# Johnson solid

The elongated square gyrobicupola (J37), a Johnson solid
This 24 equilateral triangle example is not a Johnson solid because it is not convex. (This is actually a stellation, the only one possible for the octahedron.)
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, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. 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.

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 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 (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.

## Names

The names are listed below and are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few (pyramids, cupolae, and rotunda), together with the Platonic and Archimedean solids, prisms, and antiprisms.

• Bi- means that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, they can be joined so that like faces (ortho-) or unlike faces (gyro-) meet. In this nomenclature, an octahedron would be a square bipyramid, a cuboctahedron would be a triangular gyrobicupola, and an icosidodecahedron would be a pentagonal gyrobirotunda.
• Elongated means that a prism has been joined to the base of the solid in question or between the bases of the solids in question. A rhombicuboctahedron would be an elongated square orthobicupola.
• Gyroelongated means that an antiprism has been joined to the base of the solid in question or between the bases of the solids in question. An icosahedron would be a gyroelongated pentagonal bipyramid.
• Augmented means that a pyramid or cupola has been joined to a face of the solid in question.
• Diminished means that a pyramid or cupola has been removed from the solid in question.
• Gyrate means that a cupola on the solid in question has been rotated so that different edges match up, as in the difference between ortho- and gyrobicupolae.

The last three operations — augmentation, diminution, and gyration — can be performed more than once on a large enough solid. We add bi- to the name of the operation to indicate that it has been performed twice. (A bigyrate solid has had two of its cupolae rotated.) We add tri- to indicate that it has been performed three times. (A tridiminished solid has had three of its pyramids or cupolae removed.)

Sometimes, bi- alone is not specific enough. We must distinguish between a solid that has had two parallel faces altered and one that has had two oblique faces altered. When the faces altered are parallel, we add para- to the name of the operation. (A parabiaugmented solid has had two parallel faces augmented.) When they are not, we add meta- to the name of the operation. (A metabiaugmented solid has had 2 oblique faces augmented.)

The last few Johnson solids have names based on certain polygon complexes that they are assembled from. These names are defined by Johnson as follows:[2]

If we define a lune as a complex of two triangles attached to opposite sides of a square, the prefix spheno- refers to a wedgelike complex formed by two adjacent lunes. The prefix dispheno- denotes two such complexes, while hebespheno- indicates a blunter complex of two lunes separated by a third lune. The suffix -corona refers to a crownlike complex of eight triangles, and -megacorona, to a larger such complex of 12 triangles. The suffix -cingulum indicates a belt of 12 triangles.

## Enumeration

Further information: List of 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.

Pyramids
Regular J1 J2
Triangular pyramid
(Tetrahedron)
Square pyramid Pentagonal pyramid

### Cupola 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
Triangular prism Triangular cupola Square cupola Pentagonal cupola Pentagonal rotunda
Related uniform polyhedra
Cuboctahedron Rhombicuboctahedron Rhombicosidodecahedron Icosidodecahedron

### Elongated and gyroelongated pyramids

The next five Johnson solids are elongated and gyroelongated pyramids. These represent the composite or augmentation of two polyhedra. 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
(or augmented prisms)
Gyroelongated pyramids
(or augmented antiprisms)
J7 J8 J9 Coplanar J10 J11
Elongated triangular pyramid Elongated square pyramid Elongated pentagonal pyramid Gyroelongated triangular pyramid Gyroelongated square pyramid Gyroelongated pentagonal pyramid
Augmented triangular prism Augmented cube Augmented pentagonal prism Augmented octahedron Augmented square antiprism Augmented pentagonal antiprism
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 next six Johnson solids are bipyramids, elongated bipyramids, and gyroelongated bipyramids:

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
(rhombohedron)
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

### Elongated cupolae and rotundae

Elongated cupola Elongated rotunda Gyroelongated cupola Gyroelongated rotunda
Coplanar J18 J19 J20 J21 Concave J22 J23 J24 J25
Elongated digonal cupola Elongated triangular cupola Elongated square cupola Elongated pentagonal cupola Elongated pentagonal rotunda Gyroelongated digonal cupola 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 a semiregular polyhedron (in this case an Archimedean solid), so it is not a Johnson solid.

Orthobicupola Gyrobicupola
Coplanar J27 J28 J30 J26 Semiregular J29 J31
Digonal orthobicupola Triangular orthobicupola Square orthobicupola Pentagonal orthobicupola Digonal gyrobicupola
gyrobifastigium
Triangular gyrobicupola
(cuboctahedron)
Square gyrobicupola Pentagonal gyrobicupola
Augmented from polyhedron

### Cupola-rotundae and birotundae

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

Elongated orthobicupola Elongated gyrobicupola
Coplanar J35 Semiregular J38 Coplanar J36 J37 J39
Elongated digonal orthobicupola Elongated triangular orthobicupola Elongated square orthobicupola
(rhombicuboctahedron)
Elongated pentagonal orthobicupola Elongated digonal gyrobicupola Elongated triangular gyrobicupola Elongated square gyrobicupola Elongated pentagonal gyrobicupola

### Elongated cupola-rotundae and birotundae

Elongated cupolarotunda Elongated birotunda
J40 J41 J42 J43
Elongated pentagonal orthocupolarotunda Elongated pentagonal gyrocupolarotunda Elongated pentagonal orthobirotunda Elongated pentagonal gyrobirotunda

### Gyroelongated bicupolae, cupola-rotundae, and birotundae

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola Gyroelongated cupolarotunda Gyroelongated birotunda
Concave J44 J45 J46 J47 J48
Gyroelongated digonal bicupola 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 triangular prisms

J7
(repeat)
J49 J50 J51
Elongated triangular pyramid Augmented triangular prism Biaugmented triangular prism Triaugmented triangular prism
Augumented from polyhedra
Triangular prism
tetrahedron
Triangular prism
Square pyramid

### Augmented pentagonal and hexagonal prisms

Augmented pentagonal prisms Augmented hexagonal prisms
J52 J53 J54 J55 J56 J57
Augmented pentagonal prism Biaugmented pentagonal prism Augmented hexagonal prism Parabiaugmented hexagonal prism Metabiaugmented hexagonal prism Triaugmented hexagonal prism
Augumented from polyhedra
Pentagonal prism
Square pyramid
Hexagonal prism
Square pyramid

### Augmented dodecahedra

Regular J58 J59 J60 J61
Dodecahedron Augmented dodecahedron Parabiaugmented dodecahedron Metabiaugmented dodecahedron Triaugmented dodecahedron
Augumented from polyhedra
Dodecahedron and pentagonal pyramid

### Diminished icosahedra

J63 J62 J11
(Repeated)
Regular J64
Tridiminished icosahedron Metabidiminished icosahedron Diminished icosahedron
(Gyroelongated pentagonal pyramid)
Icosahedron Augmented tridiminished icosahedron
Augumented from polyhedra
Tridiminished icosahedron, pentagonal pyramid and tetrahedron

### Augmented truncated tetrahedra and truncated cubes

J65 J66 J67
Augmented truncated tetrahedron Augmented truncated cube Biaugmented truncated cube
Augumented from polyhedra
truncated tetrahedron
triangular cupola
truncated cube
square cupola

### Snub antiprisms

The snub antiprisms can be constructed as an alternation of a truncated antiprism. Two are Johnson solids, one is a regular, and the rest can not be constructed with regular triangles.

J84 Regular J85 Irregular
Johnson solid Regular Johnson solid Concave...

Snub_disphenoid
ss{2,4}

icosahedron
ss{2,6}

snub square antiprism
ss{2,8}

ss{2,10}...

### 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 pentagonal-faced Johnson solids

Eleven Johnson solids have only triangle 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:

 J4 Square cupola J19 Elongated square cupola J23 Gyroelongated square cupola J66 Augmented truncated cube J67 Biaugmented truncated cube

## 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
27
J4
J19
J37
Icosahedron Icosidodecahedron
J2
J63
J62
J11
J6
J34
 J5 J76 J80 J81 J83
 J72 J73 J74 J75 J77 J78 J79 J82