An order of convex polyhedras whose faces are regular polygons.

[sweety439]

Reference: https://oeis.org/A180916

Like Johnson solids, such convex polyhedras (including: Platonic solids, uniform prisms, uniform antiprisms, Archimedean solids, and Johnson solids), although there are infinitely many such convex polyhedras, there is a system to sort them:

1. By number of faces.
2. If the number of faces are same, then: Platonic solid --> prism --> antiprism --> Archimedean solid --> non-uniform Johnson solid (also: regular polyhedron --> semiregular polyhedron --> non-uniform Johnson solid (the convex polyhedras whose faces are regular polygons but not regular polyhedrons nor semiregular polyhedrons), and the convex regular polyhedrons are the Platonic solids, and the convex semiregular polyhedrons are the uniform prisms, the uniform antiprisms, and the Archimedean solids, sorted by: prism --> antiprism --> Archimedean solid)
3. If two or more Archimedean solids have the same number of faces, then sorted by the order of Archimedean solids in Wikipedia, if two or more Johnson solids have the same number of faces, then sorted by the indices given by Johnson.

Since the convex polyhedras with given number of faces is already finite, the number of faces is already can be the main order of the convex polyhedras, I do not think it is necessary to use the number of edges or the number of vertices to sort (of course, for a given number of faces, these two orders are the same, since by Euler formula, F+V-E=2)

[sweety439]

Such order will be:

1. Regular tetrahedron (the Platonic solid with 4 faces) (f=4)
2. Uniform triangular prism (the uniform prism with 3-gonal) (f=5)
3. Equilateral square pyramid (the Johnson solid J1) (f=5)
4. Cube (the Platonic solid with 6 faces) (the uniform prism with 4-gonal) (f=6)
5. Pentagonal pyramid (the Johnson solid J2) (f=6)
6. Triangular bipyramid (the Johnson solid J12) (f=6)
7. Uniform pentagonal prism (the uniform prism with 5-gonal) (f=7)
8. Elongated triangular pyramid (the Johnson solid J7) (f=7)
9. Regular octahedron (the Platonic solid with 8 faces) (the uniform antiprism with 3-gonal) (f=8)
10. Hexagonal prism (the uniform prism with 6-gonal) (f=8)
11. Truncated tetrahedron (an Archimedean solid) (f=8)
12. Triangular cupola (the Johnson solid J3) (f=8)
13. Gyrobifastigium (the Johnson solid J26) (f=8)
14. Augmented triangular prism (the Johnson solid J49) (f=8)
15. Tridiminished icosahedron (the Johnson solid J63) (f=8)
16. Heptagonal prism (the uniform prism with 7-gonal) (f=9)
17. Elongated square pyramid (the Johnson solid J8) (f=9)
18. Elongated triangular bipyramid (the Johnson solid J14) (f=9)
19. Octagonal prism (the uniform prism with 8-gonal) (f=10)
20. Square antiprism (the uniform antiprism with 4-gonal) (f=10)
21. Square cupola (the Johnson solid J4) (f=10)
22. Pentagonal bipyramid (the Johnson solid J13) (f=10)
23. Augmented pentagonal prism (the Johnson solid J52) (f=10)
24. Augmented tridiminished icosahedron (the Johnson solid J64) (f=10)
25. Enneagonal prism (the uniform prism with 9-gonal) (f=11)
26. Elongated pentagonal pyramid (the Johnson solid J9) (f=11)
27. Biaugmented triangular prism (the Johnson solid J50) (f=11)
28. Augmented hexagonal prism (the Johnson solid J54) (f=11)
29. Regular dodecahedron (the Platonic solid with 12 faces) (f=12)
30. Decagonal prism (the uniform prism with 10-gonal) (f=12)
31. Pentagonal antiprism (the uniform antiprism with 5-gonal) (f=12)
32. Pentagonal cupola (the Johnson solid J5) (f=12)
33. Elongated square bipyramid (the Johnson solid J15) (f=12)
34. Metabidiminished icosahedron (the Johnson solid J62) (f=12)
35. Snub disphenoid (the Johnson solid J84) (f=12)
36. Hendecagonal prism (the uniform prism with 11-gonal) (f=13)
37. Gyroelongated square pyramid (the Johnson solid J10) (f=13)
38. Biaugmented pentagonal prism (the Johnson solid J53) (f=13)
39. Dodecagonal prism (the uniform prism with 12-gonal) (f=14)
40. Hexagonal antiprism (the uniform antiprism with 6-gonal) (f=14)
41. Cuboctahedron (an Archimedean solid) (f=14)
42. Truncated cube (an Archimedean solid) (f=14)
43. Truncated octahedron (an Archimedean solid) (f=14)
44. Elongated triangular cupola (the Johnson solid J18) (f=14)
45. Triangular orthobicupola (the Johnson solid J27) (f=14)
46. Triaugmented triangular prism (the Johnson solid J51) (f=14)
47. Parabiaugmented hexagonal prism (the Johnson solid J55) (f=14)
48. Metabiaugmented hexagonal prism (the Johnson solid J56) (f=14)
49. Augmented truncated tetrahedron (the Johnson solid J65) (f=14)
50. Sphenocorona (the Johnson solid J86) (f=14)
51. Bilunabirotunda (the Johnson solid J91) (f=14)
...
187. Rhombicosidodecahedron (an Archimedean solid) (f=62)
188. Truncated icosidodecahedron (an Archimedean solid) (f=62)
189. Triaugmented truncated dodecahedron (the Johnson solid J71) (f=62)
190. Gyrate rhombicosidodecahedron (the Johnson solid J72) (f=62)
191. Parabigyrate rhombicosidodecahedron (the Johnson solid J73) (f=62)
192. Metabigyrate rhombicosidodecahedron (the Johnson solid J74) (f=62)
193. Trigyrate rhombicosidodecahedron (the Johnson solid J75) (f=62)
...
239. Snub dodecahedron (an Archimedean solid) (f=92)
...

[sweety439]

Since all such solids (i.e. solids with regular polygon faces) have edges lengths all equal (since they are connect), we can ask ....

* If the edges of these solids have length 1, then the volume of these solids is? Also the surface area of these solids?
* The dihedral angles and solid angles for the vertices of these solids?
* The dual polyhedron of these solids? (There seems to be no name for the dual polyhedron of Johnson solids)
* The Schläfli symbol and Coxeter–Dynkin diagram for these solids?
* The symmetry group of these solids?
* The Schlegel diagram of these solids and their dual polyhedrons, also the number of (undirected) Euler path, Euler cycle, Hamiltonian path, Hamiltonian cycle, of the Schlegel diagram of them?

[sweety439]

There is an order of finite groups: by order, then by index from the small groups library, starting at 1

1. C1 (trivial group) (cyclic group with order 1) (ord=1)
2. C2 (cyclic group with order 2) (ord=2)
3. C3 (cyclic group with order 3) = A3 (Alternating group on 3 letters) (ord=3)
4. C4 (cyclic group with order 4) (ord=4)
5. C2xC2 (Klein 4-group) (ord=4)
6. C5 (cyclic group with order 5) (ord=5)
7. S3 (Symmetric group on 3 letters) (ord=6)
8. C6 (cyclic group with order 6) (ord=6)
9. C7 (cyclic group with order 7) (ord=7)
10. C8 (cyclic group with order 8) (ord=8)
11. C4xC2 (ord=8)
12. D4 (Dihedral group) (ord=8)
13. Q8 (Quaternion group) (ord=8)
14. C2xC2xC2 (ord=8)
15. C9 (cyclic group with order 9) (ord=9)
16. C3xC3 (ord=9)
17. D5 (Dihedral group) (ord=10)
18. C10 (cyclic group with order 10) (ord=10)
19. C11 (cyclic group with order 11) (ord=11)
20. Dic3 (Dicyclic group) (ord=12)
21. C12 (cyclic group with order 12) (ord=12)
22. A4 (Alternating group on 4 letters) (ord=12)
23. D6 (Dihedral group) (ord=12)
24. C6xC2 (ord=12)
25. C13 (cyclic group with order 13) (ord=13)
26. D7 (Dihedral group) (ord=14)
27. C14 (cyclic group with order 14) (ord=14)
28. C15 (cyclic group with order 15) (ord=15)
29. C16 (cyclic group with order 16) (ord=16)
30. C4xC4 (ord=16)
31. The semidirect product of C2xC2 and C4 (ord=16)
32. The semidirect product of C4 and C4 (ord=16)
33. C8xC2 (ord=16)
34. M4(2) (Modular maximal-cyclic group) (ord=16)
35. D8 (Dihedral group) (ord=16)
36. SD16 (Semidihedral group) (ord=16)
37. Q16 (Generalised quaternion group) (ord=16)
38. C4xC4xC2 (ord=16)
39. D4xC2 (ord=16)
40. Q8xC2 (ord=16)
41. Central product of C4 and D4 (ord=16)
42. C2xC2xC2xC2 (ord=16)
...

However, I do not think that the sort in the small groups library is reasonable, e.g. S3 (=D3) is before C6, D4 is after C8, D5 is before C10, and D6 is after C12, ..., I think that the cyclic group should be before all other groups with same order, and Abelian groups should be before non-Abelian groups with same order (Dn can be after the Abelian groups and before all other non-Abelian groups with order 2*n), so I think the more reasonable order is:

1. C1 (ord=1)
2. C2 (ord=2)
3. C3 (ord=3)
4. C4 (ord=4, cyclic)
5. C2xC2 (ord=4, not cyclic)
6. C5 (ord=5, cyclic)
7. C6 (ord=6, Abelian)
8. S3 (ord=6, not Abelian)
9. C7 (ord=7)
10. C8 (ord=8, cyclic, Abelian)
11. C4xC2 (ord=8, not cyclic, Abelian, since C4 is before C2, thus C4xC2 should also before C2xC2xC2)
12. C2xC2xC2 (ord=8, not cyclic, Abelian, since C4 is before C2, thus C4xC2 should also before C2xC2xC2)
13. D4 (ord=8, not cyclic, not Abelian, Dn should before all other non-Abelian groups with order 2*n)
14. Q8 (ord=8, not cyclic, not Abelian, Dn should before all other non-Abelian groups with order 2*n)
15. C9 (ord=9, cyclic)
16. C3xC3 (ord=9, not cyclic)
17. C10 (ord=10, Abelian)
18. D5 (ord=10, not Abelian)
19. C11 (ord=11)
20. C12 (ord=12, cyclic, Abelian)
21. C6xC2 (ord=12, not cyclic, Abelian)
22. D6 (ord=12, not cyclic, not Abelian, Dn should before all other non-Abelian groups with order 2*n)
23. Dic3 (ord=12, not cyclic, not Abelian, Dn should before all other non-Abelian groups with order 2*n)
24. A4 (since it is the only group with order 12 which is not supersolvable, so it should after all other groups with order 12, one of our rules is: cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group, thus cyclic groups must before non-cyclic groups, nilpotent groups must before non-nilpotent groups, etc., thus, e.g. the group A5 is the only group with order 60 which is not solvable, so it should after all other groups with order 60, and its index is 312)
...

[sweety439]

This is the partial sum of A180916, i.e. number of convex polyhedra with <=n faces that are all regular polygons (this sequence is currently not in OEIS), like A063756 is the partial sum of A000001, i.e. number of groups of order <= n.

We can use this sequence to find the index of a given convex polyhedra whose faces are regular polygons, see https://en.wikipedia.org/wiki/Talk:J..._not_required), like https://en.wikipedia.org/wiki/List_o...abelian_groups, we can use A063756 to find the index of a given finite group, identifier when groups are numbered by order, o, then by index, i, from the small groups library (broken link: from wayback machine cached copy), starting at 1. (see https://people.maths.bris.ac.uk/~matyd/GroupNames/ for the ID of finite groups in the small groups library)