Johnson solid
Convex polyhedron with regular faces
From Wikipedia, the free encyclopedia
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid,[1] is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedra. There are 92 solids with such a property: 47 are composed of the elementary pyramids, cupolas, and rotunda assembled in various ways together with prisms and antiprisms. 36 more are formed by modifying uniform polyhedra; Either augmenting with primitives, diminishing, or gyrating. Finally, 9 others can be formed in a variety of ways.
Definition and background
A Johnson solid is a convex polyhedron whose faces are all regular polygons.[2] The convex polyhedron means as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points.[3] Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors require that Johnson solids are not uniform. This means that a Johnson solid is neither a Platonic solid, Archimedean solid, prism, nor antiprism.[4][5] A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a near-miss Johnson solid.[6]
The solids were named after the mathematicians Norman Johnson and Victor Zalgaller.[7] Johnson (1966) published a list including 92 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 92, but he did conjecture that there were no others.[8] Zalgaller (1969) proved that Johnson's list was complete.[9]
Naming scheme
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:[10]
- 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.[10]
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:[10]
- 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.
Enumeration
- invalid, - Platonic, - Archimedean, - Gyrated sections.
| Odd Ones Out | |||
|---|---|---|---|
| 26 Gyrobifastigium |
84 Snub disphenoid |
85 Snub square antiprism |
90 Disphenocingulum |
| Corona family | |||
|---|---|---|---|
| 86 Sphenocorona |
87 Augmented sphenocorona |
88 Sphenomegacorona |
89 Hebesphenomegacorona |
| Rotundoid | |
|---|---|
| 91 Bilunabirotunda |
92 Triangular hebesphenorotunda |