# Johnson polyhedra

## polyhedra with Regular polygon Faces

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

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

#### 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.

#### 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.

#### Complex Elementary Forms

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

Created 17 Oct. 1997, Last Update 1 March 1999