E7 polytope

Orthographic projections in the E₇ Coxeter plane

321

231

132

In 7-dimensional geometry, there are 127 uniform polytopes with E₇ symmetry. The three simplest forms are the 3₂₁, 2₃₁, and 1₃₂ polytopes, composed of 56, 126, and 576 vertices respectively.

They can be visualized as symmetric orthographic projections in Coxeter planes of the E₇ Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 127 polytopes can be made in the E₇, E₆, D₆, D₅, D₄, D₃, A₆, A₅, A₄, A₃, A₂ Coxeter planes. A_k has k+1 symmetry, D_k has 2(k-1) symmetry, and E₆ and E₇ have 12, 18 symmetry respectively.

For 10 of 127 polytopes (7 single rings, and 3 truncations), they are shown in these 9 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

#	Coxeter plane graphs								Coxeter diagram Schläfli symbol Names
	E7 [18]	E6	A6 [7x2]	A5 [6]	A4 / D6 [10]	D5 [8]	A2 / D4 [6]	A3 / D3 [4]
1		80px	80px	80px	80px	80px	80px		231 (laq)
2		80px	80px	80px	80px	80px	80px		Rectified 231 (rolaq)
3		80px	80px	80px	80px	80px	80px		Rectified 132 (rolin)
4		80px	80px	80px	80px	80px	80px	80px	132 (lin)
5		80px	80px	80px	80px	80px	80px		Birectified 321 (branq)
6		80px	80px	80px	80px	80px	80px		Rectified 321 (ranq)
7		80px	80px	80px	80px	80px	80px		321 (naq)
8		80px	80px	80px	80px	80px	80px		Truncated 231 (talq)
9		80px	80px	80px	80px	80px	80px		Truncated 132 (tilin)
10		80px	80px	80px	80px	80px	80px		Truncated 321 (tanq)

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. "7D uniform polytopes (polyexa)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.