User:Tomruen/Uniform polyhedron table
Table of vertex figures by Wythoff symbol sets on the sphere and plane:
| (p q r) | q|pr | p|qr | r|pq | qr|p | pr|q | qp|r | pqr| | |pqr |
|---|---|---|---|---|---|---|---|---|
Dihedral[edit] | ||||||||
| (2 2 2) | [di:(2)^2] | - | - | [hos:4.4] | - | - |  4.4.4 |
 3.3.3 |
| (3 2 2) | [di:(3)^2] | [di:(2)^3] | - | [hos:6.6] |  3.4.4 |
- |  6.4.4 |
 3.3.3.3 |
| (4 2 2) | [di:(4)^2] | [di:(2)^4] | - | [hos:8.8] |  4.4.4 |
- |  8.4.4 |
 3.3.3.4 |
| (5 2 2) | [di:(5)^2] | [di:(2)^5] | - | [hos:10.10] |  5.4.4 |
- |  10.4.4 |
3.3.3.5 |
| (6 2 2) | [di:(6)^2] | [di:(2)^6] | - | [hos:12.12] | 6.4.4 |
- |  12.4.4 |
 3.3.3.6 |
| (7 2 2) | [di:(7)^2] | [di:(2)^7] | - | [hos:14.14] | 7.4.4 | - | 14.4.4 | 3.3.3.7 |
| (8 2 2) | [di:(8)^2] | [di:(2)^8] | - | [hos:16.16] | 8.4.4 |
- | 16.4.4 |  3.3.3.8 |
| (9 2 2) | [di:(9)^2] | [di:(2)^9] | - | [hos:18.18] | 9.4.4 | - | 18.4.4 | 3.3.3.9 |
| (10 2 2) | [di:(10)^2] | [di:(2)^10] | - | [hos:20.20] | 10.4.4 |
- | 20.4.4 |  3.3.3.10 |
| (11 2 2) | [di:(11)^2] | [di:(2)^11] | - | [hos:22.22] | 11.4.4 | - | 22.4.4 | 3.3.3.11 |
| (12 2 2) | [di:(12)^2] | [di:(2)^12] | - | [hos:24.24] | 12.4.4 |
- | 24.4.4 |  3.3.3.12 |
| (5/2 2 2) | [di:(5/2)^2] | [di:(2)^5/2] | - | [hos:5.5] |  5/2.4.4 |
- | 5.4.4 |  3.3.3.5/2 |
| (5/3 2 2) | - | - | - | - | - | - | - |  3.3.3.5/3 |
| (7/2 2 2) | [di:(7/2)^2] | [di:(2)^7/2] | - | [hos:7.7] | 7/2.4.4 | - | 7.4.4 | 3.3.3.7/2 |
| (7/3 2 2) | [di:(7/3)^2] | [di:(2)^7/3] | - | [hos:14/3.14/3] | 7/3.4.4 | - | 14/3.4.4 | 3.3.3.7/3 |
| (8/3 2 2) | [di:(8/3)^2] | [di:(2)^8/3] | - | [hos:16/3.16/3] | 8/3.4.4 | - | 16/3.4.4 | 3.3.3.8/3 |
| (9/2 2 2) | [di:(9/2)^2] | [di:(2)^9/2] | - | [hos:9.9] | 9/2.4.4 | - | 9.4.4 | 3.3.3.9/2 |
| (9/4 2 2) | [di:(9/4)^2] | [di:(2)^9/4] | - | [hos:9/2.9/2] | 9/4.4.4 | - | 9/2.4.4 | 3.3.3.9/4 |
| (10/3 2 2) | [di:(10/3)^2] | [di:(2)^10/3] | - | [hos:20/3.20/3] | 10/3.4.4 | - | 20/3.4.4 | 3.3.3.10/3 |
| (11/2 2 2) | [di:(11/2)^2] | [di:(2)^11/2] | - | [hos:11.11] | 11/2.4.4 | - | 11.4.4 | 3.3.3.11/2 |
| (11/3 2 2) | [di:(11/3)^2] | [di:(2)^11/3] | - | [hos:22/3.22/3] | 11/3.4.4 | - | 22/3.4.4 | 3.3.3.11/3 |
| (11/4 2 2) | [di:(11/4)^2] | [di:(2)^11/4] | - | [hos:11/2.11/2] | 11/4.4.4 | - | 11/2.4.4 | 3.3.3.11/4 |
| (11/5 2 2) | [di:(11/5)^2] | [di:(2)^11/5] | - | [hos:22/5.22/5] | 11/5.4.4 | - | 22/5.4.4 | 3.3.3.11/5 |
| (12/5 2 2) | [di:(12/5)^2] | [di:(2)^12/5] | - | [hos:24/5.24/5] | 12/5.4.4 | - | 24/5.4.4 | 3.3.3.12/5 |
Tetrahedral[edit] | ||||||||
| (3 3 2) | (3)^3 | - |  (3.3)^2 |
 3.6.6 |
- |  3.4.3.4 |
 6.6.43 | |
| (3 3/2 2) | (3)^3/2 | (3/2)^3 | (3.3/2)^2 | 3/2.6.6 | 3.6/2.6/2 |  3/2.4.3.4 |
6.6/2.4 | 3.3ĕẼMWὂᾟοΥἚ
Octahedral[edit] |
| (4 3 2) |  (4)^3 |
(3)^4 |
 (4.3)^2 |
 3.8.8 |
 4.6.6 |
 3.4.4.4 |
 8.6.4 |
 3.3.3.3.4 |
| (3 3 3/2) | (3.3/2)^3 | - | - |  3.6.3/2.6 |
- | 3.6/2.3.6/2 |
6.6.6/2 |
3.3/2.3.3.3.3 |
| (4 4 3/2) | (4.3/2)^4 | - | (4.4)^3/2 | 4.8.3/2.8 | - | 4.6/2.4.6/2 | 8.8.6/2 | 3.3/2.3.4.3.4 |
| (4 4/3 3) | (4.3)^4/3 | (4/3.3)^4 | (4.4/3)^3 | 4/3.8.3.8 | 4.8/3.3.8/3 | 4/3.6.4.6 | 8.8/3.6 | 3.3.3.4/3.3.4 |
| (4 3/2 2) | (4)^3/2 | (3/2)^4 | (4.3/2)^2 | 3/2.8.8 | 4.6/2.6/2 |  3/2.4.4.4 |
8.6/2.4 | 3.3.3/2.3.4 |
| (4/3 3 2) | (4/3)^3 | (3)^4/3 | (4/3.3)^2 | 3.8/3.8/3 | 4/3.6.6 | 3.4.4/3.4 | 8/3.6.4 | 3.3.3.3.4/3 |
| (4/3 3/2 2) | (4/3)^3/2 | (3/2)^4/3 | (4/3.3/2)^2 | 3/2.8/3.8/3 | 4/3.6/2.6/2 | 3/2.4.4/3.4 | 8/3.6/2.4 | 3.3.3/2.3.4/3 |
Icosahedral[edit] | ||||||||
| (5 3 2) |  (5)^3 |
 (3)^5 |
 (5.3)^2 |
 3.10.10 |
 5.6.6 |
 3.4.5.4 |
 10.6.4 |
 3.3.3.3.5 |
| (5/2 3 3) | (5/2.3)^3 | (3.3)^5/2 | - | 3.10/2.3.10/2 | 5/2.6.3.6 | - | 10/2.6.6 | 3.3.3.3.3.5/2 |
| (5 5 3/2) | (5.3/2)^5 | - | (5.5)^3/2 | 5.10.3/2.10 | - | 5.6/2.5.6/2 | 10.10.6/2 | 3.3/2.3.5.3.5 |
| (5 5/2 2) | (5)^5/2 | (5/2)^5 | (5.5/2)^2 | 5/2.10.10 | 5.10/2.10/2 | 5/2.4.5.4 | 10.10/2.4 | 3.3.5/2.3.5 |
| (5 5/3 3) | (5.3)^5/3 | (5/3.3)^5 | (5.5/3)^3 | 5/3.10.3.10 | 5.10/3.3.10/3 | 5/3.6.5.6 | 10.10/3.6 | 3.3.3.5/3.3.5 |
| (5 3 3/2) | (5.3/2)^3 | (3.3/2)^5 | (5.3)^3/2 | 3.10.3/2.10 | 5.6.3/2.6 | 3.6/2.5.6/2 | 10.6.6/2 | 3.3/2.3.3.3.5 |
| (5 3 3/2) | (5.3/2)^3 | (3.3/2)^5 | (5.3)^3/2 | 3.10.3/2.10 | 5.6.3/2.6 | 3.6/2.5.6/2 | 10.6.6/2 | 3.3/2.3.3.3.5 |
| (5 5 5/4) | (5.5/4)^5 | - | - | 5.10.5/4.10 | - | 5.10/4.5.10/4 | 10.10.10/4 | 3.5/4.3.5.3.5 |
| (5/2 3 2) | (5/2)^3 | (3)^5/2 | (5/2.3)^2 | 3.10/2.10/2 | 5/2.6.6 | 3.4.5/2.4 | 10/2.6.4 | 3.3.3.3.5/2 |
| (5 5/3 2) | (5)^5/3 | (5/3)^5 | (5.5/3)^2 | 5/3.10.10 | 5.10/3.10/3 | 5/3.4.5.4 | 10.10/3.4 | 3.3.5/3.3.5 |
| (5/2 5/3 3) | (5/2.3)^5/3 | (5/3.3)^5/2 | (5/2.5/3)^3 | 5/3.10/2.3.10/2 | 5/2.10/3.3.10/3 | 5/3.6.5/2.6 | 10/2.10/3.6 | 3.3.3.5/3.3.5/2 |
| (5 5/4 3) | (5.3)^5/4 | (5/4.3)^5 | (5.5/4)^3 | 5/4.10.3.10 | 5.10/4.3.10/4 | 5/4.6.5.6 | 10.10/4.6 | 3.3.3.5/4.3.5 |
| (5/3 3 2) | (5/3)^3 | (3)^5/3 | (5/3.3)^2 | 3.10/3.10/3 | 5/3.6.6 | 3.4.5/3.4 | 10/3.6.4 | 3.3.3.3.5/3 |
| (5/2 5/3 5/3) | (5/2.5/3)^5/3 | (5/3.5/3)^5/2 | - | 5/3.10/2.5/3.10/2 | 5/2.10/3.5/3.10/3 | - | 10/2.10/3.10/3 | 3.5/3.3.5/3.3.5/2 |
| (5/3 3 3) | (5/3.3)^3 | (3.3)^5/3 | - | 3.10/3.3.10/3 | 5/3.6.3.6 | - | 10/3.6.6 | 3.3.3.3.3.5/3 |
| (5/2 3/2 3/2) | (5/2.3/2)^3/2 | (3/2.3/2)^5/2 | - | 3/2.10/2.3/2.10/2 | 5/2.6/2.3/2.6/2 | - | 10/2.6/2.6/2 | 3.3/2.3.3/2.3.5/2 |
| (5/3 3/2 2) | (5/3)^3/2 | (3/2)^5/3 | (5/3.3/2)^2 | 3/2.10/3.10/3 | 5/3.6/2.6/2 | 3/2.4.5/3.4 | 10/3.6/2.4 | 3.3.3/2.3.5/3 |
| (5 5 5/3) | (5.5/3)^5 | - | - | 5.10.5/3.10 | - | 5.10/3.5.10/3 | 10.10.10/3 | 3.5/3.3.5.3.5 |
| (5 5/3 3) | (5.3)^5/3 | (5/3.3)^5 | (5.5/3)^3 | 5/3.10.3.10 | 5.10/3.3.10/3 | 5/3.6.5.6 | 10.10/3.6 | 3.3.3.5/3.3.5 |
Euclidean[edit] | ||||||||
| (4 4 2) | (4)^4 | - | (4.4)^2 | 4.8.8 | - | 4.4.4.4 | 8.8.4 | 3.3.4.3.4 |
| (3 3 3) | (3.3)^3 | - | - | 3.6.3.6 | - | - | 6.6.6 | 3.3.3.3.3.3 |
| (6 3 2) | (6)^3 | (3)^6 | (6.3)^2 | 3.12.12 | 6.6.6 | 3.4.6.4 | 12.6.4 | 3.3.3.3.6 |
Hyperbolic[edit] | ||||||||
| (7 3 2) | (7)^3 | (3)^7 | (7.3)^2 | 3.14.14 | 7.6.6 | 3.4.7.4 | 14.6.4 | 3.3.3.3.7 |
| (5 4 2) | (5)^4 | (4)^5 | (5.4)^2 | 4.10.10 | 5.8.8 | 4.4.5.4 | 10.8.4 | 3.3.4.3.5 |
| (6 4 2) | (6)^4 | (4)^6 | (6.4)^2 | 4.12.12 | 6.8.8 | 4.4.6.4 | 12.8.4 | 3.3.4.3.6 |
| (5 5 2) | (5)^5 | - | (5.5)^2 | 5.10.10 | - | 5.4.5.4 | 10.10.4 | 3.3.5.3.5 |
| (8 3 2) | (8)^3 | (3)^8 | (8.3)^2 | 3.16.16 | 8.6.6 | 3.4.8.4 | 16.6.4 | 3.3.3.3.8 |
| (3 3/2 2) | (3)^3/2 | (3/2)^3 | (3.3/2)^2 | 3/2.6.6 | 3.3.3 | 3/2.4.3.4 | 6.3.4 | 3.3.3/2.3.3 |
| (5 5/2 2) | (5)^5/2 | (5/2)^5 | (5.5/2)^2 | 5/2.10.10 | 5.5.5 | 5/2.4.5.4 | 10.5.4 | 3.3.5/2.3.5 |
| (7 7/2 2) | (7)^7/2 | (7/2)^7 | (7.7/2)^2 | 7/2.14.14 | 7.7.7 | 7/2.4.7.4 | 14.7.4 | 3.3.7/2.3.7 |
| (9 9/2 2) | (9)^9/2 | (9/2)^9 | (9.9/2)^2 | 9/2.18.18 | 9.9.9 | 9/2.4.9.4 | 18.9.4 | 3.3.9/2.3.9 |