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[2] are regular polygons and that is not a uniform polyhedron.[3][4] There are 92 such solids:
- 47 composed of the elementary pyramids, cupolas, and rotunda assembled in various ways together with prisms and antiprisms;
- 36 formed by modifying uniform polyhedra, by augmenting with primitives, diminishing, or gyrating; and
- 9 others.
Definition and background
A convex polyhedron is the convex hull of a finite set of points in 3-dimensional space, not all in a plane.[5] Its boundary is a finite union of polygons, no two in the same plane; those polygons are called the faces. A Johnson solid is a convex polyhedron[2] whose faces are all regular polygons,[6] but not a uniform polyhedron;[3][4] the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms.
The solids are named after Norman Johnson and Victor Zalgaller.[7] Johnson (1966) published a list of 92 such solids and assigned them their names and numbers. Zalgaller (1969)[8] proved Johnson's conjecture[9] that there were none beyond these 92.
A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a near-miss Johnson solid.[10]
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:[11]
- 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.[11]
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:[11]
- 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 |