 Johnson solid

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 squarebased 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 degree5 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 vertexuniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 triangle. However, it is not vertextransitive, 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 rotunda), together with the Platonic and Archimedean solids, prisms, and antiprisms.
 Bi means that two copies of the solid in question are joined basetobase. 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
J_{n} Solid name Net Image V E F F_{3} F_{4} F_{5} F_{6} F_{8} F_{10} Symmetry group 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} 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 bipyramid 5 9 6 6 D_{3h} 13 Pentagonal bipyramid 7 15 10 10 D_{5h} 14 Elongated triangular bipyramid 8 15 9 6 3 D_{3h} 15 Elongated square bipyramid
(or biaugmented cube)10 20 12 8 4 D_{4h} 16 Elongated pentagonal bipyramid 12 25 15 10 5 D_{5h} 17 Gyroelongated square bipyramid 10 24 16 16 D_{4d} 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} 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} 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} 65 Augmented truncated tetrahedron 15 27 14 8 3 3 C_{3v} 66 Augmented truncated cube 28 48 22 12 5 5 C_{4v} 67 Biaugmented truncated cube 32 60 30 16 10 4 D_{4h} 68 Augmented truncated dodecahedron 65 105 42 25 5 1 11 C_{5v} 69 Parabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 D_{5d} 70 Metabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 C_{2v} 71 Triaugmented truncated dodecahedron 75 135 62 35 15 3 9 C_{3v} 72 Gyrate rhombicosidodecahedron 60 120 62 20 30 12 C_{5v} 73 Parabigyrate rhombicosidodecahedron 60 120 62 20 30 12 D_{5d} 74 Metabigyrate rhombicosidodecahedron 60 120 62 20 30 12 C_{2v} 75 Trigyrate rhombicosidodecahedron 60 120 62 20 30 12 C_{3v} 76 Diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C_{5v} 77 Paragyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C_{5v} 78 Metagyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C_{s} 79 Bigyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C_{s} 80 Parabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 D_{5d} 81 Metabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 C_{2v} 82 Gyrate bidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 C_{s} 83 Tridiminished rhombicosidodecahedron 45 75 32 5 15 9 3 C_{3v} 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} Legend:
 J_{n} – Johnson Solid Number
 Net – Flattened (unfolded) image
 V – Number of Vertices
 E – Number of Edges
 F – Number of Faces (total)
 F_{3} – Number of 3sided Faces
 F_{4} – Number of 4sided Faces
 F_{5} – Number of 5sided Faces
 F_{6} – Number of 6sided Faces
 F_{8} – Number of 8sided Faces
 F_{10} – Number of 10sided Faces
See also
 Nearmiss Johnson solid
 Catalan solid
 Toroidal polyhedron
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, 8395.
 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 of Johnson Solids by Jim McNeill
 VRML models of Johnson Solids by Vladimir Bulatov
Categories: Johnson solids
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