Johnson solid

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas, and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not.

Definition and background

A Johnson solid is a convex polyhedron whose faces are all regular polygons. The convex polyhedron means as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not uniform. This means that a Johnson solid is not a Platonic solid, Archimedean solid, prism, or antiprism. A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a near-miss Johnson solid.

The solids were named after the mathematicians Norman Johnson and Victor Zalgaller. published a list including ninety-two solids—excluding the five Platonic solids, the thirteen Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms—and gave them their names and numbers. He did not prove that there were only ninety-two, but he did conjecture that there were no others. proved that Johnson's list was complete.

Naming and enumeration

The naming of Johnson solids follows a flexible and precise descriptive formula that allows many solids to be named in multiple different ways without compromising the accuracy of each name as a description. Most Johnson solids can be constructed from the first few solids ( pyramids, cupolae, and a rotunda), together with the Platonic and Archimedean solids, prisms, and antiprisms; the center of a particular solid's name will reflect these ingredients. From there, a series of prefixes are attached to the word to indicate additions, rotations, and transformations:
* ''Bi-'' indicates that two copies of the solid are joined base-to-base. For cupolae and rotundas, the solids can be joined so that either like faces (''ortho-'') or unlike faces (''gyro-'') meet. Using this nomenclature, a pentagonal bipyramid is a solid constructed by attaching two bases of pentagonal pyramids. Triangular orthobicupola is constructed by two triangular cupolas along their bases.
* ''Elongated'' indicates a prism is joined to the base of the solid, or between the bases; ''gyroelongated'' indicates an antiprism. ''Augmented'' indicates another polyhedron, namely a pyramid or cupola, is joined to one or more faces of the solid in question.
*''Diminished'' indicates a pyramid or cupola is removed from one or more faces of the solid in question.
*'' Gyrate'' indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, as in the difference between ortho- and gyrobicupolae.

The last three operations—''augmentation'', ''diminution'', and ''gyration''—can be performed multiple times for certain large solids. ''Bi-'' & ''Tri-'' indicate a double and triple operation respectively. For example, a ''bigyrate'' solid has two rotated cupolae, and a ''tridiminished'' solid has three removed pyramids or cupolae. In certain large solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. ''Para-'' indicates the former, that the solid in question has altered parallel faces, and ''meta-'' the latter, altered oblique faces. For example, a ''parabiaugmented'' solid has had two parallel faces augmented, and a ''metabigyrate'' solid has had two oblique faces gyrated.

The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson with the following nomenclature:
*A ''lune'' is a complex of two triangles attached to opposite sides of a square.
*''Spheno''- indicates a wedgelike complex formed by two adjacent lunes. ''Dispheno-'' indicates two such complexes.
*''Hebespheno''- indicates a blunt complex of two lunes separated by a third lune.
*''Corona'' is a crownlike complex of eight triangles.
*''Megacorona'' is a larger crownlike complex of twelve triangles.
*The suffix -''cingulum'' indicates a belt of twelve triangles.

The enumeration of Johnson solids may be denoted as $J_n$, where $n$ denoted the list's enumeration (an example is $J_1$ denoted the first Johnson solid, the equilateral square pyramid). The following is the list of ninety-two Johnson solids, with the enumeration followed according to the list of :

Some of the Johnson solids may be categorized as elementary polyhedra. This means the polyhedron cannot be separated by a plane to create two small convex polyhedra with regular faces; examples of Johnson solids are the first six Johnson solids—square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda— tridiminished icosahedron, parabidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron, snub disphenoid, snub square antiprism, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum, bilunabirotunda, and triangular hebesphenorotunda. The other Johnson solids are composite polyhedron because they are constructed by attaching some elementary polyhedra.

Properties

As the definition above, a Johnson solid is a convex polyhedron with regular polygons as their faces. However, there are several properties possessed by each of them.
* All but five of the 92 Johnson solids are known to have the Rupert property, meaning that it is possible for a larger copy of themselves to pass through a hole inside of them. The five which are not known to have this property are: gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, and paragyrate diminished rhombicosidodecahedron.
* From all of the Johnson solids, the elongated square gyrobicupola (also called the pseudorhombicuboctahedron) is unique in being locally vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one 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.

See also

* List of Johnson solids
* Near-miss Johnson solid
* Blind polytope

References

External links

*
Paper Models of Polyhedra
Many links

by George W. Hart.

*

by Jim McNeill
VRML models of Johnson Solids
by Vladimir Bulatov
CRF polychora discovery project
attempts to discove
CRF polychora
(''C''onvex 4-dimensional polytopes with ''R''egular polygons as 2-dimensional ''F''aces), a generalization of the Johnson solids to 4-dimensional space
*https://levskaya.github.io/polyhedronisme/ a generator of polyhedrons and Conway operations applied to them, including Johnson solids.

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