Johnson solid

Short description: 92 non-uniform convex polyhedra, with each face a regular polygon

The elongated square gyrobicupola (J37), a Johnson solid This 24 equilateral triangle example is not a Johnson solid because it is not convex This 24-square example is not a Johnson solid because it is not strictly convex (has 180° dihedral angles)

In geometry, a Johnson solid is a strictly convex polyhedron 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 (J1); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a "Johnson solid".

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 (J2) is an example that 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 which are not uniform (i.e., not a Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 Johnson solids (excluding the 5 Platonic solids, the 13 Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms), 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 (J37), 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.

## Contents

* 1 Names
* 2 Enumeration
* 2.1 Pyramids, cupolae, and rotundas
* 2.1.1 Pyramids
* 2.1.2 Cupolae and rotunda
* 2.2 Modified pyramids
* 2.2.1 Elongated and gyroelongated pyramids
* 2.2.2 Bipyramids
* 2.3 Modified cupolae and rotundas
* 2.3.1 Elongated and gyroelongated cupolae and rotundas
* 2.3.2 Bicupolae
* 2.3.3 Cupola-rotundas and birotundas
* 2.3.4 Elongated bicupolae
* 2.3.5 Elongated cupola-rotundas and birotundas
* 2.3.6 Gyroelongated bicupolae, cupola-rotundas, and birotundas
* 2.4 Augmented prisms
* 2.5 Modified Platonic solids
* 2.5.1 Augmented dodecahedra
* 2.5.2 Diminished and augmented diminished icosahedra
* 2.6 Modified Archimedean solids
* 2.6.1 Augmented Archimedean solids
* 2.6.2 Gyrate and diminished rhombicosidodecahedra
* 2.6.3 Other gyrate and diminished archimedean solids
* 2.7 Elementary solids
* 2.7.1 Snub antiprisms
* 2.7.2 Others
* 3 Classification by types of faces
* 3.1 Triangle-faced Johnson solids
* 3.2 Triangle and square-faced Johnson solids
* 3.3 Triangle and pentagon-faced Johnson solids
* 3.4 Triangle, square, and pentagon-faced Johnson solids
* 3.5 Triangle, square, and hexagon-faced Johnson solids
* 3.6 Triangle, square, and octagon-faced Johnson solids
* 3.7 Triangle, pentagon, and decagon-faced Johnson solids
* 3.8 Triangle, square, pentagon, and hexagon-faced Johnson solids
* 3.9 Triangle, square, pentagon, and decagon-faced Johnson solids
* 4 Circumscribable Johnson solids
* 5 See also
* 6 References
* 7 External links

## 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 rotundas), 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 rotundas, 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[4<>], a cuboctahedron as a triangular gyrobicupola[3cc*], and an icosidodecahedron as a pentagonal gyrobirotunda[5rr*].
* 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[z] 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

Further information: List of Johnson solids

### Pyramids, cupolae, and rotundas

The first 6 Johnson solids are pyramids, cupolae, or rotundas 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. They represent sections of regular polyhedra.

Regular 3> T  | J1 4> | J2 5>

|
|

Triangular pyramid
(Tetrahedron)  | Square pyramid | Pentagonal pyramid
|  |
|  |
Related regular polyhedra
Tetrahedron | Octahedron | Icosahedron
|  |

#### Cupolae and rotunda

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

Cupola | Rotunda

|

Uniform  | J3 3c aC-  | J4 4c  | J5 5c  | J6 5r aD-
Fastigium
(Digonal cupola)
(Triangular prism)  | Triangular cupola | Square cupola | Pentagonal cupola | Pentagonal rotunda
|  |  |  |
|  |  |  |
Related uniform polyhedra
Rhombohedron | 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 3=> | J8 4=> | J9 5=> | Coplanar  | J10 4z> | J11 5z> I-
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 3<> | Regular  | J13 5<> | J14 3<=> | J15 4<=> | J16 5<=> | Coplanar  | J17 4<z> | 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 rotundas

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

#### Elongated and gyroelongated cupolae and rotundas

Elongated cupola | Elongated rotunda  | Gyroelongated cupola | Gyroelongated rotunda

|
|
|

Coplanar  | J18 3c=  | J19 4c= eC-  | J20 5c=  | J21 5r=  | Concave  | J22 3cz  | J23 4cz  | J24 5cz  | J25 5rz
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 3cc  | J28 4cc  | J30 5cc  | J26 2cc*  | Semiregular  | J29 4cc*  | J31 5cc*
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-rotundas and birotundas

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

Cupola-rotunda  | Birotunda

|

J32 5cr  | J33 5cr*  | J34 5rr aD*  | Semiregular
Pentagonal orthocupolarotunda | Pentagonal gyrocupolarotunda | Pentagonal orthobirotunda | Pentagonal gyrobirotunda
(icosidodecahedron)
|  |  |
|  |  |
Augmented 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 3c=c  | Semiregular  | J38 5c=c  | Coplanar  | J36 3c=c*  | J37 4c=c* eC*  | J39 5c=c*
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-rotundas and birotundas

Elongated cupola-rotunda  | Elongated birotunda

|

J40 5c=r  | J41 5c=r*  | J42 5r=r  | J43 5r=r*
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
40px  |

#### Gyroelongated bicupolae, cupola-rotundas, and birotundas

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola | Gyroelongated cupola-rotunda  | Gyroelongated birotunda

|
|

Concave  | J44 3czc  | J45 4czc  | J46 5czc  | J47 5czr  | J48 5rzr
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
|  |  |  |  40px  |

### 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 3=+  | J50 3=++  | J51 3=+++  | J52 5=+  | J53 5=++  | J54 6=+  | J55 6=++  | J56 6=+x  | J57 6=+++
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
|  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |
Augmented from polyhedra
Triangular prism
Square pyramid | Pentagonal prism
Square pyramid | Hexagonal prism
Square pyramid
|  |

J8 and J15 would also fit here, as an augmented square prism and biaugmented square prism.

### Modified Platonic solids

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

#### Augmented dodecahedra

J58 D+  | J59 D++  | J60 D+x  | J61 D+++

|
|
|

Augmented dodecahedron | Parabiaugmented dodecahedron | Metabiaugmented dodecahedron | Triaugmented dodecahedron
|  |  |
|  |  |
Augmented from polyhedra
Dodecahedron and pentagonal pyramid

#### Diminished and augmented diminished icosahedra

Diminished icosahedron  | Augmented tridiminished icosahedron

|

J11
(Repeated)  | Uniform  | J62 I-/  | J63 I
  | J64 I
+
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 tT+  | J66 tC+  | J67 tC++  | J68 tD+  | J69 tD++  | J70 tD+x  | J71 tD+++
Augmented truncated tetrahedron | Augmented truncated cube | Biaugmented truncated cube | Augmented truncated dodecahedron | Parabiaugmented truncated dodecahedron | Metabiaugmented truncated dodecahedron | Triaugmented truncated dodecahedron
|  |  |  |  |  |
|  |  |  |  |  |
Augmented from polyhedra
truncated tetrahedron
triangular cupola | truncated cube
square cupola | truncated dodecahedron
pentagonal cupola
|  |

#### Gyrate and diminished rhombicosidodecahedra

Gyrate rhombicosidodecahedra

J72 eD*  | J73 eD**  | J74 eD*'  | J75 eD***
Gyrate rhombicosidodecahedron | Parabigyrate rhombicosidodecahedron | Metabigyrate rhombicosidodecahedron | Trigyrate rhombicosidodecahedron
|  |  |
|  |  |
Diminished rhombicosidodecahedra
J76 eD-  | J80 eD--  | J81 eD-/  | J83 eD

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
|  |  |
|  |  |

J37 would also appear here as a duplicate (it is a gyrate rhombicuboctahedron).

#### Other gyrate and diminished archimedean solids

Other archimedean solids can be gyrated and diminished, but they all result in previously counted solids.

J27  | J3  | J34  | J6  | J37  | J19  | Uniform

|
|
|
|
|
|

Gyrate cuboctahedron
(triangular orthobicupola)  | Diminished cuboctahedron
(triangular cupola)  | Gyrate icosidodecahedron
(pentagonal orthobirotunda)  | Diminished icosidodecahedron
(pentagonal rotunda)  | Gyrate rhombicuboctahedron
(elongated square gyrobicupola)  | Diminished rhombicuboctahedron
(elongated square cupola)  | Bidiminished rhombicuboctahedron
(octagonal prism)
|  |  |  |  |  |
|  |  |  |  |  |
Gyrated or diminished from polyhedra
Cuboctahedron  | Icosidodecahedron  | Rhombicuboctahedron
|  |

### 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:

J12 Triangular bipyramid
J13 Pentagonal bipyramid
J17 Gyroelongated square bipyramid

J51 Triaugmented triangular prism
J84 Snub disphenoid

|

### Triangle and square-faced Johnson solids

Twenty four Johnson solids have only triangle or square faces:

J1 Square pyramid
J7 Elongated triangular pyramid
J8 Elongated square pyramid
J10 Gyroelongated square pyramid
J14 Elongated triangular bipyramid
J15 Elongated square bipyramid
J16 Elongated pentagonal bipyramid
J26 Gyrobifastigium

J27 Triangular orthobicupola
J28 Square orthobicupola
J29 Square gyrobicupola
J35 Elongated triangular orthobicupola
J36 Elongated triangular gyrobicupola
J37 Elongated square gyrobicupola
J44 Gyroelongated triangular bicupola
J45 Gyroelongated square bicupola

J49 Augmented triangular prism
J50 Biaugmented triangular prism
J85 Snub square antiprism
J86 Sphenocorona
J87 Augmented sphenocorona
J88 Sphenomegacorona
J89 Hebesphenomegacorona
J90 Disphenocingulum

|
|

### Triangle and pentagon-faced Johnson solids

Eleven Johnson solids have only triangle and pentagon faces:

J2 Pentagonal pyramid
J11 Gyroelongated pentagonal pyramid
J34 Pentagonal orthobirotunda
J48 Gyroelongated pentagonal birotunda
J58 Augmented dodecahedron
J59 Parabiaugmented dodecahedron

J60 Metabiaugmented dodecahedron
J61 Triaugmented dodecahedron
J62 Metabidiminished icosahedron
J63 Tridiminished icosahedron
J64 Augmented tridiminished icosahedron

|

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

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

J09 Elongated pentagonal pyramid
J30 Pentagonal orthobicupola
J31 Pentagonal gyrobicupola
J32 Pentagonal orthocupolarotunda
J33 Pentagonal gyrocupolarotunda
J38 Elongated pentagonal orthobicupola
J39 Elongated pentagonal gyrobicupola
J40 Elongated pentagonal orthocupolarotunda
J41 Elongated pentagonal gyrocupolarotunda
J42 Elongated pentagonal orthobirotunda

J43 Elongated pentagonal gyrobirotunda
J46 Gyroelongated pentagonal bicupola
J47 Gyroelongated pentagonal cupolarotunda
J52 Augmented pentagonal prism
J53 Biaugmented pentagonal prism
J72 Gyrate rhombicosidodecahedron
J73 Parabigyrate rhombicosidodecahedron
J74 Metabigyrate rhombicosidodecahedron
J75 Trigyrate rhombicosidodecahedron
J91 Bilunabirotunda

|

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

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

J3 Triangular cupola
J18 Elongated triangular cupola
J22 Gyroelongated triangular cupola
J54 Augmented hexagonal prism

J55 Parabiaugmented hexagonal prism
J56 Metabiaugmented hexagonal prism
J57 Triaugmented hexagonal prism
J65 Augmented truncated tetrahedron

|

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

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

J4 Square cupola
J19 Elongated square cupola
J23 Gyroelongated square cupola

J66 Augmented truncated cube
J67 Biaugmented truncated cube

|

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

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

J06 Pentagonal rotunda
J25 Gyroelongated pentagonal rotunda

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

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

J92 Triangular hebesphenorotunda

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

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

J05 Pentagonal cupola
J20 Elongated pentagonal cupola
J21 Elongated pentagonal rotunda
J24 Gyroelongated pentagonal cupola
J68 Augmented truncated dodecahedron
J69 Parabiaugmented truncated dodecahedron
J70 Metabiaugmented truncated dodecahedron
J71 Triaugmented truncated dodecahedron

J76 Diminished rhombicosidodecahedron
J77 Paragyrate diminished rhombicosidodecahedron
J78 Metagyrate diminished rhombicosidodecahedron
J79 Bigyrate diminished rhombicosidodecahedron
J80 Parabidiminished rhombicosidodecahedron
J81 Metabidiminished rhombicosidodecahedron
J82 Gyrate bidiminished rhombicosidodecahedron
J83 Tridiminished rhombicosidodecahedron

|

## 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 polyhedra by gyration, diminishment, or dissection.[3]

Octahedron | Cuboctahedron | Rhombicuboctahedron

|
|

| J3
| J27
| J4
| J19
| J37

Icosahedron | Icosidodecahedron

|

| J11
| J62
| J63
| J6
| J34

Rhombicosidodecahedron

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

J78
| J79
| J80
| J81
| J82
| J83

## See also

* Near-miss Johnson solid
* Catalan solid
* Toroidal polyhedron

## 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. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
* Zalgaller, Victor A. (1967). "Convex Polyhedra with Regular Faces" (in ru). Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 2: 1–221. ISSN 0373-2703. The first proof that there are only 92 Johnson solids. English translation: Zalgaller, Victor A. (1969). "Convex Polyhedra with Regular Faces". Seminars in Mathematics, V. A. Steklov Math. Inst., Leningrad (Consultants Bureau) 2. ISSN 0080-8873.
* Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 3 Further Convex polyhedra
* Timofeenko, A.V. (2009). "The Non-Platonic and Non-Archimedean Noncomposite Polyhedra" (in en). J. Math. Sci. 162 (5): 710–729. doi:10.1007/s10958-009-9655-0. [1]

olyhedra." J. Math. Sci. 162, 710-729, 2009.

1. ↑ GWH. "Pseudo Rhombicuboctahedra". http://www.georgehart.com/virtual-polyhedra/pseudo-rhombicuboctahedra.html.
2. ↑ George Hart (quoting Johnson) (1996). "Johnson Solids". http://www.georgehart.com/virtual-polyhedra/johnson-info.html.
3. ↑ Klitzing, Dr. Richard. "Johnson solids et al.". http://bendwavy.org/klitzing/explain/johnson.htm.

## External links

* Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières". Structural Topology (6): 83–95. https://upcommons.upc.edu/bitstream/handle/2099/890/st6-11-a7.pdf.
* 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". http://mathworld.wolfram.com/JohnsonSolid.html.
* 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 (Convex 4-dimensional polytopes with Regular polygons as 2-dimensional Faces), 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.

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Johnson solids

Pyramids, cupolae and rotundae|

* square pyramid
* pentagonal pyramid
* triangular cupola
* square cupola
* pentagonal cupola
* pentagonal rotunda

Modified pyramids|

* elongated triangular pyramid
* elongated square pyramid
* elongated pentagonal pyramid
* gyroelongated square pyramid
* gyroelongated pentagonal pyramid
* triangular bipyramid
* square bipyramid
* pentagonal bipyramid
* elongated triangular bipyramid
* elongated square bipyramid
* elongated pentagonal bipyramid
* gyroelongated square bipyramid

Modified cupolae and rotundae|

* elongated triangular cupola
* elongated square cupola
* elongated pentagonal cupola
* elongated pentagonal rotunda
* gyroelongated triangular cupola
* gyroelongated square cupola
* gyroelongated pentagonal cupola
* gyroelongated pentagonal rotunda
* gyrobifastigium
* triangular orthobicupola
* square orthobicupola
* square gyrobicupola
* pentagonal orthobicupola
* pentagonal gyrobicupola
* pentagonal orthocupolarotunda
* pentagonal gyrocupolarotunda
* pentagonal orthobirotunda
* elongated triangular orthobicupola
* elongated triangular gyrobicupola
* elongated square gyrobicupola
* elongated pentagonal orthobicupola
* elongated pentagonal gyrobicupola
* elongated pentagonal orthocupolarotunda
* elongated pentagonal gyrocupolarotunda
* elongated pentagonal orthobirotunda
* elongated pentagonal gyrobirotunda
* gyroelongated triangular bicupola
* gyroelongated square bicupola
* gyroelongated pentagonal bicupola
* gyroelongated pentagonal cupolarotunda
* gyroelongated pentagonal birotunda

Augmented prisms|

* 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

Modified Platonic solids|

* augmented dodecahedron
* parabiaugmented dodecahedron
* metabiaugmented dodecahedron
* triaugmented dodecahedron
* metabidiminished icosahedron
* tridiminished icosahedron
* augmented tridiminished icosahedron

Modified Archimedean solids|

* augmented truncated tetrahedron
* augmented truncated cube
* biaugmented truncated cube
* augmented truncated dodecahedron
* parabiaugmented truncated dodecahedron
* metabiaugmented truncated dodecahedron
* triaugmented truncated dodecahedron
* gyrate rhombicosidodecahedron
* parabigyrate rhombicosidodecahedron
* metabigyrate rhombicosidodecahedron
* trigyrate rhombicosidodecahedron
* diminished rhombicosidodecahedron
* paragyrate diminished rhombicosidodecahedron
* metagyrate diminished rhombicosidodecahedron
* bigyrate diminished rhombicosidodecahedron
* parabidiminished rhombicosidodecahedron
* metabidiminished rhombicosidodecahedron
* gyrate bidiminished rhombicosidodecahedron
* tridiminished rhombicosidodecahedron

Elementary solids|

* snub disphenoid
* snub square antiprism
* sphenocorona
* augmented sphenocorona
* sphenomegacorona
* hebesphenomegacorona
* disphenocingulum
* bilunabirotunda
* triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)

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