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 (J_{1}); 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 (J_{2}) 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 (J_{37}) 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.
Contents
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 rotundae), 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
Prismatoids and rotundae
J_{n} | Solid name | Net | Image | V | E | F | F_{3} | F_{4} | F_{5} | F_{6} | F_{8} | F_{10} | Symmetry |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | Square pyramid | 5 | 8 | 5 | 4 | 1 | C_{4v} | ||||||
2 | Pentagonal pyramid | 6 | 10 | 6 | 5 | 1 | C_{5v} | ||||||
3 | Triangular cupola | 9 | 15 | 8 | 4 | 3 | 1 | C_{3v} | |||||
4 | Square cupola | 12 | 20 | 10 | 4 | 5 | 1 | C_{4v} | |||||
5 | Pentagonal cupola | 15 | 25 | 12 | 5 | 5 | 1 | 1 | C_{5v} | ||||
6 | Pentagonal rotunda | 20 | 35 | 17 | 10 | 6 | 1 | C_{5v} |
Modified pyramids and dipyramids
J_{n} | Solid name | Net | Image | V | E | F | F_{3} | F_{4} | F_{5} | F_{6} | F_{8} | F_{10} | Symmetry |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7 | Elongated triangular pyramid (or elongated tetrahedron) | 7 | 12 | 7 | 4 | 3 | C_{3v} | ||||||
8 | Elongated square pyramid (or augmented cube) | 9 | 16 | 9 | 4 | 5 | C_{4v} | ||||||
9 | Elongated pentagonal pyramid | 11 | 20 | 11 | 5 | 5 | 1 | C_{5v} | |||||
10 | Gyroelongated square pyramid | 9 | 20 | 13 | 12 | 1 | C_{4v} | ||||||
11 | Gyroelongated pentagonal pyramid (or diminished icosahedron) | 11 | 25 | 16 | 15 | 1 | C_{5v} | ||||||
12 | Triangular dipyramid | 5 | 9 | 6 | 6 | D_{3h} | |||||||
13 | Pentagonal dipyramid | 7 | 15 | 10 | 10 | D_{5h} | |||||||
14 | Elongated triangular dipyramid | 8 | 15 | 9 | 6 | 3 | D_{3h} | ||||||
15 | Elongated square dipyramid (or biaugmented cube) |
10 | 20 | 12 | 8 | 4 | D_{4h} | ||||||
16 | Elongated pentagonal dipyramid | 12 | 25 | 15 | 10 | 5 | D_{5h} | ||||||
17 | Gyroelongated square dipyramid | 10 | 24 | 16 | 16 | D_{4d} |
Modified cupolas and rotunda
- elongated cupola
- elongated rotunda
- elongated birotunda
- elongated cupolarotunda
- elongated bicupola
- gyroelongated cupola
- gyroelongated rotunda
- bicupola
- cupolarotunda
- gyroelongated bicupola
- gyroelongated birotunda
- gyroelongated cupolarotunda
J_{n} | Solid name | Net | Image | V | E | F | F_{3} | F_{4} | F_{5} | F_{6} | F_{8} | F_{10} | Symmetry |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
18 | Elongated triangular cupola | 15 | 27 | 14 | 4 | 9 | 1 | C_{3v} | |||||
19 | Elongated square cupola (diminished rhombicuboctahedron) |
20 | 36 | 18 | 4 | 13 | 1 | C_{4v} | |||||
20 | Elongated pentagonal cupola | 25 | 45 | 22 | 5 | 15 | 1 | 1 | C_{5v} | ||||
21 | Elongated pentagonal rotunda | 30 | 55 | 27 | 10 | 10 | 6 | 1 | C_{5v} | ||||
22 | Gyroelongated triangular cupola | 15 | 33 | 20 | 16 | 3 | 1 | C_{3v} | |||||
23 | Gyroelongated square cupola | 20 | 44 | 26 | 20 | 5 | 1 | C_{4v} | |||||
24 | Gyroelongated pentagonal cupola | 25 | 55 | 32 | 25 | 5 | 1 | 1 | C_{5v} | ||||
25 | Gyroelongated pentagonal rotunda | 30 | 65 | 37 | 30 | 6 | 1 | C_{5v} | |||||
26 | Gyrobifastigium | 8 | 14 | 8 | 4 | 4 | D_{2d} | ||||||
27 | Triangular orthobicupola (gyrate cuboctahedron) |
12 | 24 | 14 | 8 | 6 | D_{3h} | ||||||
28 | Square orthobicupola | 16 | 32 | 18 | 8 | 10 | D_{4h} | ||||||
29 | Square gyrobicupola | 16 | 32 | 18 | 8 | 10 | D_{4d} | ||||||
30 | Pentagonal orthobicupola | 20 | 40 | 22 | 10 | 10 | 2 | D_{5h} | |||||
31 | Pentagonal gyrobicupola | 20 | 40 | 22 | 10 | 10 | 2 | D_{5d} | |||||
32 | Pentagonal orthocupolarotunda | 25 | 50 | 27 | 15 | 5 | 7 | C_{5v} | |||||
33 | Pentagonal gyrocupolarotunda | 25 | 50 | 27 | 15 | 5 | 7 | C_{5v} | |||||
34 | Pentagonal orthobirotunda (gyrate icosidodecahedron) |
30 | 60 | 32 | 20 | 12 | D_{5h} | ||||||
35 | Elongated triangular orthobicupola | 18 | 36 | 20 | 8 | 12 | D_{3h} | ||||||
36 | Elongated triangular gyrobicupola | 18 | 36 | 20 | 8 | 12 | D_{3d} | ||||||
37 | Elongated square gyrobicupola (gyrate rhombicuboctahedron) |
24 | 48 | 26 | 8 | 18 | D_{4d} | ||||||
38 | Elongated pentagonal orthobicupola | 30 | 60 | 32 | 10 | 20 | 2 | D_{5h} | |||||
39 | Elongated pentagonal gyrobicupola | 30 | 60 | 32 | 10 | 20 | 2 | D_{5d} | |||||
40 | Elongated pentagonal orthocupolarotunda | 35 | 70 | 37 | 15 | 15 | 7 | C_{5v} | |||||
41 | Elongated pentagonal gyrocupolarotunda | 35 | 70 | 37 | 15 | 15 | 7 | C_{5v} | |||||
42 | Elongated pentagonal orthobirotunda | 40 | 80 | 42 | 20 | 10 | 12 | D_{5h} | |||||
43 | Elongated pentagonal gyrobirotunda | 40 | 80 | 42 | 20 | 10 | 12 | D_{5d} | |||||
44 | Gyroelongated triangular bicupola (2 chiral forms) |
18 | 42 | 26 | 20 | 6 | D_{3} | ||||||
45 | Gyroelongated square bicupola (2 chiral forms) |
24 | 56 | 34 | 24 | 10 | D_{4} | ||||||
46 | Gyroelongated pentagonal bicupola (2 chiral forms) |
30 | 70 | 42 | 30 | 10 | 2 | D_{5} | |||||
47 | Gyroelongated pentagonal cupolarotunda (2 chiral forms) |
35 | 80 | 47 | 35 | 5 | 7 | C_{5} | |||||
48 | Gyroelongated pentagonal birotunda (2 chiral forms) |
40 | 90 | 52 | 40 | 12 | D_{5} |
Augmented prisms
J_{n} | Solid name | Net | Image | V | E | F | F_{3} | F_{4} | F_{5} | F_{6} | F_{8} | F_{10} | Symmetry |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49 | Augmented triangular prism | 7 | 13 | 8 | 6 | 2 | C_{2v} | ||||||
50 | Biaugmented triangular prism | 8 | 17 | 11 | 10 | 1 | C_{2v} | ||||||
51 | Triaugmented triangular prism | 9 | 21 | 14 | 14 | D_{3h} | |||||||
52 | Augmented pentagonal prism | 11 | 19 | 10 | 4 | 4 | 2 | C_{2v} | |||||
53 | Biaugmented pentagonal prism | 12 | 23 | 13 | 8 | 3 | 2 | C_{2v} | |||||
54 | Augmented hexagonal prism | 13 | 22 | 11 | 4 | 5 | 2 | C_{2v} | |||||
55 | Parabiaugmented hexagonal prism | 14 | 26 | 14 | 8 | 4 | 2 | D_{2h} | |||||
56 | Metabiaugmented hexagonal prism | 14 | 26 | 14 | 8 | 4 | 2 | C_{2v} | |||||
57 | Triaugmented hexagonal prism | 15 | 30 | 17 | 12 | 3 | 2 | D_{3h} |
Modified Platonic solids
- Augmented dodecahedrons
- Diminished icosahedrons
J_{n} | Solid name | Net | Image | V | E | F | F_{3} | F_{4} | F_{5} | F_{6} | F_{8} | F_{10} | Symmetry |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
58 | Augmented dodecahedron | 21 | 35 | 16 | 5 | 11 | C_{5v} | ||||||
59 | Parabiaugmented dodecahedron | 22 | 40 | 20 | 10 | 10 | D_{5d} | ||||||
60 | Metabiaugmented dodecahedron | 22 | 40 | 20 | 10 | 10 | C_{2v} | ||||||
61 | Triaugmented dodecahedron | 23 | 45 | 24 | 15 | 9 | C_{3v} | ||||||
62 | Metabidiminished icosahedron | 10 | 20 | 12 | 10 | 2 | C_{2v} | ||||||
63 | Tridiminished icosahedron | 9 | 15 | 8 | 5 | 3 | C_{3v} | ||||||
64 | Augmented tridiminished icosahedron | 10 | 18 | 10 | 7 | 3 | C_{3v} |
Modified Archimedean solids
- augmented truncated tetrahedron
- augmented truncated cube
- augmented truncated dodecahedron
- gyrate rhombicosadodecahedron
- diminished rhombicosadodecahedron
- gyrate diminished rhombicosadodecahedron
- diminished rhombicosadodecahedron
- gyrate diminished rhombicosadodecahedron
- diminished rhombicosadodecahedron
Miscellaneous
J_{n} | Solid name | Net | Image | V | E | F | F_{3} | F_{4} | F_{5} | F_{6} | F_{8} | F_{10} | Symmetry |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
84 | Snub disphenoid (Siamese dodecahedron) |
8 | 18 | 12 | 12 | D_{2d} | |||||||
85 | Snub square antiprism | 16 | 40 | 26 | 24 | 2 | D_{4d} | ||||||
86 | Sphenocorona | 10 | 22 | 14 | 12 | 2 | C_{2v} | ||||||
87 | Augmented sphenocorona | 11 | 26 | 17 | 16 | 1 | C_{s} | ||||||
88 | Sphenomegacorona | 12 | 28 | 18 | 16 | 2 | C_{2v} | ||||||
89 | Hebesphenomegacorona | 14 | 33 | 21 | 18 | 3 | C_{2v} | ||||||
90 | Disphenocingulum | 16 | 38 | 24 | 20 | 4 | D_{2d} | ||||||
91 | Bilunabirotunda | 14 | 26 | 14 | 8 | 2 | 4 | D_{2h} | |||||
92 | Triangular hebesphenorotunda | 18 | 36 | 20 | 13 | 3 | 3 | 1 | C_{3v} |
See also
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.
External links
- Sylvain Gagnon, "Convex polyhedra with regular faces", Structural Topology, No. 6, 1982, 83-95.
- Paper Models of Polyhedra Many links
- Johnson Solids by George W. Hart.
- Images of all 92 solids, categorized, on one page
- Weisstein, Eric W., "Johnson Solid" from MathWorld.
- VRML models
- VRML models of Johnson Solids by Vladimir Bulatov
ca:Políedre de Johnson eo:Solido de Johnson it:Solido di Johnson nl:Johnson-lichaam pt:Sólidos de Johnson th:ทรงตันจอห์นสัน