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]	Coxeter diagram Schläfli symbol Names
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.

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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