Johnson solid

A Johnson solid is a strictly convex regular-faced polyhedron that is not uniform. They are named after Norman W. Johnson, who in 1966 first listed all 92 such polyhedra. Before him, Duncan Sommerville discovered the subset of them that are circumscribable. In 1969, Victor Zalgaller proved that the list was complete.

Even though they are not allowed to be uniform, Johnson solids can have just one type of polygon for their faces, as the triangular bipyramid does, or have only one vertex figure, as the elongated square gyrobicupola does.

Johnson solids only have triangles, squares, pentagons, hexagons, octagons, or decagons as faces. All are symmetrical.

Near-miss Johnson solids
A large amount of near-miss Johnson solids may also be constructed. These polyhedra are convex and all of their faces are either regular or almost regular. They may also use polygons unavailable to the proper Johnson solids, such as the heptagon, enneagon, hendecagon, or dodecagon.

Pyramids, cupolas, and the pentagonal rotunda
A pyramid is formed by connecting a point and an n-gon with a band of triangles that all meet at the point. The triangular pyramid is a tetrahedron, and is too symmetric to be a Johnson solid. A hexagonal pyramid would be planar.

A cupola is formed by connecting an n-gon and a 2n-gon with a band of alternating triangles and squares. A hexagonal cupola would be planar.

The pentagonal rotunda is unique. It is formed by a pentagon and a decagon connected by two sets of triangles and one set of pentagons. It has no CRF analogues based on other polygons.

Each of these polyhedra is related to a uniform polyhedron: the square pyramid with the octahedron, the pentagonal pyramid with the icosahedron, the triangular cupola with the cuboctahedron, the square cupola with the small rhombicuboctahedron, the pentagonal cupola with the small rhombicosidodecahedron and the pentagonal rotunda with the icosidodecahedron.

Elongations
Some solids can be "elongated" by attaching a prism to one of their faces, usually the largest face. The elongated square cupola can also be constructed as a diminished rhombicuboctahedron.

Gyroelongations
"Gyroelongation" adds an antiprism to said face instead of a prism.

The triangular pyramid cannot be gyroelongated because its faces would be coplanar to those of the added antiprism, and the resulting polyhedron would not be strictly convex.

The gyroelongated pentagonal pyramid can also be constructed as a diminished icosahedron.

Bipyramids, bicupolas, birotundas, and cupolarotundas
Two pyramids, cupolas, or rotundas can be joined together, typically by their largest face. (Joining them by another face would create a nonconvex polyhedron.) The pentagonal cupola can join with the pentagonal rotunda due to both being decagon-based.

Bicupolas, birotundas, and the pentagonal cupola-rotunda compound can be aligned in two different ways, with the "top" faces (the ones parallel to the "base") either aligned with one another or 180° out of alignment. When aligned, the compound is called "ortho-," and when out of alignment, the compound is called "gyro-."

The gyrobifastigium is a "digonal gyrobicupola." The "digonal cupola" is just a triangular prism, thought of as turned on its side; a band of triangles and squares joining a 2-gon and a 4-gon.

The square bipyramid is not included because it is an octahedron, which is regular and therefore not a Johnson solid.

The triangular gyrobicupola and the pentagonal gyrobirotunda are not included because they are the uniform cuboctahedron and icosidodecahedron, respectively. Their "ortho" forms can be constructed as gyrations of those polyhedra.

Elongations and gyroelongations of the pairs
Elongations and gyroelongations of the above pairs place a prism or antiprism in between the two parts.

The elongated square gyrobicupola can be constructed as a gyrate rhombicuboctahedron.

The elongated square orthobicupola is not included because it is a uniform small rhombicuboctahedron. The gyroelongated pentagonal bipyramid is not included because it is a regular icosahedron.

The gyrobifastigium cannot be elongated, nor can the triangular bipyramid be gyroelongated, because some adjacent faces would be coplanar.

Augmentations
Polyhedra can be "augmented" by adding pyramids or cupolae to their faces. However, too many augmentations will make the solid nonconvex, giving each polyhedron an upper limit of possible augmentations. Most polyhedra do not admit augmentations (while maintaining strict convexity).

A pair of augmentations is referred to as "para-" if on parallel faces of a polyhedron, and as "meta-" if not (following the conventions used in similar situations in organic chemistry - "ortho" augmentations would be non-convex). If multiple augmentations on a polyhedron can only be arranged in one way (while maintaining convexity of the resulting polyhedron), like in the biaugmented truncated cube, no such clarification is needed.

The various possible "augmented square prisms" are not shown here because they would be the same as elongated square (bi)pyramids, which have already been covered.

Diminishings
"Diminishing" is the opposite of augmenting: it removes one or more pyramids or cupolae from a solid, leaving behind regular polygonal face(s). A "diminished icosahedron" would be a gyroelongated pentagonal pyramid, and a "parabidiminished icosahedron" would be a pentagonal antiprism.

Gyrations and diminishings of small rhombicosidodecahedron
A solid that can be diminished can be "gyrated" instead if the diminishing would remove a cupola (or rotunda). The gyration rotates this cupola.

Gyrating a pyramid would not change the solid. Gyrating a cuboctahedron or icosidodecahedron would produce a triangular orthobicupola or pentagonal orthobirotunda, respectively. Gyrating a small rhombicuboctahedron would produce an elongated square gyrobicupola, and diminishing it would produce an elongated square cupola.

The elementary Johnson solids
All Johnson solids up to this point were made by "cutting and pasting" pieces of Platonic solids, Archimedean solids, prisms, and antiprisms. The last few Johnson solids do not result from such simple manipulations, although some of them have subtle relationships with other solids. Only the augmented sphenocorona can be constructed by augmentation.

Symmetry
The following Johnson solids have reflection planes:

A1×I×I: J78, J79, J82, J87

K2×I: J49, J50, J52, J53, J54, J56, J60, J62, J70, J74, J81, J86, J88, J89

A2×I: J3, J7, J18, J22, J61, J63, J64, J65, J71, J75, J83, J92

B2×I: J1, J4, J8, J10, J19, J23, J66

H2×I: J2, J5, J6, J9, J11, J20, J21, J24, J25, J32, J33, J40, J41, J58, J68, J72, J76, J77

K3: J55, J91

B2×A1/2: J26, J84, J90

A2×A1: J12, J14, J27, J35, J51, J57

B2×A1: J15, J28, J67

H2×A1: J13, J16, J30, J34, J38, J42

G2×A1/2: J36

I2(8)×A1/2: J17, J29, J37, J85

I2(10)×A1/2: J31, J39, J43, J59, J69, J73, J80

The following Johnson solids do not have reflection planes:

H2+×I: J47

A2×A1+: J44

B2×A1+: J45

H2×A1+: J46, J48

Face types
The following Johnson solids only have triangular faces: J12, J13, J17, J51, J84

The following Johnson solids have triangular and square faces: J1, J7, J8, J10, J14, J15, J16, J26, J27, J28, J29, J35, J36, J37, J44, J45, J49, J50, J85, J86, J87, J88, J89, J90

The following Johnson solids have triangular and pentagonal faces: J2, J11, J34, J48, J58, J59, J60, J61, J62, J63, J64

The following Johnson solids have triangular, square and pentagonal faces: J9, J30, J31, J32, J33, J38, J39, J40, J41, J42, J43, J46, J47, J52, J53, J72, J73, J74, J75, J91

The following Johnson solids have triangular, square and hexagonal faces: J3, J18, J22, J54, J55, J56, J57, J65

The following Johnson solids have triangular, square and octagonal faces: J4, J19, J23, J66, J67

The following Johnson solids have triangular, pentagonal and decagonal faces: J6, J25

The following Johnson solids have triangular, square, pentagonal and decagonal faces: J5, J20, J21, J24, J68, J69, J70, J71, J76, J77, J78, J79, J80, J81, J82, J83

Only one Johnson solid has triangular, square, pentagonal and hexagonal faces: J92

Inspheres and circumspheres
The following Johnson solids can be inscribed in spheres:

Radius 0.71: J1

Radius 0.95: J2, J11, J62, J63

Radius 1: J3, J27

Radius 1.40: J4, J19, J37

Radius 1.62: J6, J34

Radius 2.23: J5, J72, J73, J74, J75, J76, J77, J78, J79, J80, J81, J82, J83

The following Johnson solids can be circumscribed around spheres:

Radius ?: J1

Radius ?: J2

Radius 0.27: J12

Radius 0.42: J13

In higher dimensions
The higher-dimensional generalizations of the Johnson solids include the non-uniform CRF polytopes, where faces must be regular, and the non-uniform Blind polytopes, where facets (e.g. cells in a 4-polytope) are regular. The Blind polytopes have been completely enumerated, while the CRF polytopes are a much broader class.