Johnson Solid
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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 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 such that 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 is not uniform (i.e., not a Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they use "Johnson solid" of it.

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 is 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 Platinic solids , the 13 Archimedean solids, the infinity 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.

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. They represent sections of regular polyhedra.

Regular J1 J2
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.

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.

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.

Modified cupolae and rotundae

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

Bicupolae

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

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)
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.

Gyroelongated bicupolae, cupola-rotunda, and birotunda

These Johnson solids have 2 chiral forms.

Augmented prisms

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

Modified Platonic solids

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

Diminished and augmented diminished icosahedra

Diminished icosahedron 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.

Gyrate and diminished rhombicosidodecahedra

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
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)
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

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 pentagonal-faced Johnson solids

Twenty Johnson solids have only triangle, square and pentagonal 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 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

Triangle, pentagon, and decagonal-faced Johnson solids

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

 J06 Pentagonal rotunda J25 Gyroelongated pentagonal rotunda

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

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

 J92 Triangular hebesphenorotunda

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

Sixteen Johnson solids have only triangle, square, pentagon and decagonal 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 polyhedron by gyration, diminishment, or dissection.[3]

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. (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. 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. Zbl 0177.24802.
• Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 3 Further Convex polyhedra
1. ^ GWH. "Pseudo Rhombicuboctahedra". www.georgehart.com. Retrieved 2018.
2. ^ George Hart (quoting Johnson) (1996). "Johnson Solids". Virtual Polyhedra. Retrieved 2014.
3. ^ Klitzing, Dr. Richard. "Johnson solids et al". bendwavy.org. Retrieved 2018.