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:
- square pyramid
- pentagonal pyramid
- triangular cupola
- square cupola
- pentagonal cupola
- pentagonal rotunda
- elongated triangular pyramid
- elongated square pyramid
- elongated pentagonal pyramid
- gyroelongated square pyramid
- gyroelongated pentagonal pyramid
- triangular dipyramid
- pentagonal dipyramid
- elongated triangular dipyramid
- elongated square dipyramid
- elongated pentagonal dipyramid
- gyroelongated square dipyramid
- elongated triangular cupola
- elongated square cupola
- elongated pentagonal cupola
- elongated pentagonal rotunda
- gyroelongated triangular cupola
- gyroelongated square cupola
- gyroelongated pentagonal cupola
- gyroelongated pentagonal rotunda
- gyrobifastigium
- triangular orthobicupola
- square orthobicupola
- square gyrobicupola
- pentagonal orthobicupola
- pentagonal gyrobicupola
- pentagonal orthocupolarontunda
- pentagonal gyrocupolarotunda
- pentagonal orthobirotunda
- elongated triangular orthobicupola
- elongated triangular gyrobicupola
- elongated square gyrobicupola
- elongated pentagonal orthobicupola
- elongated pentagonal gyrobicupola
- elongated pentagonal orthocupolarotunda
- elongated pentagonal gyrocupolarotunda
- elongated pentagonal orthobirotunda
- elongated pentagonal gyrobirotunda
- gyroelongated triangular bicupola
- gyroelongated square bicupola
- gyroelongated pentagonal bicupola
- gyroelongated pentagonal cupolarotunda
- gyroelongated pentagonal birotunda
- augmented triangular prism
- biaugmented triangular prism
- triaugmented triangular prism
- augmented pentagonal prism
- biaugmented pentagonal prism
- augmented hexagonal prism
- parabiaugmented hexagonal prism
- metabiaugmented hexagonal prism
- triaugmented hexagonal prism
- augmented dodecahedron
- parabiaugmented dodecahedron
- metabiaugmented dodecahedron
- triaugmented dodecahedron
- metabidiminished icosahedron
- tridiminished icosahedron
- augmented tridiminished icosahedron
- augmented truncated tetrahedron
- augmented truncated cube
- biaugmented truncated cube
- augmented truncated dodecahedron
- parabiaugmented truncated dodecahedron
- metabiaugmented truncated dodecahedron
- triaugmented truncated dodecahedron
- gyrate rhombicosidodecahedron
- parabigyrate rhombicosidodecahedron
- metabigyrate rhombicosidodecahedron
- trigyrate rhombicosidodecahedron
- diminished rhombicosidodecahedron
- paragyrate diminished rhombicosidodecahedron
- metagyrate diminished rhombicosidodecahedron
- bigyrate diminished rhombicosidodecahedron
- parabidiminished rhombicosidodecahedron
- metabidiminished rhombicosidodecahedron
- gyrate bidiminished rhombicosidodecahedron
- tridiminished rhombicosidodecahedron
- snub disphenoid
- snub square antiprism
- sphenocorona
- augmented sphenocorona
- sphenomegacorona
- hebesphenomegacorona
- disphenocingulum
- bilunabirotunda
- 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:
http://www.georgehart.com/virtual-polyhedra/johnson-info.html