Johnson solid
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Elongated_square_gyrobicupola.png
In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. There is no requirement that each face must be the same polygon. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1); it has one square face and four 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 on any regular polygon's being 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-regular: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle.
The names and Johnson numbers for the solids are:
- square pyramid
- pentagonal pyramid
- triangular cupola
- square cupola
- pentagonal cupola
- pentagonal rotunda
- elongated triangular pyramid
- elongated square pyramid
- elongated pentagonal pyramid
- gyroelongated square pyramid
- gyroelongated pentagonal pyramid
- triangular dipyramid
- pentagonal dipyramid
- elongated triangular dipyramid
- elongated square dipyramid
- elongated pentagonal dipyramid
- gyroelongated square dipyramid
- elongated triangular cupola
- elongated square cupola
- elongated pentagonal cupola
- elongated pentagonal rotunda
- gyroelongated triangular cupola
- gyroelongated square cupola
- gyroelongated pentagonal cupola
- gyroelongated pentagonal rotunda
- gyrobifastigium
- triangular orthobicupola
- square orthobicupola
- square gyrobicupola
- pentagonal orthobicupola
- pentagonal gyrobicupola
- pentagonal orthocupolarotunda
- pentagonal gyrocupolarotunda
- pentagonal orthobirotunda
- elongated triangular orthobicupola
- elongated triangular gyrobicupola
- elongated square gyrobicupola
- elongated pentagonal orthobicupola
- elongated pentagonal gyrobicupola
- elongated pentagonal orthocupolarotunda
- elongated pentagonal gyrocupolarotunda
- elongated pentagonal orthobirotunda
- elongated pentagonal gyrobirotunda
- gyroelongated triangular bicupola
- gyroelongated square bicupola
- gyroelongated pentagonal bicupola
- gyroelongated pentagonal cupolarotunda
- gyroelongated pentagonal birotunda
- augmented triangular prism
- biaugmented triangular prism
- triaugmented triangular prism
- augmented pentagonal prism
- biaugmented pentagonal prism
- augmented hexagonal prism
- parabiaugmented hexagonal prism
- metabiaugmented hexagonal prism
- triaugmented hexagonal prism
- augmented dodecahedron
- parabiaugmented dodecahedron
- metabiaugmented dodecahedron
- triaugmented dodecahedron
- metabidiminished icosahedron
- tridiminished icosahedron
- augmented tridiminished icosahedron
- augmented truncated tetrahedron
- augmented truncated cube
- biaugmented truncated cube
- augmented truncated dodecahedron
- parabiaugmented truncated dodecahedron
- metabiaugmented truncated dodecahedron
- triaugmented truncated dodecahedron
- gyrate rhombicosidodecahedron
- parabigyrate rhombicosidodecahedron
- metabigyrate rhombicosidodecahedron
- trigyrate rhombicosidodecahedron
- diminished rhombicosidodecahedron
- paragyrate diminished rhombicosidodecahedron
- metagyrate diminished rhombicosidodecahedron
- bigyrate diminished rhombicosidodecahedron
- parabidiminished rhombicosidodecahedron
- metabidiminished rhombicosidodecahedron
- gyrate bidiminished rhombicosidodecahedron
- tridiminished rhombicosidodecahedron
- snub disphenoid
- snub square antiprism
- sphenocorona
- augmented sphenocorona
- sphenomegacorona
- hebesphenomegacorona
- disphenocingulum
- bilunabirotunda
- triangular hebesphenorotunda
The names 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.
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.
- Eric W. Weisstein. Johnson Solid (http://mathworld.wolfram.com/JohnsonSolid.html) at MathWorld.
- Template:Book reference The first proof that there are only 92 Johnson solids.
External links
- Paper Models of Polyhedra (http://www.korthalsaltes.com/) Many links
- Johnson Solids (http://www.georgehart.com/virtual-polyhedra/johnson-info.html) by George W. Hart.
- Images of all 92 solids, categorized, on one page (http://www.uwgb.edu/dutchs/symmetry/johnsonp.htm)it:Solido di Johnson