Johnson Polyhedra

Polyhedra with Regular Polygon Faces

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
First-time Visitors: Please visit Site Map and Disclaimer. Use "Back" to return here.

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.

Definitions and Symbols

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.

Names for Parts of Solids

NameSymbolDefinition
PrismPnRegular n-sided prism: two n-gons joined by a belt of n squares
AntiprismSnRegular n-sided antiprism: two n-gons joined by a belt of 2n triangles
PyramidYn 3-, 4- or 5-gon base with n triangles meeting at a point
  • Y3 is a tetrahedron
  • Y4 is half of an octahedron
  • Y5 is a pentagonal cap from an icosahedron
CupolaQn n-gon on top, 2n-gon base, alternating squares and triangles on sides.
  • Q3 is half of a cuboctahedron
  • Q4 is an octagonal cap from a rhombicuboctahedron
  • Q5 is a decagonal cap from a rhombicosadodecahedron
RotundaRPiece of an icosidodecahedron consisting of triangles and pentagons.
LuneLThree-face unit consisting of a square and opposing triangles
CoronaUn-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

Symbols for Platonic and Archimedean Solids

NameSymbol
DodecahedronD5
IcosahedronI5
RhombicosidodecahedronE5
Truncated solidTn
--TetrahedronT3
--CubeT4
--DodecahedronT5

Terms for Operations

TermDefinitionSymbol
AugmentedCupola or pyramid built onto the face of a solidQn, Yn, etc. for added piece
ElongatedSolid lengthened by inserting a prism or adding to one endPn for added prism
GyroelongatedSolid lengthened by inserting antiprism or adding to one endSn for added antiprism
GyratePiece (usually cupola) twisted relative to its normal orientationg in front of symbol for gyrated piece
DiminishedPiece (usually cupola) removed from solidMinus sign in front of symbol for removed piece
Para-Operation performed on two faces or pieces on opposite sides of solidp in front of symbol for changed piece
MetaOperation performed on two faces or pieces not on opposite sides of solidm 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).

Illustrations of the Johnson Polyhedra

Elementary Pieces Derived from Other Polyhedra

Elongated Pyramids

Elongated Biyramids

Elongated Cupolas and Rotundas

Bicupolas and Birotundas

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.

Elongated Bicupolas

Elongated Rotunda Compounds

Gyroelongated Rotunda Compounds

Augmented Triangular and Pentagonal Prisms

Augmented Hexagonal Prisms

Augmented Dodecahedra

Diminished Icosahedra

Augmented Truncated Tetrahedron and Cube

Augmented Truncated Dodecahedron

Gyrate Rhombicosidodecahedron

Diminished Rhombicosidodecahedron

Bi- and Tri-Diminished Rhombicosidodecahedron

Complex Elementary Forms



Return to Symmetry Index
Return to Professor Dutch's Home page

Created 17 Oct. 1997, Last Update 1 March 1999

Not an official UW Green Bay site