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.[3]Template:R/superscript[4]Template:R/superscript There are 92 such solids:
- 48 composed of the primitive pyramids, cupolas, and rotundas assembled in various ways together with prisms and antiprisms;
- 35 formed by modifying uniform polyhedra, by augmenting, diminishing, or gyrating with primitives; and
- 9 which are not derived from "cut-and-paste" manipulations of uniform solids.
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.[5]Template:R/superscript 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]Template:R/superscript but not a uniform polyhedron;[3]Template:R/superscript[4]Template:R/superscript the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms.
The solids are named after Norman Johnson and Victor Zalgaller.[7]Template:R/superscript 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".[8]Template:R/superscript proved Johnson's conjecture[9]Template:R/superscript 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]Template:R/superscript
Naming and construction of solids
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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. The names of the Johnson solids are described in the following sections.
Pyramids, cupolas, rotundas
The first 48 Johnson solids are constructed from pyramids, cupolas, or rotundas, combined with prisms or antiprisms. The following prefixes are attached to the word to indicate specific combinations of shapes:[11]Template:R/superscript
- Bi- indicates that two copies of the solid are joined base-to-base.
- For cupolas and rotundas, ortho- indicates that like faces meet.
- For cupolas and rotundas, gyro- indicates that unlike faces meet.
- Elongated indicates a prism is joined to the base of the solid, or between the bases.
- Gyroelongated indicates an antiprism is joined to the base of the solid, or between the bases.
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.
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| Fastigium | |
|---|---|
| gyrobi- | 26 Gyrobifastigium File:Gyrobifastigium.png |
Modified uniform polyhedra
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The next 35 Johnson solids are constructed by modifying uniform polyhedra such as prisms, Platonic, or Archimedean solids by adding, subtracting, or rotating pyramids or cupolas. The following prefixes are attached to the word to indicate additions, subtractions, or rotations:[11]Template:R/superscript
- Augmented indicates a pyramid or cupola is added 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.
The 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 cupolas, and a tridiminished solid has three removed pyramids or cupolas. 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 cupolas gyrated.[11]Template:R/superscript
Non cut-and-paste
The last 9 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]Template:R/superscript
- A lune is a figure of two triangles attached to opposite sides of a square. Prefixes indicating a complex of lunes are:
- Spheno- is a wedgelike complex of two adjacent lunes. Dispheno- indicates two such complexes.
- Hebespheno- is a blunt complex of three adjacent lunes.
- Suffixes indicating a complex of triangles are:
- -corona is a crownlike complex of eight triangles.
- -megacorona is a larger crownlike complex of twelve triangles.
- -cingulum is a belt of twelve triangles.
- Suffix -rotunda indicates a complex of two or three pentagons with triangles between them, bearing a structural resemblance to the pentagonal rotunda.
Notable subsets
Deltahedra
Script error: No such module "labelled list hatnote". 5 Johnson solids are deltahedra, with only triangle faces.
Chiral solids
Script error: No such module "labelled list hatnote". The 5 gyroelongated bicupolas or birotundas are chiral and have distinct left-handed and right-handed forms.
Circumscribable solids
Script error: No such module "labelled list hatnote". 25 of the Johnson solids have vertices that exist on the surface of a sphere. All of them can be seen to be related to a Platonic or Archimedean solid by gyration, diminishment, or dissection.[12]
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.