// Workers AI · dad joke modeWhat did the E7 polytope say? "I'm a complex shape.

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 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet.

#	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									231 (laq)
2									Rectified 231 (rolaq)
3									Rectified 132 (rolin)
4									132 (lin)
5									Birectified 321 (branq)
6									Rectified 321 (ranq)
7									321 (naq)
8									Truncated 231 (talq)
9									Truncated 132 (tilin)
10									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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
  • (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) with acronyms".

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations