Johnson solid

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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[edit]

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[edit]

Further information: List of Johnson solids

Pyramids[edit]

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
Tetrahedron.png Square pyramid.png Pentagonal pyramid.png
Tetrahedron flat.svg Johnson solid 1 net.png Johnson solid 2 net.png

Cupola and rotunda[edit]

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
Triangular prism wedge.png Triangular cupola.png Square cupola.png Pentagonal cupola.png Pentagonal rotunda.png
Johnson solid 3 net.png Johnson solid 4 net.png Johnson solid 5 net.png Johnson solid 6 net.png
Related uniform polyhedra
Cuboctahedron Rhombicuboctahedron Rhombicosidodecahedron Icosidodecahedron
Cuboctahedron.png Small rhombicuboctahedron.png Small rhombicosidodecahedron.png Icosidodecahedron.png

Elongated and gyroelongated pyramids[edit]

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 parallelograms, 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
Elongated triangular pyramid.png Elongated square pyramid.png Elongated pentagonal pyramid.png Augmented octahedron.png Gyroelongated square pyramid.png Gyroelongated pentagonal pyramid.png
Johnson solid 7 net.png Johnson solid 8 net.png Johnson solid 9 net.png Augmented octahedgon net.png Johnson solid 10 net.png Johnson solid 11 net.png
Augmented from polyhedra
tetrahedron
triangular prism
square pyramid
cube
pentagonal pyramid
pentagonal prism
tetrahedron
octahedron
square pyramid
square antiprism
pentagonal pyramid
pentagonal antiprism
Tetrahedron.png Triangular prism.png Square pyramid.png Hexahedron.png Pentagonal pyramid.png Pentagonal prism.png Tetrahedron.png Octahedron.png Square pyramid.png Square antiprism.png Pentagonal pyramid.png Pentagonal antiprism.png

Bipyramids[edit]

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 Gyroelongated square bipyramid Gyroelongated pentagonal bipyramid
(icosahedron)
Triangular dipyramid.png Octahedron.png Pentagonal dipyramid.png Elongated triangular dipyramid.png Elongated square dipyramid.png Elongated pentagonal dipyramid.png Gyroelongated triangular bipyramid.png Gyroelongated square dipyramid.png Pentagonal gyroelongated bipyramid.png
Johnson solid 12 net.png Octahedron flat.svg Johnson solid 13 net.png Johnson solid 14 net.png Johnson solid 15 net.png Johnson solid 16 net.png Johnson solid 17 net.png Icosahedron flat.svg
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
Tetrahedron.png Square pyramid.png Pentagonal pyramid.png Tetrahedron.pngTriangular prism.png Square pyramid.pngHexahedron.png Pentagonal pyramid.pngPentagonal prism.png Tetrahedron.pngOctahedron.png Square pyramid.pngSquare antiprism.png Pentagonal pyramid.pngPentagonal antiprism.png

Elongated cupola and rotunda[edit]

Elongated cupola Elongated rotunda Gyroelongated cupola Gyroelongated rotunda
Coplanar J18 J19 J20 J21 J22 Coplanar 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
Elongated digonal cupola.png Elongated triangular cupola.png Elongated square cupola.png Elongated pentagonal cupola.svg Elongated pentagonal rotunda.png Gyroelongated digonal cupola.png Gyroelongated triangular cupola.png Gyroelongated square cupola.png Gyroelongated pentagonal cupola.png Gyroelongated pentagonal rotunda.png
Johnson solid 18 net.png Johnson solid 19 net.png Johnson solid 20 net.png Johnson solid 21 net.png Johnson solid 22 net.png Johnson solid 23 net.png Johnson solid 24 net.png Johnson solid 25 net.png
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
Tetragonal prism.pngTriangular prism.png Hexagonal prism.pngTriangular cupola.png Octagonal prism.pngSquare cupola.png Decagonal prism.pngPentagonal cupola.png Decagonal prism.pngPentagonal rotunda.png Square antiprism.pngTriangular prism.png Hexagonal antiprism.pngTriangular cupola.png Octagonal antiprism.pngSquare cupola.png Decagonal antiprism.pngPentagonal cupola.png Decagonal antiprism.pngPentagonal rotunda.png

Bicupola[edit]

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
Digonal orthobicupola.png Triangular orthobicupola.png Square orthobicupola.png Pentagonal orthobicupola.png Gyrobifastigium.png Cuboctahedron.png Square gyrobicupola.png Pentagonal gyrobicupola.png
Johnson solid 27 net.png Johnson solid 28 net.png Johnson solid 30 net.png Johnson solid 26 net.png Cuboctahedron flat.svg Johnson solid 29 net.png Johnson solid 31 net.png
Augmented from polyhedron
Triangular prism.png Triangular cupola.png Square cupola.png Pentagonal cupola.png Triangular prism.png Triangular cupola.png Square cupola.png Pentagonal cupola.png

Cupola-rotunda and birotunda[edit]

Cupola-rotunda Birotunda
J32 J33 J34 Semiregular
Pentagonal orthocupolarotunda Pentagonal gyrocupolarotunda Pentagonal orthobirotunda Pentagonal gyrobirotunda
icosidodecahedron
Pentagonal orthocupolarotunda.png Pentagonal gyrocupolarotunda.png Pentagonal orthobirotunda.png Icosidodecahedron.png
Johnson solid 32 net.png Johnson solid 33 net.png Johnson solid 34 net.png Icosidodecahedron flat.svg
Augumented from polyhedra
Pentagonal cupola
Pentagonal rotunda
Pentagonal rotunda
Pentagonal cupola.pngPentagonal rotunda.png Pentagonal rotunda.png

Elongated bicupolae[edit]

Elongated orthobicupola Elongated gyrobicupola
Coplanar J35 Semiregular J38 Coplanar J36 J37 J39
Elongated diagonal orthobicupola Elongated triangular orthobicupola Elongated square orthobicupola
(rhombicuboctahedron)
Elongated pentagonal orthobicupola Elongated diagonal gyrobicupola Elongated triangular gyrobicupola Elongated square gyrobicupola Elongated pentagonal gyrobicupola
Elongated digonal orthobicupola.png Elongated triangular orthobicupola.png Small rhombicuboctahedron.png Elongated pentagonal orthobicupola.png Elongated digonal gyrobicupola.png Elongated triangular gyrobicupola.png Elongated square gyrobicupola.png Elongated pentagonal gyrobicupola.png
Johnson solid 35 net.png Rhombicuboctahedron flat.png Johnson solid 38 net.png Johnson solid 36 net.png Johnson solid 37 net.png Johnson solid 39 net.png

Elongated cupola-rotunda and birotundae[edit]

Elongated cupolarotunda Elongated birotunda
J40 J41 J42 J43
Elongated pentagonal orthocupolarotunda Elongated pentagonal gyrocupolarotunda Elongated pentagonal orthobirotunda Elongated pentagonal gyrobirotunda
Elongated pentagonal orthocupolarotunda.png Elongated pentagonal gyrocupolarotunda.png Elongated pentagonal orthobirotunda.png Elongated pentagonal gyrobirotunda.png
Johnson solid 40 net.png Johnson solid 41 net.png Johnson solid 42 net.png Johnson solid 43 net.png

Gyroelongated bicupolae, cupola-rotunda, and birotunda[edit]

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
Gyroelongated digonal bicupola.png Gyroelongated triangular bicupola.png Gyroelongated square bicupola.png Gyroelongated pentagonal bicupola.png Gyroelongated pentagonal cupolarotunda.png Gyroelongated pentagonal birotunda.png
Johnson solid 44 net.png Johnson solid 45 net.png Johnson solid 46 net.png Johnson solid 47 net.png Johnson solid 48 net.png
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
Triangular prism.pngSquare antiprism.png Triangular cupola.pngHexagonal antiprism.png Square cupola.pngOctagonal antiprism.png Pentagonal cupola.pngDecagonal antiprism.png Pentagonal cupola.pngPentagonal rotunda.pngDecagonal antiprism.png Pentagonal rotunda.pngDecagonal antiprism.png

Augmented triangular prisms[edit]

J7
(repeat)
J49 J50 J51
Elongated triangular pyramid Augmented triangular prism Biaugmented triangular prism Triaugmented triangular prism
Elongated triangular pyramid.png Augmented triangular prism.png Biaugmented triangular prism.png Triaugmented triangular prism.png
Johnson solid 7 net.png Johnson solid 49 net.png Johnson solid 50 net.png Johnson solid 51 net.png
Augumented from polyhedra
Triangular prism
tetrahedron
Triangular prism
Square pyramid
Triangular prism.png Tetrahedron.png Triangular prism.png Square pyramid.png

Augmented pentagonal and hexagonal prisms[edit]

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
Augmented pentagonal prism.png Biaugmented pentagonal prism.png Augmented hexagonal prism.png Parabiaugmented hexagonal prism.png Metabiaugmented hexagonal prism.png Triaugmented hexagonal prism.png
Johnson solid 52 net.png Johnson solid 53 net.png Johnson solid 54 net.png Johnson solid 55 net.png Johnson solid 56 net.png Johnson solid 57 net.png
Augumented from polyhedra
Pentagonal prism
Square pyramid
Hexagonal prism
Square pyramid
Pentagonal prism.png Square pyramid.png Hexagonal prism.png Square pyramid.png

Augmented dodecahedron[edit]

Regular J58 J59 J60 J61
Dodecahedron Augmented dodecahedron Parabiaugmented dodecahedron Metabiaugmented dodecahedron Triaugmented dodecahedron
Dodecahedron.png Augmented dodecahedron.png Parabiaugmented dodecahedron.png Metabiaugmented dodecahedron.png Triaugmented dodecahedron.png
Dodecahedron flat.svg Johnson solid 58 net.png Johnson solid 59 net.png Johnson solid 60 net.png Johnson solid 61 net.png
Augumented from polyhedra
Dodecahedron and pentagonal pyramid
Dodecahedron.png Pentagonal pyramid.png

Diminished icosahedra[edit]

J63 J62 J11
(Repeated)
Regular J64
Tridiminished icosahedron Metabidiminished icosahedron Diminished icosahedron
(Gyroelongated pentagonal pyramid)
Icosahedron Augmented tridiminished icosahedron
Tridiminished icosahedron.png Metabidiminished icosahedron.png Gyroelongated pentagonal pyramid.png Icosahedron.png Augmented tridiminished icosahedron.png
Johnson solid 63 net.png Johnson solid 62 net.png Johnson solid 11 net.png Icosahedron flat.svg Johnson solid 64 net.png
Augumented from polyhedra
Tridiminished icosahedron, pentagonal pyramid and tetrahedron
Tridiminished icosahedron.pngPentagonal pyramid.pngTetrahedron.png

Augmented truncated tetrahedra and cubes[edit]

J65 J66 J67
Augmented truncated tetrahedron Augmented truncated cube Biaugmented truncated cube
Augmented truncated tetrahedron.png Augmented truncated cube.png Biaugmented truncated cube.png
Johnson solid 65 net.png Johnson solid 66 net.png Johnson solid 67 net.png
Augumented from polyhedra
truncated tetrahedron
triangular cupola
truncated cube
square cupola
Truncated tetrahedron.pngTriangular cupola.png Truncated hexahedron.pngSquare cupola.png

Augmented truncated dodecahedron[edit]

Semiregular J68 J69 J70 J71
Truncated dodecahedron Augmented truncated dodecahedron Parabiaugmented truncated dodecahedron Metabiaugmented truncated dodecahedron Triaugmented truncated dodecahedron
Truncated dodecahedron.png Augmented truncated dodecahedron.png Parabiaugmented truncated dodecahedron.png Metabiaugmented truncated dodecahedron.png Triaugmented truncated dodecahedron.png
Truncated dodecahedron flat.png Johnson solid 68 net.png Johnson solid 69 net.png Johnson solid 70 net.png Johnson solid 71 net.png

Gyrate rhombicosidodecahedron[edit]

J72 J73 J74 J75
Gyrate rhombicosidodecahedron Parabigyrate rhombicosidodecahedron Metabigyrate rhombicosidodecahedron Trigyrate rhombicosidodecahedron
Gyrate rhombicosidodecahedron.png Parabigyrate rhombicosidodecahedron.png Metabigyrate rhombicosidodecahedron.png Trigyrate rhombicosidodecahedron.png
Johnson solid 72 net.png Johnson solid 73 net.png Johnson solid 74 net.png Johnson solid 75 net.png

Diminished rhombicosidodecahedron[edit]

J76 J77 J78 J79
Diminished rhombicosidodecahedron Paragyrate diminished rhombicosidodecahedron Metagyrate diminished rhombicosidodecahedron Bigyrate diminished rhombicosidodecahedron
Diminished rhombicosidodecahedron.png Paragyrate diminished rhombicosidodecahedron.png Metagyrate diminished rhombicosidodecahedron.png Bigyrate diminished rhombicosidodecahedron.png
Johnson solid 76 net.png Johnson solid 77 net.png Johnson solid 78 net.png Johnson solid 79 net.png
 
J80 J81 J82 J83
Parabidiminished rhombicosidodecahedron Metabidiminished rhombicosidodecahedron Gyrate bidiminished rhombicosidodecahedron Tridiminished rhombicosidodecahedron
Parabidiminished rhombicosidodecahedron.png Metabidiminished rhombicosidodecahedron.png Gyrate bidiminished rhombicosidodecahedron.png Tridiminished rhombicosidodecahedron.png
Johnson solid 80 net.png Johnson solid 81 net.png Johnson solid 82 net.png Johnson solid 83 net.png

Others[edit]

J84 J85 J86 J87 J88
Snub disphenoid Snub square antiprism Sphenocorona Augmented sphenocorona Sphenomegacorona
Snub disphenoid.png Snub square antiprism.png Sphenocorona.png Augmented sphenocorona.png Sphenomegacorona.png
Johnson solid 84 net.png Johnson solid 85 net.png Johnson solid 86 net.png Johnson solid 87 net.png Johnson solid 88 net.png
J89 J90 J91 J92
Hebesphenomegacorona Disphenocingulum Bilunabirotunda Triangular hebesphenorotunda
Hebesphenomegacorona.png Disphenocingulum.png Bilunabirotunda.png Triangular hebesphenorotunda.png
Johnson solid 89 net.png Johnson solid 90 net.png Johnson solid 91 net.png Johnson solid 92 net.png

Classification by types of faces[edit]

Triangle-faced Johnson solids[edit]

Five solids are deltahedra, with all equilateral triangle faces:

J12 Triangular bipyramid Triangular dipyramid.png
J13 Pentagonal bipyramid Pentagonal dipyramid.png
J17 Gyroelongated square bipyramid Gyroelongated square dipyramid.png
J51 Triaugmented triangular prism Triaugmented triangular prism.png
J84 Snub disphenoid Snub disphenoid.png

Triangle and square-faced Johnson solids[edit]

Twenty four Johnson solids have only triangle or square faces:

J1 Square pyramid Square pyramid.png
J7 Elongated triangular pyramid Elongated triangular pyramid.png
J8 Elongated square pyramid Elongated square pyramid.png
J10 Gyroelongated square pyramid Gyroelongated square pyramid.png
J14 Elongated triangular bipyramid Elongated triangular dipyramid.png
J15 Elongated square bipyramid Elongated square dipyramid.png
J16 Elongated pentagonal bipyramid Elongated pentagonal dipyramid.png
J26 Gyrobifastigium Gyrobifastigium.png
J27 Triangular orthobicupola Triangular orthobicupola.png
J28 Square orthobicupola Square orthobicupola.png
J29 Square gyrobicupola Square gyrobicupola.png
J35 Elongated triangular orthobicupola Elongated triangular orthobicupola.png
J36 Elongated triangular gyrobicupola Elongated triangular gyrobicupola.png
J37 Elongated square gyrobicupola Elongated square gyrobicupola.png
J44 Gyroelongated triangular bicupola Gyroelongated triangular bicupola.png
J45 Gyroelongated square bicupola Gyroelongated square bicupola.png
J49 Augmented triangular prism Augmented triangular prism.png
J50 Biaugmented triangular prism Biaugmented triangular prism.png
J85 Snub square antiprism Snub square antiprism.png
J86 Sphenocorona Sphenocorona.png
J87 Augmented sphenocorona Augmented sphenocorona.png
J88 Sphenomegacorona Sphenomegacorona.png
J89 Hebesphenomegacorona Hebesphenomegacorona.png
J90 Disphenocingulum Disphenocingulum.png

Triangle and pentagonal-faced Johnson solids[edit]

Eleven Johnson solids have only triangle and pentagonal faces:

J2 Pentagonal pyramid Pentagonal pyramid.png
J11 Gyroelongated pentagonal pyramid Gyroelongated pentagonal pyramid.png
J34 Pentagonal orthobirotunda Pentagonal orthobirotunda.png
J48 Gyroelongated pentagonal birotunda Gyroelongated pentagonal birotunda.png
J58 Augmented dodecahedron Augmented dodecahedron.png
J59 Parabiaugmented dodecahedron Parabiaugmented dodecahedron.png
J60 Metabiaugmented dodecahedron Metabiaugmented dodecahedron.png
J61 Triaugmented dodecahedron Triaugmented dodecahedron.png
J62 Metabidiminished icosahedron Metabidiminished icosahedron.png
J63 Tridiminished icosahedron Tridiminished icosahedron.png
J64 Augmented tridiminished icosahedron Augmented tridiminished icosahedron.png

Triangle, square and hexagonal-faced Johnson solids[edit]

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

J3 Triangular cupola Triangular cupola.png
J18 Elongated triangular cupola Elongated triangular cupola.png
J22 Gyroelongated triangular cupola Gyroelongated triangular cupola.png
J54 Augmented hexagonal prism Augmented hexagonal prism.png
J55 Parabiaugmented hexagonal prism Parabiaugmented hexagonal prism.png
J56 Metabiaugmented hexagonal prism Metabiaugmented hexagonal prism.png
J57 Triaugmented hexagonal prism Triaugmented hexagonal prism.png
J65 Augmented truncated tetrahedron Augmented truncated tetrahedron.png

Triangle, square and octagonal-faced Johnson solids[edit]

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

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

Circumscribable Johnson solids[edit]

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
Square pyramid.png
J3
Triangular cupola.png
27
Triangular orthobicupola.png
J4
Square cupola.png
J19
Elongated square cupola.png
J37
Elongated square gyrobicupola.png
Icosahedron Icosidodecahedron
J2
Pentagonal pyramid.png
J63
Tridiminished icosahedron.png
J62
Metabidiminished icosahedron.png
J11
Gyroelongated pentagonal pyramid.png
J6
Pentagonal rotunda.png
J34
Pentagonal orthobirotunda.png
Rhombicosidodecahedron (diminished)
J5
Pentagonal cupola.png
J76
Diminished rhombicosidodecahedron.png
J80
Parabidiminished rhombicosidodecahedron.png
J81
Metabidiminished rhombicosidodecahedron.png
J83
Tridiminished rhombicosidodecahedron.png
Rhombicosidodecahedron (+gyration)
J72
Gyrate rhombicosidodecahedron.png
J73
Parabigyrate rhombicosidodecahedron.png
J74
Metabigyrate rhombicosidodecahedron.png
J75
Trigyrate rhombicosidodecahedron.png
J77
Paragyrate diminished rhombicosidodecahedron.png
J78
Metagyrate diminished rhombicosidodecahedron.png
J79
Bigyrate diminished rhombicosidodecahedron.png
J82
Gyrate bidiminished rhombicosidodecahedron.png

See also[edit]

References[edit]

  • 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
  1. ^ http://www.georgehart.com/virtual-polyhedra/pseudo-rhombicuboctahedra.html
  2. ^ George Hart (quoting Johnson) (1996). "Johnson Solids". Virtual Polyhedra. Retrieved 5 February 2014. 
  3. ^ http://bendwavy.org/klitzing/explain/johnson.htm

External links[edit]