Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform, 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 ; 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 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, also called the pseudorhombicuboctahedron, 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, 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 or unlike faces 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.
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
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.| Regular | J1 | J2 |
| Triangular pyramid | Square pyramid | Pentagonal pyramid |
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.Elongated and gyroelongated cupolae and rotundae
Bicupolae
The triangular gyrobicupola is an Archimedean solid, so it is not a Johnson solid.Cupola-rotundae and birotunda
The pentagonal gyrobirotunda is an Archimedean solid, so it is not a Johnson solid.Elongated bicupolae
The elongated square orthobicupola is an Archimedean solid, so it is not a Johnson solid.Elongated cupola-rotundae and birotundae
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.Augmented dodecahedra
Diminished and augmented icosahedra
Modified Archimedean solids
Johnson solids 65 to 83 are built by augmenting, diminishing or gyrating Archimedean solids.Augmented Archimedean solids
Gyrate and diminished rhombicosidodecahedra
J37 would also appear here as a duplicate.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 | Icosahedron ss | Snub square antiprism ss |
| Digonal gyrobianticupola | Triangular gyrobianticupola | Square gyrobianticupola |
Others
Classification by types of faces
Triangle-faced Johnson solids
Five Johnson solids are deltahedra, with all equilateral triangle faces:Triangle and square-faced Johnson solidsTwenty four Johnson solids have only triangle or square faces:
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