# Johnson solid

In geometry, a **Johnson solid** is a strictly convex polyhedron, which is not uniform (i.e., not a Platonic solid, Archimedean solid, prism, or antiprism), and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (*J*_{1}); it has 1 square face and 4 triangular faces.

As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (*J*_{2}) is an example that actually has a degree-5 vertex.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

Of the Johnson solids, the elongated square gyrobicupola (*J*_{37}), also called the pseudorhombicuboctahedron,[1] is unique in being locally vertex-uniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 triangle. However, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid.

## Names

The naming of Johnson solids follows a flexible and precise descriptive formula, such that many solids can be named in different ways without compromising their accuracy as a description. Most Johnson solids can be constructed from the first few (pyramids, cupolae, and rotunda), together with the Platonic and Archimedean solids, prisms, and antiprisms; the centre of a particular solid's name will reflect these ingredients. From there, a series of prefixes are attached to the word to indicate additions, rotations and transformations:

**Bi-**indicates that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, the solids can be joined so that either like faces (**ortho-**) or unlike faces (**gyro-**) meet. Using this nomenclature, an octahedron can be described as a*square bipyramid*, a cuboctahedron as a*triangular gyrobicupola*, and an icosidodecahedron as a*pentagonal gyrobirotunda*.**Elongated**indicates a prism is joined to the base of the solid in question, or between the bases in the case of Bi- solids. A rhombicuboctahedron can thus be described as an*elongated square orthobicupola*.**Gyroelongated**indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids. An icosahedron can thus be described as a*gyroelongated pentagonal bipyramid*.**Augmented**indicates another polyhedron, namely a pyramid or cupola, is joined to one or more faces of the solid in question.**Diminished**indicates a pyramid or cupola is removed from one or more faces of the solid in question.**Gyrate**indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, as in the difference between ortho- and gyrobicupolae.

The last three operations – *augmentation*, *diminution*, and *gyration* – can be performed multiple times for certain large solids. *Bi-* & *Tri-* indicate a double and triple operation respectively. For example, a *bigyrate* solid has two rotated cupolae, and a *tridiminished* solid has three removed pyramids or cupolae.

In certain large solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. *Para-* indicates the former, that the solid in question has altered parallel faces, and *Meta-* the latter, altered oblique faces. For example, a *parabiaugmented* solid has had two parallel faces augmented, and a *metabigyrate* solid has had 2 oblique faces gyrated.

The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson[2] with the following nomenclature:

- A
*lune*is a complex of two triangles attached to opposite sides of a square. *Spheno*- indicates a wedgelike complex formed by two adjacent lunes.*Dispheno-*indicates two such complexes.*Hebespheno*- indicates a blunt complex of two lunes separated by a third lune.*Corona*is a crownlike complex of eight triangles.*Megacorona*is a larger crownlike complex of 12 triangles.- The suffix -
*cingulum*indicates a belt of 12 triangles.

## Enumeration

### Pyramids, cupolae and rotundae

The first 6 Johnson solids are pyramids, cupolae, or rotundae with at most 5 lateral faces. Pyramids and cupolae with 6 or more lateral faces are coplanar and are hence not Johnson solids.

#### Pyramids

The first two Johnson solids, J1 and J2, are pyramids. The *triangular pyramid* is the regular tetrahedron, so it is not a Johnson solid.

Regular | J1 | J2 |
---|---|---|

Triangular pyramid (Tetrahedron) |
Square pyramid | Pentagonal pyramid |

#### Cupolae and rotunda

The next four Johnson solids are three cupolae and one rotunda. They represent sections of uniform polyhedra.

Cupola | Rotunda | |||
---|---|---|---|---|

Uniform | J3 | J4 | J5 | J6 |

Fastigium (Triangular prism) |
Triangular cupola | Square cupola | Pentagonal cupola | Pentagonal rotunda |

Related uniform polyhedra | ||||

Cuboctahedron | Rhombicuboctahedron | Rhombicosidodecahedron | Icosidodecahedron | |

### Modified pyramids

Johnson solids 7 to 17 are derived from pyramids.

#### Elongated and gyroelongated pyramids

In the gyroelongated triangular pyramid, three pairs of adjacent triangles are coplanar and form non-square rhombi, so it is not a Johnson solid.

Elongated pyramids | Gyroelongated pyramids | ||||
---|---|---|---|---|---|

J7 | J8 | J9 | Coplanar | J10 | J11 |

Elongated triangular pyramid | Elongated square pyramid | Elongated pentagonal pyramid | Gyroelongated triangular pyramid (diminished trigonal trapezohedron) |
Gyroelongated square pyramid | Gyroelongated pentagonal pyramid |

Augmented from polyhedra | |||||

tetrahedron triangular prism |
square pyramid cube |
pentagonal pyramid pentagonal prism |
tetrahedron octahedron |
square pyramid square antiprism |
pentagonal pyramid pentagonal antiprism |

#### Bipyramids

The *square bipyramid* is the regular octahedron, while the *gyroelongated pentagonal bipyramid* is the regular icosahedron, so they are not Johnson solids. In the gyroelongated triangular bipyramid, six pairs of adjacent triangles are coplanar and form non-square rhombi, so it is also not a Johnson solid.

Bipyramids | Elongated bipyramids | Gyroelongated bipyramids | ||||||
---|---|---|---|---|---|---|---|---|

J12 | Regular | J13 | J14 | J15 | J16 | Coplanar | J17 | Regular |

Triangular bipyramid | Square bipyramid (octahedron) |
Pentagonal bipyramid | Elongated triangular bipyramid | Elongated square bipyramid | Elongated pentagonal bipyramid | Gyroelongated triangular bipyramid (trigonal trapezohedron) |
Gyroelongated square bipyramid | Gyroelongated pentagonal bipyramid (icosahedron) |

Augmented from polyhedra | ||||||||

tetrahedron | square pyramid | pentagonal pyramid | tetrahedron triangular prism |
square pyramid cube |
pentagonal pyramid pentagonal prism |
tetrahedron Octahedron |
square pyramid square antiprism |
pentagonal pyramid pentagonal antiprism |

### Modified cupolae and rotundae

Johnson solids 18 to 48 are derived from cupolae and rotundae.

#### Elongated and gyroelongated cupolae and rotundae

Elongated cupola | Elongated rotunda | Gyroelongated cupola | Gyroelongated rotunda | ||||||
---|---|---|---|---|---|---|---|---|---|

Coplanar | J18 | J19 | J20 | J21 | Concave | J22 | J23 | J24 | J25 |

Elongated fastigium | Elongated triangular cupola | Elongated square cupola | Elongated pentagonal cupola | Elongated pentagonal rotunda | Gyroelongated fastigium | Gyroelongated triangular cupola | Gyroelongated square cupola | Gyroelongated pentagonal cupola | Gyroelongated pentagonal rotunda |

Augmented from polyhedra | |||||||||

Square prism Triangular prism |
Hexagonal prism Triangular cupola |
Octagonal prism Square cupola |
Decagonal prism Pentagonal cupola |
Decagonal prism Pentagonal rotunda |
square antiprism Triangular prism |
Hexagonal antiprism Triangular cupola |
Octagonal antiprism Square cupola |
Decagonal antiprism Pentagonal cupola |
Decagonal antiprism Pentagonal rotunda |

#### Bicupolae

The triangular gyrobicupola is an Archimedean solid (in this case the cuboctahedron), so it is not a Johnson solid.

Orthobicupola | Gyrobicupola | ||||||
---|---|---|---|---|---|---|---|

Coplanar | J27 | J28 | J30 | J26 | Semiregular | J29 | J31 |

Orthobifastigium | Triangular orthobicupola | Square orthobicupola | Pentagonal orthobicupola | Gyrobifastigium | Triangular gyrobicupola (cuboctahedron) |
Square gyrobicupola | Pentagonal gyrobicupola |

Augmented from polyhedron | |||||||

Triangular prism | Triangular cupola | Square cupola | Pentagonal cupola | Triangular prism | Triangular cupola | Square cupola | Pentagonal cupola |

#### Cupola-rotundae and birotunda

The pentagonal gyrobirotunda is an Archimedean solid (in this case the icosidodecahedron), so it is not a Johnson solid.

Cupola-rotunda | Birotunda | ||
---|---|---|---|

J32 | J33 | J34 | Semiregular |

Pentagonal orthocupolarotunda | Pentagonal gyrocupolarotunda | Pentagonal orthobirotunda | Pentagonal gyrobirotunda (icosidodecahedron) |

Augumented from polyhedra | |||

Pentagonal cupola Pentagonal rotunda |
Pentagonal rotunda | ||

#### Elongated bicupolae

The elongated square orthobicupola is an Archimedean solid (in this case the rhombicuboctahedron), so it is not a Johnson solid.

Elongated orthobicupola | Elongated gyrobicupola | ||||||
---|---|---|---|---|---|---|---|

Coplanar | J35 | Semiregular | J38 | Coplanar | J36 | J37 | J39 |

Elongated orthobifastigium | Elongated triangular orthobicupola | Elongated square orthobicupola (rhombicuboctahedron) |
Elongated pentagonal orthobicupola | Elongated gyrobifastigium | Elongated triangular gyrobicupola | Elongated square gyrobicupola | Elongated pentagonal gyrobicupola |

Augmented from polyhedra | |||||||

Square prism Triangular prism |
Hexagonal prism Triangular cupola |
Octagonal prism Square cupola |
Decagonal prism Pentagonal cupola |
Square prism Triangular prism |
Hexagonal prism Triangular cupola |
Octagonal prism Square cupola |
Decagonal prism Pentagonal cupola |

#### Elongated cupola-rotundae and birotundae

Elongated cupola-rotunda | Elongated birotunda | ||
---|---|---|---|

J40 | J41 | J42 | J43 |

Elongated pentagonal orthocupolarotunda | Elongated pentagonal gyrocupolarotunda | Elongated pentagonal orthobirotunda | Elongated pentagonal gyrobirotunda |

Augmented from polyhedra | |||

Decagonal prism Pentagonal cupola Pentagonal rotunda |
Decagonal prism Pentagonal rotunda | ||

#### Gyroelongated bicupolae, cupola-rotunda, and birotunda

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola | Gyroelongated cupola-rotunda | Gyroelongated birotunda | |||
---|---|---|---|---|---|

Concave | J44 | J45 | J46 | J47 | J48 |

Gyroelongated bifastigium | Gyroelongated triangular bicupola | Gyroelongated square bicupola | Gyroelongated pentagonal bicupola | Gyroelongated pentagonal cupolarotunda | Gyroelongated pentagonal birotunda |

Augmented from polyhedra | |||||

Triangular prism Square antiprism |
Triangular cupola Hexagonal antiprism |
Square cupola Octagonal antiprism |
Pentagonal cupola Decagonal antiprism |
Pentagonal cupola Pentagonal rotunda Decagonal antiprism |
Pentagonal rotunda Decagonal antiprism |

### Augmented prisms

Johnson solids 49 to 57 are built by augmenting the sides of prisms with square pyramids.

Augmented triangular prisms | Augmented pentagonal prisms | Augmented hexagonal prisms | ||||||
---|---|---|---|---|---|---|---|---|

J49 | J50 | J51 | J52 | J53 | J54 | J55 | J56 | J57 |

Augmented triangular prism | Biaugmented triangular prism | Triaugmented triangular prism | Augmented pentagonal prism | Biaugmented pentagonal prism | Augmented hexagonal prism | Parabiaugmented hexagonal prism | Metabiaugmented hexagonal prism | Triaugmented hexagonal prism |

Augumented from polyhedra | ||||||||

Triangular prism Square pyramid |
Pentagonal prism Square pyramid |
Hexagonal prism Square pyramid | ||||||

### Modified Platonic solids

Johnson solids 58 to 64 are built by augmenting or diminishing Platonic solids.

#### Augmented dodecahedra

J58 | J59 | J60 | J61 |
---|---|---|---|

Augmented dodecahedron | Parabiaugmented dodecahedron | Metabiaugmented dodecahedron | Triaugmented dodecahedron |

Augumented from polyhedra | |||

Dodecahedron and pentagonal pyramid | |||

#### Diminished and augmented icosahedra

Diminished icosahedra | Augmented tridiminished icosahedron | |||
---|---|---|---|---|

J11 (Repeated) |
Uniform | J62 | J63 | J64 |

Diminished icosahedron (Gyroelongated pentagonal pyramid) |
Parabidiminished icosahedron (Pentagonal antiprism) |
Metabidiminished icosahedron | Tridiminished icosahedron | Augmented tridiminished icosahedron |

### Modified Archimedean solids

Johnson solids 65 to 83 are built by augmenting, diminishing or gyrating Archimedean solids.

#### Augmented Archimedean solids

Augmented truncated tetrahedron | Augmented truncated cubes | Augmented truncated dodecahedra | ||||
---|---|---|---|---|---|---|

J65 | J66 | J67 | J68 | J69 | J70 | J71 |

Augmented truncated tetrahedron | Augmented truncated cube | Biaugmented truncated cube | Augmented truncated dodecahedron | Parabiaugmented truncated dodecahedron | Metabiaugmented truncated dodecahedron | Triaugmented truncated dodecahedron |

Augumented from polyhedra | ||||||

truncated tetrahedron triangular cupola |
truncated cube square cupola |
truncated dodecahedron pentagonal cupola | ||||

#### Gyrate and diminished rhombicosidodecahedra

Gyrate rhombicosidodecahedra | |||
---|---|---|---|

J72 | J73 | J74 | J75 |

Gyrate rhombicosidodecahedron | Parabigyrate rhombicosidodecahedron | Metabigyrate rhombicosidodecahedron | Trigyrate rhombicosidodecahedron |

Diminished rhombicosidodecahedra | |||

J76 | J80 | J81 | J83 |

Diminished rhombicosidodecahedron | Parabidiminished rhombicosidodecahedron | Metabidiminished rhombicosidodecahedron | Tridiminished rhombicosidodecahedron |

Gyrate diminished rhombicosidodecahedra | |||

J77 | J78 | J79 | J82 |

Paragyrate diminished rhombicosidodecahedron | Metagyrate diminished rhombicosidodecahedron | Bigyrate diminished rhombicosidodecahedron | Gyrate bidiminished rhombicosidodecahedron |

### Elementary solids

Johnson solids 84 to 92 are not derived from "cut-and-paste" manipulations of uniform solids.

#### Snub antiprisms

The snub antiprisms can be constructed as an alternation of a truncated antiprism. The gyrobianticupolae are another construction for the snub antiprisms. Only snub antiprisms with at most 4 sides can be constructed from regular polygons. The snub triangular antiprism is the regular icosahedron, so it is not a Johnson solid.

J84 | Regular | J85 |
---|---|---|

Snub disphenoid ss{2,4} |
Icosahedron ss{2,6} |
Snub square antiprism ss{2,8} |

Digonal gyrobianticupola | Triangular gyrobianticupola | Square gyrobianticupola |

#### Others

J86 | J87 | J88 | |
---|---|---|---|

Sphenocorona | Augmented sphenocorona | Sphenomegacorona | |

J89 | J90 | J91 | J92 |

Hebesphenomegacorona | Disphenocingulum | Bilunabirotunda | Triangular hebesphenorotunda |

## Classification by types of faces

### Triangle-faced Johnson solids

Five Johnson solids are deltahedra, with all equilateral triangle faces:

### Triangle and square-faced Johnson solids

Twenty four Johnson solids have only triangle or square faces:

### Triangle and pentagonal-faced Johnson solids

Eleven Johnson solids have only triangle and pentagonal faces:

### Triangle, square, and pentagonal-faced Johnson solids

Twenty Johnson solids have only triangle, square and pentagonal faces:

### Triangle, square, and hexagonal-faced Johnson solids

Eight Johnson solids have only triangle, square and hexagonal faces:

### Triangle, square, and octagonal-faced Johnson solids

Five Johnson solids have only triangle, square and octagonal faces:

### Triangle, pentagon, and decagonal-faced Johnson solids

Two Johnson solids have only triangle, pentagon and decagonal faces:

### Triangle, square, pentagon, and hexagonal-faced Johnson solids

Only one Johnson solid has triangle, square, pentagon and hexagonal faces:

### Triangle, square, pentagon, and decagonal-faced Johnson solids

Sixteen Johnson solids have only triangle, square, pentagon and decagonal faces:

## Circumscribable Johnson solids

25 of the Johnson solids have vertices that exist on the surface of a sphere: 1–6,11,19,27,34,37,62,63,72–83. All of them can be seen to be related to a regular or uniform polyhedron by gyration, diminishment, or dissection.[3]

Octahedron | Cuboctahedron | Rhombicuboctahedron | |||
---|---|---|---|---|---|

J1 |
J3 |
J27 |
J4 |
J19 |
J37 |

Icosahedron | Icosidodecahedron | ||||
---|---|---|---|---|---|

J2 |
J11 |
J62 |
J63 |
J6 |
J34 |

Rhombicosidodecahedron | ||||||
---|---|---|---|---|---|---|

J5 |
J72 |
J73 |
J74 |
J75 |
J76 |
J77 |

J78 |
J79 |
J80 |
J81 |
J82 |
J83 |

## Dual Polyhedrons

As of now, the website https://levskaya.github.io/polyhedronisme/ supports Conway operations applied to Johnson solids. For example, see https://levskaya.github.io/polyhedronisme/?recipe=C1000dJ73&palette=%2300ffff%20%23ff7f7f for the dual polyhedron of J73.

## References

- Johnson, Norman W. (1966). "Convex Solids with Regular Faces".
*Canadian Journal of Mathematics*.**18**: 169–200. doi:10.4153/cjm-1966-021-8. ISSN 0008-414X. Zbl 0132.14603. Contains the original enumeration of the 92 solids and the conjecture that there are no others. - Zalgaller, Victor A. (1969).
*Convex Polyhedra with Regular Faces*. Consultants Bureau. Zbl 0177.24802. No ISBN. The first proof that there are only 92 Johnson solids: see also Zalgaller, Victor A. (1967). "Convex Polyhedra with Regular Faces".*Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova*(in Russian).**2**: 1–221. ISSN 0373-2703. Zbl 0165.56302. - Anthony Pugh (1976).
*Polyhedra: A visual approach*. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 3 Further Convex polyhedra

- GWH. "Pseudo Rhombicuboctahedra".
*www.georgehart.com*. Retrieved 17 April 2018. - George Hart (quoting Johnson) (1996). "Johnson Solids".
*Virtual Polyhedra*. Retrieved 5 February 2014. - Klitzing, Dr. Richard. "Johnson solids et al".
*bendwavy.org*. Retrieved 17 April 2018.

## External links

- Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières" [Convex polyhedra with regular faces] (PDF).
*Structural Topology*(6): 83–95. - Paper Models of Polyhedra Many links
- Johnson Solids by George W. Hart.
- Images of all 92 solids, categorized, on one page
- Weisstein, Eric W. "Johnson Solid".
*MathWorld*. - VRML models of Johnson Solids by Jim McNeill
- VRML models of Johnson Solids by Vladimir Bulatov
- CRF polychora discovery project attempts to discover CRF polychora, a generalization of the Johnson solids to 4-dimensional space
- https://levskaya.github.io/polyhedronisme/ a generator of polyhedrons and Conway operations applied to them, including Johnson solids.