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 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 Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”.

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 are 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 Platonic solids, the 13 Archimedean solids, the infinitely 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.

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:

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:

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.

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.

The next four Johnson solids are three cupolae and one rotunda. They represent sections of uniform 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.

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.

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

The pentagonal gyrobirotunda is an Archimedean solid (in this case the icosidodecahedron), so it is not a Johnson solid.

The elongated square orthobicupola is an Archimedean solid (in this case the rhombicuboctahedron), so it is not a Johnson solid.

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

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

Johnson solids 65 to 83 are built by augmenting, diminishing or gyrating Archimedean solids.

J37 would also appear here as a duplicate (it is a gyrate rhombicuboctahedron).

Other archimedean solids can be gyrated and diminished, but they all result in previously counted solids.

Johnson solids 84 to 92 are not derived from "cut-and-paste" manipulations of uniform solids.

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.

Five Johnson solids are deltahedra, with all equilateral triangle faces:

Twenty Johnson solids have only triangle, square and pentagonal faces:

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

Sixteen Johnson solids have only triangle, square, pentagon and decagonal faces:

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]