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, 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 (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.


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.

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.


Further information: List of Johnson solids


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
Square pyramid Pentagonal pyramid

Cupolæ 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
triangular prism
square pyramid
pentagonal pyramid
pentagonal prism
square pyramid
square antiprism
pentagonal pyramid
pentagonal antiprism


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
Pentagonal bipyramid Elongated triangular bipyramid Elongated square bipyramid Elongated pentagonal bipyramid Gyroelongated triangular bipyramid
Gyroelongated square bipyramid Gyroelongated pentagonal bipyramid
Augmented from polyhedra
tetrahedron square pyramid pentagonal pyramid tetrahedron
triangular prism
square pyramid
pentagonal pyramid
pentagonal prism
square pyramid
square antiprism
pentagonal pyramid
pentagonal antiprism

Elongated cupolæ and rotundæ

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


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
Triangular gyrobicupola
Square gyrobicupola Pentagonal gyrobicupola
Augmented from polyhedron

Cupola-rotundæ and birotunda

Cupola-rotunda Birotunda
J32 J33 J34 Semiregular
Pentagonal orthocupolarotunda Pentagonal gyrocupolarotunda Pentagonal orthobirotunda Pentagonal gyrobirotunda
Augumented from polyhedra
Pentagonal cupola
Pentagonal rotunda
Pentagonal rotunda

Elongated bicupolæ

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

Elongated cupola-rotundæ and birotundæ

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

Gyroelongated bicupolæ, cupola-rotunda, and birotunda

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

J49 J50 J51
Elongated triangular pyramid Augmented triangular prism Biaugmented triangular prism Triaugmented triangular prism
Augumented from polyhedra
Triangular prism
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
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 tetrahedron 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

Augmented truncated dodecahedra

Semiregular J68 J69 J70 J71
Truncated dodecahedron Augmented truncated dodecahedron Parabiaugmented truncated dodecahedron Metabiaugmented truncated dodecahedron Triaugmented truncated dodecahedron

Gyrate rhombicosidodecahedra

J72 J73 J74 J75
Gyrate rhombicosidodecahedron Parabigyrate rhombicosidodecahedron Metabigyrate rhombicosidodecahedron Trigyrate rhombicosidodecahedron

Diminished rhombicosidodecahedra

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

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


snub square antiprism



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:

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 hexagonal-faced Johnson solids

Eight Johnson solids have only triangle, square and hexagonal 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 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
Icosahedron Icosidodecahedron
Rhombicosidodecahedron (diminished)
Rhombicosidodecahedron (+gyration)

See also


External links

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