂₁ Rectified pentacontatetrapeton (ram)
style="text-align:center; background:#e0f0e0;" \|8		80px	80px	80px	80px	80px	80px	{{CDD\|nodea\|3a\|nodea_1\|3a\|branch\|3a\|nodea_1\|3a\|nodea}} Birectified 1₂₂ Birectified pentacontatetrapeton (barm)
style="text-align:center; background:#e0f0e0;" \|9		80px	80px	80px	80px	80px	80px	{{CDD\|nodea\|3a\|nodea\|3a\|branch_11\|3a\|nodea\|3a\|nodea}} 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html 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
{{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}}

Category:6-polytopes