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.
Name  Symbol  Definition 

Prism  Pn  Regular nsided prism: two ngons joined by a belt of n squares 
Antiprism  Sn  Regular nsided antiprism: two ngons joined by a belt of 2n triangles 
Pyramid  Yn  3, 4 or 5gon base with n triangles meeting at a point

Cupola  Qn  ngon on top, 2ngon base, alternating squares and triangles on sides.

Rotunda  R  Piece of an icosidodecahedron consisting of triangles and pentagons. 
Lune  L  Threeface unit consisting of a square and opposing triangles 
Corona  U  ngon 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 
Name  Symbol 

Dodecahedron  D5 
Icosahedron  I5 
Rhombicosidodecahedron  E5 
Truncated solid  Tn 
Tetrahedron  T3 
Cube  T4 
Dodecahedron  T5 
Term  Definition  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 2gonal cupola, hence the notation Q2, the only case where it is used.
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Created 17 Oct. 1997, Last Update 1 March 1999
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