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Elongated square gyrobicupola

The elongated square gyrobicupola (J37), a Johnson solid

Stella octangula

This 24 equilateral triangle example is not a Johnson solid because it is not convex. (This particular regular polyhedron is actually a stellation, the only possible one for the octahedron.)

Partial cubic honeycomb

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 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

Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry
1 Square pyramid Johnson solid 1 net Square pyramid 5 8 5 4 1 C4v
2 Pentagonal pyramid Johnson solid 2 net Pentagonal pyramid 6 10 6 5 1 C5v
3 Triangular cupola Johnson solid 3 net Triangular cupola 9 15 8 4 3 1 C3v
4 Square cupola Johnson solid 4 net Square cupola 12 20 10 4 5 1 C4v
5 Pentagonal cupola Johnson solid 5 net Pentagonal cupola 15 25 12 5 5 1 1 C5v
6 Pentagonal rotunda Johnson solid 6 net Pentagonal rotunda 20 35 17 10 6 1 C5v

Modified pyramids and dipyramids

Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry
7 Elongated triangular pyramid (or elongated tetrahedron) Johnson solid 7 net Elongated triangular pyramid 7 12 7 4 3 C3v
8 Elongated square pyramid (or augmented cube) Johnson solid 8 net Elongated square pyramid 9 16 9 4 5 C4v
9 Elongated pentagonal pyramid Johnson solid 9 net Elongated pentagonal pyramid 11 20 11 5 5 1 C5v
10 Gyroelongated square pyramid Johnson solid 10 net Gyroelongated square pyramid 9 20 13 12 1 C4v
11 Gyroelongated pentagonal pyramid (or diminished icosahedron) Johnson solid 11 net Gyroelongated pentagonal pyramid 11 25 16 15 1 C5v
12 Triangular dipyramid Johnson solid 12 net Triangular dipyramid 5 9 6 6 D3h
13 Pentagonal dipyramid Johnson solid 13 net Pentagonal dipyramid 7 15 10 10 D5h
14 Elongated triangular dipyramid Johnson solid 14 net Elongated triangular dipyramid 8 15 9 6 3 D3h
15 Elongated square dipyramid
(or biaugmented cube)
Johnson solid 15 net Elongated square dipyramid 10 20 12 8 4 D4h
16 Elongated pentagonal dipyramid Johnson solid 16 net Elongated pentagonal dipyramid 12 25 15 10 5 D5h
17 Gyroelongated square dipyramid Johnson solid 17 net Gyroelongated square dipyramid 10 24 16 16 D4d

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
Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry
18 Elongated triangular cupola Johnson solid 18 net Elongated triangular cupola 15 27 14 4 9 1 C3v
19 Elongated square cupola
(diminished rhombicuboctahedron)
Johnson solid 19 net Elongated square cupola 20 36 18 4 13 1 C4v
20 Elongated pentagonal cupola Johnson solid 20 net Elongated pentagonal cupola 25 45 22 5 15 1 1 C5v
21 Elongated pentagonal rotunda Johnson solid 21 net Elongated pentagonal rotunda 30 55 27 10 10 6 1 C5v
22 Gyroelongated triangular cupola Johnson solid 22 net Gyroelongated triangular cupola 15 33 20 16 3 1 C3v
23 Gyroelongated square cupola Johnson solid 23 net Gyroelongated square cupola 20 44 26 20 5 1 C4v
24 Gyroelongated pentagonal cupola Johnson solid 24 net Gyroelongated pentagonal cupola 25 55 32 25 5 1 1 C5v
25 Gyroelongated pentagonal rotunda Johnson solid 25 net Gyroelongated pentagonal rotunda 30 65 37 30 6 1 C5v
26 Gyrobifastigium Johnson solid 26 net Gyrobifastigium 8 14 8 4 4 D2d
27 Triangular orthobicupola
(gyrate cuboctahedron)
Johnson solid 27 net Triangular orthobicupola 12 24 14 8 6 D3h
28 Square orthobicupola Johnson solid 28 net Square orthobicupola 16 32 18 8 10 D4h
29 Square gyrobicupola Johnson solid 29 net Square gyrobicupola 16 32 18 8 10 D4d
30 Pentagonal orthobicupola Johnson solid 30 net Pentagonal orthobicupola 20 40 22 10 10 2 D5h
31 Pentagonal gyrobicupola Johnson solid 31 net Pentagonal gyrobicupola 20 40 22 10 10 2 D5d
32 Pentagonal orthocupolarotunda Johnson solid 32 net Pentagonal orthocupolarotunda 25 50 27 15 5 7 C5v
33 Pentagonal gyrocupolarotunda Johnson solid 33 net Pentagonal gyrocupolarotunda 25 50 27 15 5 7 C5v
34 Pentagonal orthobirotunda
(gyrate icosidodecahedron)
Johnson solid 34 net Pentagonal orthobirotunda 30 60 32 20 12 D5h
35 Elongated triangular orthobicupola Johnson solid 35 net Elongated triangular orthobicupola 18 36 20 8 12 D3h
36 Elongated triangular gyrobicupola Johnson solid 36 net Elongated triangular gyrobicupola 18 36 20 8 12 D3d
37 Elongated square gyrobicupola
(gyrate rhombicuboctahedron)
Johnson solid 37 net Elongated square gyrobicupola 24 48 26 8 18 D4d
38 Elongated pentagonal orthobicupola Johnson solid 38 net Elongated pentagonal orthobicupola 30 60 32 10 20 2 D5h
39 Elongated pentagonal gyrobicupola Johnson solid 39 net Elongated pentagonal gyrobicupola 30 60 32 10 20 2 D5d
40 Elongated pentagonal orthocupolarotunda Johnson solid 40 net Elongated pentagonal orthocupolarotunda 35 70 37 15 15 7 C5v
41 Elongated pentagonal gyrocupolarotunda Johnson solid 41 net Elongated pentagonal gyrocupolarotunda 35 70 37 15 15 7 C5v
42 Elongated pentagonal orthobirotunda Johnson solid 42 net Elongated pentagonal orthobirotunda 40 80 42 20 10 12 D5h
43 Elongated pentagonal gyrobirotunda Johnson solid 43 net Elongated pentagonal gyrobirotunda 40 80 42 20 10 12 D5d
44 Gyroelongated triangular bicupola
(2 chiral forms)
Johnson solid 44 net Gyroelongated triangular bicupola 18 42 26 20 6 D3
45 Gyroelongated square bicupola
(2 chiral forms)
Johnson solid 45 net Gyroelongated square bicupola 24 56 34 24 10 D4
46 Gyroelongated pentagonal bicupola
(2 chiral forms)
Johnson solid 46 net Gyroelongated pentagonal bicupola 30 70 42 30 10 2 D5
47 Gyroelongated pentagonal cupolarotunda
(2 chiral forms)
Johnson solid 47 net Gyroelongated pentagonal cupolarotunda 35 80 47 35 5 7 C5
48 Gyroelongated pentagonal birotunda
(2 chiral forms)
Johnson solid 48 net Gyroelongated pentagonal birotunda 40 90 52 40 12 D5

Augmented prisms

Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry
49 Augmented triangular prism Johnson solid 49 net Augmented triangular prism 7 13 8 6 2 C2v
50 Biaugmented triangular prism Johnson solid 50 net Biaugmented triangular prism 8 17 11 10 1 C2v
51 Triaugmented triangular prism Johnson solid 51 net Triaugmented triangular prism 9 21 14 14 D3h
52 Augmented pentagonal prism Johnson solid 52 net Augmented pentagonal prism 11 19 10 4 4 2 C2v
53 Biaugmented pentagonal prism Johnson solid 53 net Biaugmented pentagonal prism 12 23 13 8 3 2 C2v
54 Augmented hexagonal prism Johnson solid 54 net Augmented hexagonal prism 13 22 11 4 5 2 C2v
55 Parabiaugmented hexagonal prism Johnson solid 55 net Parabiaugmented hexagonal prism 14 26 14 8 4 2 D2h
56 Metabiaugmented hexagonal prism Johnson solid 56 net Metabiaugmented hexagonal prism 14 26 14 8 4 2 C2v
57 Triaugmented hexagonal prism Johnson solid 57 net Triaugmented hexagonal prism 15 30 17 12 3 2 D3h

Modified Platonic solids

  • Augmented dodecahedrons
  • Diminished icosahedrons
Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry
58 Augmented dodecahedron Johnson solid 58 net Augmented dodecahedron 21 35 16 5 11 C5v
59 Parabiaugmented dodecahedron Johnson solid 59 net Parabiaugmented dodecahedron 22 40 20 10 10 D5d
60 Metabiaugmented dodecahedron Johnson solid 60 net Metabiaugmented dodecahedron 22 40 20 10 10 C2v
61 Triaugmented dodecahedron Johnson solid 61 net Triaugmented dodecahedron 23 45 24 15 9 C3v
62 Metabidiminished icosahedron Johnson solid 62 net Metabidiminished icosahedron 10 20 12 10 2 C2v
63 Tridiminished icosahedron Johnson solid 63 net Tridiminished icosahedron 9 15 8 5 3 C3v
64 Augmented tridiminished icosahedron Johnson solid 64 net Augmented tridiminished icosahedron 10 18 10 7 3 C3v

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
Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry
65 Augmented truncated tetrahedron Johnson solid 65 net Augmented truncated tetrahedron 15 27 14 8 3 3 C3v
66 Augmented truncated cube Johnson solid 66 net Augmented truncated cube 28 48 22 12 5 5 C4v
67 Biaugmented truncated cube Johnson solid 67 net Biaugmented truncated cube 32 60 30 16 10 4 D4h
68 Augmented truncated dodecahedron Johnson solid 68 net Augmented truncated dodecahedron 65 105 42 25 5 1 11 C5v
69 Parabiaugmented truncated dodecahedron Johnson solid 69 net Parabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 D5d
70 Metabiaugmented truncated dodecahedron Johnson solid 70 net Metabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 C2v
71 Triaugmented truncated dodecahedron Johnson solid 71 net Triaugmented truncated dodecahedron 75 135 62 35 15 3 9 C3v
72 Gyrate rhombicosidodecahedron Johnson solid 72 net Gyrate rhombicosidodecahedron 60 120 62 20 30 12 C5v
73 Parabigyrate rhombicosidodecahedron Johnson solid 73 net Parabigyrate rhombicosidodecahedron 60 120 62 20 30 12 D5d
74 Metabigyrate rhombicosidodecahedron Johnson solid 74 net Metabigyrate rhombicosidodecahedron 60 120 62 20 30 12 C2v
75 Trigyrate rhombicosidodecahedron Johnson solid 75 net Trigyrate rhombicosidodecahedron 60 120 62 20 30 12 C3v
76 Diminished rhombicosidodecahedron Johnson solid 76 net Diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v
77 Paragyrate diminished rhombicosidodecahedron Johnson solid 77 net Paragyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v
78 Metagyrate diminished rhombicosidodecahedron Johnson solid 78 net Metagyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs
79 Bigyrate diminished rhombicosidodecahedron Johnson solid 79 net Bigyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs
80 Parabidiminished rhombicosidodecahedron Johnson solid 80 net Parabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 D5d
81 Metabidiminished rhombicosidodecahedron Johnson solid 81 net Metabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 C2v
82 Gyrate bidiminished rhombicosidodecahedron Johnson solid 82 net Gyrate bidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 Cs
83 Tridiminished rhombicosidodecahedron Johnson solid 83 net Tridiminished rhombicosidodecahedron 45 75 32 5 15 9 3 C3v

Miscellaneous

Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry
84 Snub disphenoid
(Siamese dodecahedron)
Johnson solid 84 net Snub disphenoid 8 18 12 12 D2d
85 Snub square antiprism Johnson solid 85 net Snub square antiprism 16 40 26 24 2 D4d
86 Sphenocorona Johnson solid 86 net Sphenocorona 10 22 14 12 2 C2v
87 Augmented sphenocorona Johnson solid 87 net Augmented sphenocorona 11 26 17 16 1 Cs
88 Sphenomegacorona Johnson solid 88 net Sphenomegacorona 12 28 18 16 2 C2v
89 Hebesphenomegacorona Johnson solid 89 net Hebesphenomegacorona 14 33 21 18 3 C2v
90 Disphenocingulum Johnson solid 90 net Disphenocingulum 16 38 24 20 4 D2d
91 Bilunabirotunda Johnson solid 91 net Bilunabirotunda 14 26 14 8 2 4 D2h
92 Triangular hebesphenorotunda Johnson solid 92 net Triangular hebesphenorotunda 18 36 20 13 3 3 1 C3v

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

nl:Johnson-lichaampt:Sólidos de Johnson th:ทรงตันจอห์นสัน

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