List of E7 polytopes
 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, D2, A7, A6, A5, A4, A3, A2Coxeter 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-one nodea_1, 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-Dynkin diagram Schläfli symbol Names |
||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| E7 [18] |
E6 / F4 [12] |
A6 / B7 [7x2] |
A5 [6] |
D7 [12/2] |
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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (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
- Richard Klitzing, 7D, uniform polytopes (polyexa)
Notes
| Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | BCn | Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | Hexagon | Pentagon | ||||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform polychoron | 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 | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
| Topics: Polytope families • Regular polytope • List of regular polytopes | ||||||||||||
