# Johnson solid

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A Johnson solid is a convex polyhedron each face of which is a regular polygon which is not vertex-uniform. These polyhedra are what are left once you take away the Platonic solids, Archimedean solids, prisms and antiprisms. There is no requirement that each face must be the same polygon. An example of a Johnson solid that is neither a platonic solid nor an archimedean solid is a square based pyramid; it has one square face and four triangular faces.

There are some requirements, nonetheless. To have vertices, there must be at least three of the faces meeting at a point, and the total of their angles must be less than 360 degrees; i.e the corners of the face must be less than 120 degrees. Regular polygons must have all sides of equal length, and all angles of equal degrees, so parallellopeds or trapezoids may not be used. Just as there are an infinite number of natural numbers, there are an infinite number of regular polygons. Every one of them may be used as the base of a pyramid, but the triangles used to make the pyramid are not regular.

In 1966, [Norman Johnson]? published a list which included all 92 solids, and gave them their names. He did not prove that there were only 92, but he did conjecture that there were no others. Zalgaller? in 1969 proved that Johnson's list was complete.

The names and Johnson numbers for the solids are:

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 orthocupolarontunda
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
74. metabigyrate rhombicosidodecahedron
75. trigyrate rhombicosidodecahedron
76. diminished rhombicosidodecahedron
77. paragyrate diminished rhombicosidodecahedron
78. metagyrate diminished rhombicosidodecahedron
79. bigyrate diminished rhombicosidodecahedron
80. parabidiminished rhombicosidodecahedron
81. metabidiminished rhombicosidodecahedron
82. gyrate bidiminished rhombicosidodecahedron
83. tridiminished rhombicosidodecahedron
84. snub disphenoid
85. snub square antiprism
86. sphenocorona
87. augmented sphenocorona
88. sphenomegacorona
89. hebesphenomegacorona
90. disphenocingulum
91. bilunabirotunda
92. triangular hebesphenorotunda

The names are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few (pyramids, cupolae, and rotunda?), together with the platonic and archimedean solids, prisms, antiprisms.

• Bi- means that two copies of the solid in question are joined base to base. For cupolae and rotundae, they can be joined so that like faces meet (ortho-) or unlike faces meet (gyro-). An octahedron would be a square bipyramid, a cuboctahedron would be a triangular gyrobicupola, and an icosidodecahedron would be a pentagonal gyrobirotunda.
• Elongated means that a prism has been joined to the base of the solid in question or between the bases of the solids in question. A rhombicuboctahedron would be an elongated square orthobicupola.
• Gyroelongated means that an antiprism has been joined to the base of the solid in question or between the bases of the solids in question. An icosahedron would be a gyroelongated pentagonal bipyramid.
• Augmented means that a pyramid has been joined to a face of the solid in question.
• Diminished means that a pyramid or cupola has been removed from the solid in question.
• Gyrate means that a cupola on the solid in question has been rotated so that different edges match up, as in the difference between ortho- and gyrobicupola.

References:

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