Johnson solid

From Academic Kids

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Elongated_square_gyrobicupola.png
The elongated square gyrobicupola (J37), a Johnson solid

In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. There is no requirement that each face must be the same polygon. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1); it has one square face and four triangular faces.

As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J2) is an example that actually has a degree-5 vertex.

Although there is no obvious restriction on any regular polygon's being a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

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

Of the Johnson solids, the elongated square gyrobicupola (J37) is unique in being locally vertex-regular: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle.

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 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
  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 rotundae), together with the Platonic and Archimedean solids, prisms, and 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 (ortho-) or unlike faces (gyro-) meet. In this nomenclature, 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 or cupola 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 gyrobicupolae.

References

  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Eric W. Weisstein. Johnson Solid (http://mathworld.wolfram.com/JohnsonSolid.html) at MathWorld.
  • Template:Book reference The first proof that there are only 92 Johnson solids.

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