Johnson Polyhedra

Polyhedra with Regular Polygon Faces

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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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

Name	Symbol	Definition
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 Y3 is a tetrahedron Y4 is half of an octahedron Y5 is a pentagonal cap from an icosahedron
Cupola	Qn	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
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

Symbols for Platonic and Archimedean Solids

Name	Symbol
Dodecahedron	D5
Icosahedron	I5
Rhombicosidodecahedron	E5
Truncated solid	Tn
--Tetrahedron	T3
--Cube	T4
--Dodecahedron	T5

Terms for Operations

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).

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

A diminished icosahedron is a gyroelongated pentagonal pyramid
A parabidiminished icosahedron is a pentagonal antiprism
A tridiminished icosahedron is hard to recognize as a derivative of the icosahedron. It has three pentagon faces with an apical triangle at one end and a cap of four triangles at the other.

Augmented Truncated Tetrahedron and Cube

Augmented Truncated Dodecahedron

Gyrate Rhombicosidodecahedron

Diminished Rhombicosidodecahedron

Bi- and Tri-Diminished Rhombicosidodecahedron

Complex Elementary Forms

sS4: snub square antiprism
sS2: snub disphenoid. Has fourfold inversion symmetry around the long axis.

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Created 17 Oct. 1997, Last Update 1 March 1999

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