The Johnson solids are the convex polyhedra having regular faces and equal edge lengths (with the exception of the completely
regular Platonic solids, the "semiregular"
Archimedean solids, and the two infinite families
of prisms and antiprisms).
There are 28 simple (i.e., cannot be dissected into two other regular-faced polyhedra
by a plane) regular-faced polyhedra in addition to the prisms
and antiprisms (Zalgaller 1969), and Johnson (1966)
proposed and Zalgaller (1969) proved that there exist exactly 92 Johnson solids in
all.
They are implemented in the Wolfram Language as PolyhedronData[
"Johnson", n
].
The sketelons of the Johnson solids may be termed Johnson skeleton graphs.
There is a near-Johnson solid which can be constructed by inscribing regular nonagons inside the eight triangular faces of a regular octahedron, then joining the free edges to the 24 triangles and finally the remaining edges of the triangles to six squares, with one square for each octahedral vertex. It turns out that the triangles are not quite equilateral, making the edges that bound the squares a slightly different length from that of the enneagonal edge. However, because the differences in edge lengths are so small, the flexing of an average model allows the solid to be constructed with all edges equal.
A database of solids and polyhedron vertex nets of these solids is maintained on the Sandia National Laboratories
Netlib server (https://netlib.sandia.gov/polyhedra/),
but a few errors exist in several entries. Corrected versions are implemented in
the Wolfram Language via PolyhedronData.
The following list summarizes the names of the Johnson solids and gives their images
and nets.
1. Square pyramid
2. Pentagonal pyramid
3. Triangular cupola
4. Square cupola
5. Pentagonal cupola
6. Pentagonal rotunda
7. Elongated triangular pyramid
8. Elongated square pyramid
9. Elongated pentagonal pyramid
10. Gyroelongated square pyramid
11. Gyroelongated pentagonal pyramid
12. Triangular dipyramid
13. Pentagonal dipyramid
14. Elongated triangular dipyramid
15. Elongated square dipyramid
16. Elongated pentagonal dipyramid
17. Gyroelongated square dipyramid
18. Elongated triangular cupola
19. Elongated square cupola
20. Elongated pentagonal cupola
21. Elongated pentagonal rotunda
22. Gyroelongated triangular cupola
23. Gyroelongated square cupola
24. Gyroelongated pentagonal cupola
25. Gyroelongated pentagonal rotunda
26. Gyrobifastigium
27. Triangular orthobicupola
28. Square orthobicupola
29. Square gyrobicupola
30. Pentagonal orthobicupola
31. Pentagonal gyrobicupola
32. Pentagonal orthocupolarotunda
33. Pentagonal gyrocupolarotunda
34. Pentagonal orthobirotunda
35. Elongated triangular orthobicupola
36. Elongated triangular gyrobicupola
37. Elongated square gyrobicupola
38. Elongated pentagonal orthobicupola
39. Elongated pentagonal gyrobicupola
40. Elongated pentagonal orthocupolarotunda
41. Elongated pentagonal gyrocupolarotunda
42. Elongated pentagonal orthobirotunda
43. Elongated pentagonal gyrobirotunda
44. Gyroelongated triangular bicupola
45. Gyroelongated square bicupola
46. Gyroelongated pentagonal bicupola
47. Gyroelongated pentagonal
cupolarotunda
48. Gyroelongated pentagonal birotunda
49. Augmented triangular prism
50. Biaugmented triangular prism
51. Triaugmented triangular prism
52. Augmented pentagonal prism
53. Biaugmented pentagonal prism
54. Augmented hexagonal prism
55. Parabiaugmented hexagonal prism
56. Metabiaugmented hexagonal prism
57. Triaugmented hexagonal prism
58. Augmented dodecahedron
59. Parabiaugmented dodecahedron
60. Metabiaugmented dodecahedron
61. Triaugmented dodecahedron
62. Metabidiminished icosahedron
63. Tridiminished icosahedron
64. Augmented tridiminished icosahedron
65. Augmented truncated tetrahedron
66. Augmented truncated cube
67. Biaugmented truncated cube
68. Augmented truncated dodecahedron
69. Parabiaugmented truncated
dodecahedron
70. Metabiaugmented truncated
dodecahedron
71. Triaugmented truncated dodecahedron
72. Gyrate rhombicosidodecahedron
73. Parabigyrate rhombicosidodecahedron
