E7 polytope

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Orthographic projections in the E7 Coxeter plane
Up2 3 21 t0 E7.svg
321
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t0 E7.svg
231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Up2 1 32 t0 E7.svg
132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

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[edit]

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. Ak has k+1 symmetry, Dk 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 Up2 2 31 t0 E7.svg Up2 2 31 t0 E6.svg Up2 2 31 t0 A6.svg Up2 2 31 t0 A5.svg Up2 2 31 t0 D6.svg Up2 2 31 t0 D5.svg Up2 2 31 t0 D4.svg Up2 2 31 t0 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
231 (laq)
2 Up2 2 31 t1 E7.svg Up2 2 31 t1 E6.svg Up2 2 31 t1 A6.svg Up2 2 31 t1 A5.svg Up2 2 31 t1 D6.svg Up2 2 31 t1 D5.svg Up2 2 31 t1 D4.svg Up2 2 31 t1 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Rectified 231 (rolaq)
3 Up2 1 32 t1 E7.svg Up2 1 32 t1 E6.svg Up2 1 32 t1 A6.svg Up2 1 32 t1 A5.svg Up2 1 32 t1 D6.svg Up2 1 32 t1 D5.svg Up2 1 32 t1 D4.svg Up2 1 32 t1 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 132 (rolin)
4 Up2 1 32 t0 E7.svg Up2 1 32 t0 E6.svg Up2 1 32 t0 A6.svg Up2 1 32 t0 A5.svg Up2 1 32 t0 D6.svg Up2 1 32 t0 D5.svg Up2 1 32 t0 D4.svg Up2 1 32 t0 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
132 (lin)
5 Up2 3 21 t2 E7.svg Up2 3 21 t2 E6.svg Up2 3 21 t2 A6.svg Up2 3 21 t2 A5.svg Up2 3 21 t2 D6.svg Up2 3 21 t2 D5.svg Up2 3 21 t2 D4.svg Up2 3 21 t2 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Birectified 321 (branq)
6 Up2 3 21 t1 E7.svg Up2 3 21 t1 E6.svg Up2 3 21 t1 A6.svg Up2 3 21 t1 A5.svg Up2 3 21 t1 D6.svg Up2 3 21 t1 D5.svg Up2 3 21 t1 D4.svg Up2 3 21 t1 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Rectified 321 (ranq)
7 Up2 3 21 t0 E7.svg Up2 3 21 t0 E6.svg Up2 3 21 t0 A6.svg Up2 3 21 t0 A5.svg Up2 3 21 t0 D6.svg Up2 3 21 t0 D5.svg Up2 3 21 t0 D4.svg Up2 3 21 t0 D3.svg CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
321 (naq)
8 Up2 2 31 t01 E7.svg Up2 2 31 t01 E6.svg Up2 2 31 t01 A6.svg Up2 2 31 t01 A5.svg Up2 2 31 t01 D6.svg Up2 2 31 t01 D5.svg Up2 2 31 t01 D4.svg Up2 2 31 t01 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea 1.png
Truncated 231 (talq)
9 Up2 1 32 t01 E7.svg Up2 1 32 t01 E6.svg Up2 1 32 t01 A6.svg Up2 1 32 t01 A5.svg Up2 1 32 t01 D6.svg Up2 1 32 t01 D5.svg Up2 1 32 t01 D4.svg Up2 1 32 t01 D3.svg CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 132 (tilin)
10 Up2 3 21 t01 E7.svg Up2 3 21 t01 E6.svg Up2 3 21 t01 A6.svg Up2 3 21 t01 A5.svg Up2 3 21 t01 D6.svg Up2 3 21 t01 D5.svg Up2 3 21 t01 D4.svg Up2 3 21 t01 D3.svg CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Truncated 321 (tanq)

References[edit]

  • 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)". 
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds