Johnson solid

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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.

Contents

Names [edit]

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 [edit]

Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry group
1 Square pyramid Johnson solid 1 net.png Square pyramid.png 5 8 5 4 1 C4v
(*44)
2 Pentagonal pyramid Johnson solid 2 net.png Pentagonal pyramid.png 6 10 6 5 1 C5v
(*55)
3 Triangular cupola Johnson solid 3 net.png Triangular cupola.png 9 15 8 4 3 1 C3v
(*33)
4 Square cupola Johnson solid 4 net.png Square cupola.png 12 20 10 4 5 1 C4v
(*44)
5 Pentagonal cupola Johnson solid 5 net.png Pentagonal cupola.png 15 25 12 5 5 1 1 C5v
(*55)
6 Pentagonal rotunda Johnson solid 6 net.png Pentagonal rotunda.png 20 35 17 10 6 1 C5v
(*55)
7 Elongated triangular pyramid (or elongated tetrahedron) Johnson solid 7 net.png Elongated triangular pyramid.png 7 12 7 4 3 C3v
(*33)
8 Elongated square pyramid (or augmented cube) Johnson solid 8 net.png Elongated square pyramid.png 9 16 9 4 5 C4v
(*44)
9 Elongated pentagonal pyramid Johnson solid 9 net.png Elongated pentagonal pyramid.png 11 20 11 5 5 1 C5v
(*55)
10 Gyroelongated square pyramid Johnson solid 10 net.png Gyroelongated square pyramid.png 9 20 13 12 1 C4v
(*44)
11 Gyroelongated pentagonal pyramid (or diminished icosahedron) Johnson solid 11 net.png Gyroelongated pentagonal pyramid.png 11 25 16 15 1 C5v
(*55)
12 Triangular bipyramid Johnson solid 12 net.png Triangular dipyramid.png 5 9 6 6 D3h
(*223)
13 Pentagonal bipyramid Johnson solid 13 net.png Pentagonal dipyramid.png 7 15 10 10 D5h
(*225)
14 Elongated triangular bipyramid Johnson solid 14 net.png Elongated triangular dipyramid.png 8 15 9 6 3 D3h
(*223)
15 Elongated square bipyramid
(or biaugmented cube)		 Johnson solid 15 net.png Elongated square dipyramid.png 10 20 12 8 4 D4h
(*224)
16 Elongated pentagonal bipyramid Johnson solid 16 net.png Elongated pentagonal dipyramid.png 12 25 15 10 5 D5h
(*225)
17 Gyroelongated square bipyramid Johnson solid 17 net.png Gyroelongated square dipyramid.png 10 24 16 16 D4d
(2*4)
18 Elongated triangular cupola Johnson solid 18 net.png Elongated triangular cupola.png 15 27 14 4 9 1 C3v
(*33)
19 Elongated square cupola
(diminished rhombicuboctahedron)		 Johnson solid 19 net.png Elongated square cupola.png 20 36 18 4 13 1 C4v
(*44)
20 Elongated pentagonal cupola Johnson solid 20 net.png Elongated pentagonal cupola.svg 25 45 22 5 15 1 1 C5v
(*55)
21 Elongated pentagonal rotunda Johnson solid 21 net.png Elongated pentagonal rotunda.png 30 55 27 10 10 6 1 C5v
(*55)
22 Gyroelongated triangular cupola Johnson solid 22 net.png Gyroelongated triangular cupola.png 15 33 20 16 3 1 C3v
(*33)
23 Gyroelongated square cupola Johnson solid 23 net.png Gyroelongated square cupola.png 20 44 26 20 5 1 C4v
(*44)
24 Gyroelongated pentagonal cupola Johnson solid 24 net.png Gyroelongated pentagonal cupola.png 25 55 32 25 5 1 1 C5v
(*55)
25 Gyroelongated pentagonal rotunda Johnson solid 25 net.png Gyroelongated pentagonal rotunda.png 30 65 37 30 6 1 C5v
(*55)
26 Gyrobifastigium Johnson solid 26 net.png Gyrobifastigium.png 8 14 8 4 4 D2d
(2*2)
27 Triangular orthobicupola
(gyrate cuboctahedron)		 Johnson solid 27 net.png Triangular orthobicupola.png 12 24 14 8 6 D3h
(*223)
28 Square orthobicupola Johnson solid 28 net.png Square orthobicupola.png 16 32 18 8 10 D4h
(*224)
29 Square gyrobicupola Johnson solid 29 net.png Square gyrobicupola.png 16 32 18 8 10 D4d
(2*4)
30 Pentagonal orthobicupola Johnson solid 30 net.png Pentagonal orthobicupola.png 20 40 22 10 10 2 D5h
(*225)
31 Pentagonal gyrobicupola Johnson solid 31 net.png Pentagonal gyrobicupola.png 20 40 22 10 10 2 D5d
(2*5)
32 Pentagonal orthocupolarotunda Johnson solid 32 net.png Pentagonal orthocupolarotunda.png 25 50 27 15 5 7 C5v
(*55)
33 Pentagonal gyrocupolarotunda Johnson solid 33 net.png Pentagonal gyrocupolarotunda.png 25 50 27 15 5 7 C5v
(*55)
34 Pentagonal orthobirotunda
(gyrate icosidodecahedron)		 Johnson solid 34 net.png Pentagonal orthobirotunda.png 30 60 32 20 12 D5h
(*225)
35 Elongated triangular orthobicupola Johnson solid 35 net.png Elongated triangular orthobicupola.png 18 36 20 8 12 D3h
(*223)
36 Elongated triangular gyrobicupola Johnson solid 36 net.png Elongated triangular gyrobicupola.png 18 36 20 8 12 D3d
(2*3)
37 Elongated square gyrobicupola
(gyrate rhombicuboctahedron)		 Johnson solid 37 net.png Elongated square gyrobicupola.png 24 48 26 8 18 D4d
(2*4)
38 Elongated pentagonal orthobicupola Johnson solid 38 net.png Elongated pentagonal orthobicupola.png 30 60 32 10 20 2 D5h
(*225)
39 Elongated pentagonal gyrobicupola Johnson solid 39 net.png Elongated pentagonal gyrobicupola.png 30 60 32 10 20 2 D5d
(2*5)
40 Elongated pentagonal orthocupolarotunda Johnson solid 40 net.png Elongated pentagonal orthocupolarotunda.png 35 70 37 15 15 7 C5v
(*55)
41 Elongated pentagonal gyrocupolarotunda Johnson solid 41 net.png Elongated pentagonal gyrocupolarotunda.png 35 70 37 15 15 7 C5v
(*55)
42 Elongated pentagonal orthobirotunda Johnson solid 42 net.png Elongated pentagonal orthobirotunda.png 40 80 42 20 10 12 D5h
(*225)
43 Elongated pentagonal gyrobirotunda Johnson solid 43 net.png Elongated pentagonal gyrobirotunda.png 40 80 42 20 10 12 D5d
(2*5)
44 Gyroelongated triangular bicupola
(2 chiral forms)		 Johnson solid 44 net.png Gyroelongated triangular bicupola.png 18 42 26 20 6 D3
(223)
45 Gyroelongated square bicupola
(2 chiral forms)		 Johnson solid 45 net.png Gyroelongated square bicupola.png 24 56 34 24 10 D4
(224)
46 Gyroelongated pentagonal bicupola
(2 chiral forms)		 Johnson solid 46 net.png Gyroelongated pentagonal bicupola.png 30 70 42 30 10 2 D5
(225)
47 Gyroelongated pentagonal cupolarotunda
(2 chiral forms)		 Johnson solid 47 net.png Gyroelongated pentagonal cupolarotunda.png 35 80 47 35 5 7 C5<NR>(55)
48 Gyroelongated pentagonal birotunda
(2 chiral forms)		 Johnson solid 48 net.png Gyroelongated pentagonal birotunda.png 40 90 52 40 12 D5
(225)
49 Augmented triangular prism Johnson solid 49 net.png Augmented triangular prism.png 7 13 8 6 2 C2v
(*22)
50 Biaugmented triangular prism Johnson solid 50 net.png Biaugmented triangular prism.png 8 17 11 10 1 C2v
(*22)
51 Triaugmented triangular prism Johnson solid 51 net.png Triaugmented triangular prism.png 9 21 14 14 D3h
(*223)
52 Augmented pentagonal prism Johnson solid 52 net.png Augmented pentagonal prism.png 11 19 10 4 4 2 C2v
(*22)
53 Biaugmented pentagonal prism Johnson solid 53 net.png Biaugmented pentagonal prism.png 12 23 13 8 3 2 C2v
(*22)
54 Augmented hexagonal prism Johnson solid 54 net.png Augmented hexagonal prism.png 13 22 11 4 5 2 C2v
(*22)
55 Parabiaugmented hexagonal prism Johnson solid 55 net.png Parabiaugmented hexagonal prism.png 14 26 14 8 4 2 D2h
(*222)
56 Metabiaugmented hexagonal prism Johnson solid 56 net.png Metabiaugmented hexagonal prism.png 14 26 14 8 4 2 C2v
(*22)
57 Triaugmented hexagonal prism Johnson solid 57 net.png Triaugmented hexagonal prism.png 15 30 17 12 3 2 D3h
(*223)
58 Augmented dodecahedron Johnson solid 58 net.png Augmented dodecahedron.png 21 35 16 5 11 C5v
(*55)
59 Parabiaugmented dodecahedron Johnson solid 59 net.png Parabiaugmented dodecahedron.png 22 40 20 10 10 D5d
(2*5)
60 Metabiaugmented dodecahedron Johnson solid 60 net.png Metabiaugmented dodecahedron.png 22 40 20 10 10 C2v
(*22)
61 Triaugmented dodecahedron Johnson solid 61 net.png Triaugmented dodecahedron.png 23 45 24 15 9 C3v
(*33)
62 Metabidiminished icosahedron Johnson solid 62 net.png Metabidiminished icosahedron.png 10 20 12 10 2 C2v
(*22)
63 Tridiminished icosahedron Johnson solid 63 net.png Tridiminished icosahedron.png 9 15 8 5 3 C3v
(*33)
64 Augmented tridiminished icosahedron Johnson solid 64 net.png Augmented tridiminished icosahedron.png 10 18 10 7 3 C3v
(*33)
65 Augmented truncated tetrahedron Johnson solid 65 net.png Augmented truncated tetrahedron.png 15 27 14 8 3 3 C3v
(*33)
66 Augmented truncated cube Johnson solid 66 net.png Augmented truncated cube.png 28 48 22 12 5 5 C4v
(*44)
67 Biaugmented truncated cube Johnson solid 67 net.png Biaugmented truncated cube.png 32 60 30 16 10 4 D4h
(*224)
68 Augmented truncated dodecahedron Johnson solid 68 net.png Augmented truncated dodecahedron.png 65 105 42 25 5 1 11 C5v
(*55)
69 Parabiaugmented truncated dodecahedron Johnson solid 69 net.png Parabiaugmented truncated dodecahedron.png 70 120 52 30 10 2 10 D5d
(2*5)
70 Metabiaugmented truncated dodecahedron Johnson solid 70 net.png Metabiaugmented truncated dodecahedron.png 70 120 52 30 10 2 10 C2v
(*22)
71 Triaugmented truncated dodecahedron Johnson solid 71 net.png Triaugmented truncated dodecahedron.png 75 135 62 35 15 3 9 C3v
(*33)
72 Gyrate rhombicosidodecahedron Johnson solid 72 net.png Gyrate rhombicosidodecahedron.png 60 120 62 20 30 12 C5v
(*55)
73 Parabigyrate rhombicosidodecahedron Johnson solid 73 net.png Parabigyrate rhombicosidodecahedron.png 60 120 62 20 30 12 D5d
(2*5)
74 Metabigyrate rhombicosidodecahedron Johnson solid 74 net.png Metabigyrate rhombicosidodecahedron.png 60 120 62 20 30 12 C2v
(*22)
75 Trigyrate rhombicosidodecahedron Johnson solid 75 net.png Trigyrate rhombicosidodecahedron.png 60 120 62 20 30 12 C3v
(*33)
76 Diminished rhombicosidodecahedron Johnson solid 76 net.png Diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 C5v
(*55)
77 Paragyrate diminished rhombicosidodecahedron Johnson solid 77 net.png Paragyrate diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 C5v
(*55)
78 Metagyrate diminished rhombicosidodecahedron Johnson solid 78 net.png Metagyrate diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 Cs
(*11)
79 Bigyrate diminished rhombicosidodecahedron Johnson solid 79 net.png Bigyrate diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 Cs
(*11)
80 Parabidiminished rhombicosidodecahedron Johnson solid 80 net.png Parabidiminished rhombicosidodecahedron.png 50 90 42 10 20 10 2 D5d
(2*5)
81 Metabidiminished rhombicosidodecahedron Johnson solid 81 net.png Metabidiminished rhombicosidodecahedron.png 50 90 42 10 20 10 2 C2v
(*22)
82 Gyrate bidiminished rhombicosidodecahedron Johnson solid 82 net.png Gyrate bidiminished rhombicosidodecahedron.png 50 90 42 10 20 10 2 Cs
(*11)
83 Tridiminished rhombicosidodecahedron Johnson solid 83 net.png Tridiminished rhombicosidodecahedron.png 45 75 32 5 15 9 3 C3v
(*33)
84 Snub disphenoid
(Siamese dodecahedron)		 Johnson solid 84 net.png Snub disphenoid.png 8 18 12 12 D2d
(2*2)
85 Snub square antiprism Johnson solid 85 net.png Snub square antiprism.png 16 40 26 24 2 D4d
(2*4)
86 Sphenocorona Johnson solid 86 net.png Sphenocorona.png 10 22 14 12 2 C2v
(*22)
87 Augmented sphenocorona Johnson solid 87 net.png Augmented sphenocorona.png 11 26 17 16 1 Cs
(*11)
88 Sphenomegacorona Johnson solid 88 net.png Sphenomegacorona.png 12 28 18 16 2 C2v
(*22)
89 Hebesphenomegacorona Johnson solid 89 net.png Hebesphenomegacorona.png 14 33 21 18 3 C2v
(*22)
90 Disphenocingulum Johnson solid 90 net.png Disphenocingulum.png 16 38 24 20 4 D2d
(2*2)
91 Bilunabirotunda Johnson solid 91 net.png Bilunabirotunda.png 14 26 14 8 2 4 D2h
(*222)
92 Triangular hebesphenorotunda Johnson solid 92 net.png Triangular hebesphenorotunda.png 18 36 20 13 3 3 1 C3v
(*33)

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

See also [edit]

References [edit]

  • 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 [edit]

Johnson solid navigator
Prismatoids and rotundae
Modified pyramids and dipyramids
Modified cupolae and rotundae
Augmented prisms
Modified Platonic solids
Modified Archimedean solids
Miscellaneous
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