Johnson solid
From Academic Kids

Elongated_square_gyrobicupola.png
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 squarebased pyramid with equilateral sides (J_{1}); 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 (J_{2}) is an example that actually has a degree5 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 (J_{37}) is unique in being locally vertexregular: 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:
 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 orthocupolarotunda
 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 rotundae), together with the Platonic and Archimedean solids, prisms, and antiprisms.
 Bi means that two copies of the solid in question are joined basetobase. 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.
External links
 Paper Models of Polyhedra (http://www.korthalsaltes.com/) Many links
 Johnson Solids (http://www.georgehart.com/virtualpolyhedra/johnsoninfo.html) by George W. Hart.
 Images of all 92 solids, categorized, on one page (http://www.uwgb.edu/dutchs/symmetry/johnsonp.htm)it:Solido di Johnson