Norman W. Johnson enumerated all the convex polyhedra with regular polygon faces. In addition to the Platonic and Archimedean solids, there are 92 other polyhedra.
Most of the Johnson polyhedra are derived from the Platonic and Archimedean solids by adding or removing pieces. Johnson developed a nomenclature and notation system that is used here with slight modifications. Solids are considered elementary if they cannot be dissected into simpler solids by a plane containing some set of edges.
|Prism||Pn||Regular n-sided prism: two n-gons joined by a belt of n squares|
|Antiprism||Sn||Regular n-sided antiprism: two n-gons joined by a belt of 2n triangles|
|Pyramid||Yn||3-, 4- or 5-gon base with n triangles meeting at a point
|Cupola||Qn||n-gon on top, 2n-gon base, alternating squares and triangles on sides.
|Rotunda||R||Piece of an icosidodecahedron consisting of triangles and pentagons.|
|Lune||L||Three-face unit consisting of a square and opposing triangles|
|Corona||U||n-gon with triangles on alternate edges.|
|Removed Piece||-(symbol)||A piece removed from a solid is denoted by a minus sign in front of its symbol|
|Augmented||Cupola or pyramid built onto the face of a solid||Qn, Yn, etc. for added piece|
|Elongated||Solid lengthened by inserting a prism or adding to one end||Pn for added prism|
|Gyroelongated||Solid lengthened by inserting antiprism or adding to one end||Sn for added antiprism|
|Gyrate||Piece (usually cupola) twisted relative to its normal orientation||g in front of symbol for gyrated piece|
|Diminished||Piece (usually cupola) removed from solid||Minus sign in front of symbol for removed piece|
|Para-||Operation performed on two faces or pieces on opposite sides of solid||p in front of symbol for changed piece|
|Meta||Operation performed on two faces or pieces not on opposite sides of solid||m in front of symbol for changed piece|
Bi- and tri- denote operations performed on more than one element. Johnson uses a superscript to denote such cases. In this page, when it improves clarity, element symbols are repeated. For example, a biaugmented triangular prism is denoted Y4P3Y4 (a square pyramid built onto two adjacent faces of a triangular prism). Where this notation cannot be used clearly, the symbol is enclosed in parentheses with the number of repetitions afterward. Thus, a triaugmented triangular prism is denoted (Y4)3P3 (a square pyramid built onto all three faces of a triangular prism).
gQ2Q2: gyrobifastigium. Two triangular prisms joined on their square faces. In this orientation, the prism can be considered a 2-gonal cupola, hence the notation Q2, the only case where it is used.
Created 17 Oct. 1997, Last Update 1 March 1999