Johnson solid
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;
- 35 formed by modifying uniform polyhedra, by augmenting with primitives, diminishing, or gyrating; and
- 10 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 |
See also
References
- ^ Araki, Yoshiaki; Horiyama, Takashi; Uehara, Ryuhei (2015). "Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid". In Rahman, M. Sohel; Tomita, Etsuji (eds.). WALCOM: Algorithms and Computation. Lecture Notes in Computer Science. Vol. 8973. Cham: Springer International Publishing. pp. 294–305. doi:10.1007/978-3-319-15612-5_26. ISBN 978-3-319-15612-5.
- ^ 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.
- ^ a b Todesco, Gian Marco (2020). "Hyperbolic Honeycomb". In Emmer, Michele; Abate, Marco (eds.). Imagine Math 7: Between Culture and Mathematics. Springer. p. 282. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8.
- ^ a b Williams, Kim; Monteleone, Cosino (2021). Daniele Barbaro's Perspective of 1568. Springer. p. 23. doi:10.1007/978-3-030-76687-0. ISBN 978-3-030-76687-0.
- ^ Buldygin, V. V.; Kharazishvili, A. B. (2000). Geometric Aspects of Probability Theory and Mathematical Statistics. Springer. p. 2. doi:10.1007/978-94-017-1687-1. ISBN 978-94-017-1687-1.
- ^ Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
- ^ Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5.
- ^ Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.
- ^ Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
- ^ Kaplan, Craig S.; Hart, George W. (2001). "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons" (PDF). Bridges: Mathematical Connections in Art, Music and Science: 21–28.
- ^ a b c Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
External links
- Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières" [Convex polyhedra with regular faces] (PDF). Structural Topology (6): 83–95.
- Paper Models of Polyhedra Archived 2013-02-26 at the Wayback Machine 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
- Weisstein, Eric W. "Johnson Solid". MathWorld.
- 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 Archived 2020-10-31 at the Wayback Machine (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.