“E6 polytope”的意思、由来-开放百科全书

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E6 polytope

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Graphs
References

Orthographic projections in the E₆ Coxeter plane
2₂₁ {{CDD>nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea}}	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.

#	Coxeter plane graphs						Coxeter diagram Names
#	Aut(E₆) [18/2]	E₆ [12]	D₅ [8]	D₄ / A₂ [6]	A₅ [6]	D₃ / A₃ [4]	Coxeter diagram Names
1	nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea}} 2₂₁ Icosihepta-heptacontidipeton (jak)
2	nodea\|3a\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea}} Rectified 2₂₁ Rectified icosihepta-heptacontidipeton (rojak)
3	nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea_1\|3a\|nodea}} Trirectified 2₂₁ Trirectified icosihepta-heptacontidipeton (harjak)
4	nodea_1\|3a\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea}} Truncated 2₂₁ Truncated icosihepta-heptacontidipeton (tojak)
5	nodea_1\|3a\|nodea\|3a\|branch_10\|3a\|nodea\|3a\|nodea}} 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]	Coxeter diagram Names
6	nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea\|3a\|nodea}} 1₂₂ Pentacontatetrapeton (mo)
7	nodea\|3a\|nodea\|3a\|branch_10\|3a\|nodea\|3a\|nodea}} Rectified 1₂₂ / Birectified 2₂₁ Rectified pentacontatetrapeton (ram)
8	nodea\|3a\|nodea_1\|3a\|branch\|3a\|nodea_1\|3a\|nodea}} Birectified 1₂₂ Birectified pentacontatetrapeton (barm)
9	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}} 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)}}

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