Johnson solid
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Template:Mergefrom 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.Template:R There are 92 such solids:
- 47 composed of the elementary pyramids, cupolas, and rotunda assembled in various ways together with prisms and antiprisms;
- 35 formed by modifying uniform polyhedra, by augmenting with primitives, diminishing, or gyrating; and
- 10 others.
Definition and background
Script error: No such module "Multiple image". A convex polyhedron is the convex hull of a finite set of points in 3-dimensional space, not all in a plane.Template:R 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,Template:R but not a uniform polyhedron;Template:R the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms.
The solids are named after Norman Johnson and Victor Zalgaller.Template:R Script error: No such module "Footnotes". published a list of 92 such solids and assigned them their names and numbers. Script error: No such module "Footnotes".Template:R proved Johnson's conjectureTemplate:R 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.Template:R
Naming scheme
Template:Main article Script error: No such module "Multiple image". 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:Template:R
- 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.
Script error: No such module "Multiple image". 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.Template:R
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:Template:R
- 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
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| 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 |
See also
References
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- ↑ a b By definition, each face is the intersection of the convex polyhedron with a different bounding plane, so no two faces are coplanar — any two adjacent faces form an angle less than 180 degrees. If instead a convex polyhedron is presented by giving a collection of polygons that a priori may be coplanar (e.g., by subdividing a face), one could write "strictly convex polyhedron" here to indicate the condition that no two of the polygons are coplanar, that no two meet in a 180-degree angle. This notion of "strictly convex" for polyhedra is not the same as the standard notion used for general convex sets: no convex polyhedra are strictly convex in the latter sense; see p. 263 of A. G. Khovanskii, Geometry of generalized virtual polyhedra, J. Math. Sciences 269 (2023), 256–269.
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External links
- Script error: No such module "Citation/CS1".
- Paper Models of Polyhedra Script error: No such module "webarchive". Many links
- Johnson Solids by George W. Hart.
- Visual Polyhedra, with 3D models and data for all 92 solids, by David I. McCooey.
- Images of all 92 solids, categorized, on one page
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- VRML models of Johnson Solids by Jim McNeill
- VRML models of Johnson Solids by Vladimir Bulatov
- CRF polychora discovery project attempts to discover CRF polychora Script error: No such module "webarchive". (Convex 4-dimensional polytopes with Regular polygons as 2-dimensional Faces), 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.