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Johnson solid

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons and that is not a uniform polyhedron. There are 92 such solids:

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. 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 whose faces are all regular polygons, but not a uniform polyhedron; the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms.

The solids are named after Norman Johnson and Victor Zalgaller. published a list of 92 such solids and assigned them their names and numbers. proved Johnson's conjecture 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.

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:

  • 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.

Enumeration

- invalid, - Platonic, - Archimedean, - Gyrated sections.

See also

References

External links