# 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 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) 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 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.)

## Enumeration

Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry group
1 Square pyramid 5 8 5 4 1 C4v
2 Pentagonal pyramid 6 10 6 5 1 C5v
3 Triangular cupola 9 15 8 4 3 1 C3v
4 Square cupola 12 20 10 4 5 1 C4v
5 Pentagonal cupola 15 25 12 5 5 1 1 C5v
6 Pentagonal rotunda 20 35 17 10 6 1 C5v
7 Elongated triangular pyramid (or elongated tetrahedron) 7 12 7 4 3 C3v
8 Elongated square pyramid (or augmented cube) 9 16 9 4 5 C4v
9 Elongated pentagonal pyramid 11 20 11 5 5 1 C5v
10 Gyroelongated square pyramid 9 20 13 12 1 C4v
11 Gyroelongated pentagonal pyramid (or diminished icosahedron) 11 25 16 15 1 C5v
12 Triangular bipyramid 5 9 6 6 D3h
13 Pentagonal bipyramid 7 15 10 10 D5h
14 Elongated triangular bipyramid 8 15 9 6 3 D3h
15 Elongated square bipyramid
(or biaugmented cube)
10 20 12 8 4 D4h
16 Elongated pentagonal bipyramid 12 25 15 10 5 D5h
17 Gyroelongated square bipyramid 10 24 16 16 D4d
18 Elongated triangular cupola 15 27 14 4 9 1 C3v
19 Elongated square cupola
(diminished rhombicuboctahedron)
20 36 18 4 13 1 C4v
20 Elongated pentagonal cupola 25 45 22 5 15 1 1 C5v
21 Elongated pentagonal rotunda 30 55 27 10 10 6 1 C5v
22 Gyroelongated triangular cupola 15 33 20 16 3 1 C3v
23 Gyroelongated square cupola 20 44 26 20 5 1 C4v
24 Gyroelongated pentagonal cupola 25 55 32 25 5 1 1 C5v
25 Gyroelongated pentagonal rotunda 30 65 37 30 6 1 C5v
26 Gyrobifastigium 8 14 8 4 4 D2d
27 Triangular orthobicupola
(gyrate cuboctahedron)
12 24 14 8 6 D3h
28 Square orthobicupola 16 32 18 8 10 D4h
29 Square gyrobicupola 16 32 18 8 10 D4d
30 Pentagonal orthobicupola 20 40 22 10 10 2 D5h
31 Pentagonal gyrobicupola 20 40 22 10 10 2 D5d
32 Pentagonal orthocupolarotunda 25 50 27 15 5 7 C5v
33 Pentagonal gyrocupolarotunda 25 50 27 15 5 7 C5v
34 Pentagonal orthobirotunda
(gyrate icosidodecahedron)
30 60 32 20 12 D5h
35 Elongated triangular orthobicupola 18 36 20 8 12 D3h
36 Elongated triangular gyrobicupola 18 36 20 8 12 D3d
37 Elongated square gyrobicupola
(gyrate rhombicuboctahedron)
24 48 26 8 18 D4d
38 Elongated pentagonal orthobicupola 30 60 32 10 20 2 D5h
39 Elongated pentagonal gyrobicupola 30 60 32 10 20 2 D5d
40 Elongated pentagonal orthocupolarotunda 35 70 37 15 15 7 C5v
41 Elongated pentagonal gyrocupolarotunda 35 70 37 15 15 7 C5v
42 Elongated pentagonal orthobirotunda 40 80 42 20 10 12 D5h
43 Elongated pentagonal gyrobirotunda 40 80 42 20 10 12 D5d
44 Gyroelongated triangular bicupola
(2 chiral forms)
18 42 26 20 6 D3
45 Gyroelongated square bicupola
(2 chiral forms)
24 56 34 24 10 D4
46 Gyroelongated pentagonal bicupola
(2 chiral forms)
30 70 42 30 10 2 D5
47 Gyroelongated pentagonal cupolarotunda
(2 chiral forms)
35 80 47 35 5 7 C5
48 Gyroelongated pentagonal birotunda
(2 chiral forms)
40 90 52 40 12 D5
49 Augmented triangular prism 7 13 8 6 2 C2v
50 Biaugmented triangular prism 8 17 11 10 1 C2v
51 Triaugmented triangular prism 9 21 14 14 D3h
52 Augmented pentagonal prism 11 19 10 4 4 2 C2v
53 Biaugmented pentagonal prism 12 23 13 8 3 2 C2v
54 Augmented hexagonal prism 13 22 11 4 5 2 C2v
55 Parabiaugmented hexagonal prism 14 26 14 8 4 2 D2h
56 Metabiaugmented hexagonal prism 14 26 14 8 4 2 C2v
57 Triaugmented hexagonal prism 15 30 17 12 3 2 D3h
58 Augmented dodecahedron 21 35 16 5 11 C5v
59 Parabiaugmented dodecahedron 22 40 20 10 10 D5d
60 Metabiaugmented dodecahedron 22 40 20 10 10 C2v
61 Triaugmented dodecahedron 23 45 24 15 9 C3v
62 Metabidiminished icosahedron 10 20 12 10 2 C2v
63 Tridiminished icosahedron 9 15 8 5 3 C3v
64 Augmented tridiminished icosahedron 10 18 10 7 3 C3v
65 Augmented truncated tetrahedron 15 27 14 8 3 3 C3v
66 Augmented truncated cube 28 48 22 12 5 5 C4v
67 Biaugmented truncated cube 32 60 30 16 10 4 D4h
68 Augmented truncated dodecahedron 65 105 42 25 5 1 11 C5v
69 Parabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 D5d
70 Metabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 C2v
71 Triaugmented truncated dodecahedron 75 135 62 35 15 3 9 C3v
72 Gyrate rhombicosidodecahedron 60 120 62 20 30 12 C5v
73 Parabigyrate rhombicosidodecahedron 60 120 62 20 30 12 D5d
74 Metabigyrate rhombicosidodecahedron 60 120 62 20 30 12 C2v
75 Trigyrate rhombicosidodecahedron 60 120 62 20 30 12 C3v
76 Diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v
77 Paragyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v
78 Metagyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs
79 Bigyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs
80 Parabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 D5d
81 Metabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 C2v
82 Gyrate bidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 Cs
83 Tridiminished rhombicosidodecahedron 45 75 32 5 15 9 3 C3v
84 Snub disphenoid
(Siamese dodecahedron)
8 18 12 12 D2d
85 Snub square antiprism 16 40 26 24 2 D4d
86 Sphenocorona 10 22 14 12 2 C2v
87 Augmented sphenocorona 11 26 17 16 1 Cs
88 Sphenomegacorona 12 28 18 16 2 C2v
89 Hebesphenomegacorona 14 33 21 18 3 C2v
90 Disphenocingulum 16 38 24 20 4 D2d
91 Bilunabirotunda 14 26 14 8 2 4 D2h
92 Triangular hebesphenorotunda 18 36 20 13 3 3 1 C3v

Legend:

• Jn – Johnson Solid Number
• Net – Flattened (unfolded) image
• V – Number of Vertices
• E – Number of Edges
• F – Number of Faces (total)
• F3 – Number of 3-sided Faces
• F4 – Number of 4-sided Faces
• F5 – Number of 5-sided Faces
• F6 – Number of 6-sided Faces
• F8 – Number of 8-sided Faces
• F10 – Number of 10-sided Faces

## References

• Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
• Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN.  The first proof that there are only 92 Johnson solids.

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