Johnson solid
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][4] There are 92 such solids:
- 48 composed of the elementary pyramids, cupolas, and rotundas assembled in various ways together with prisms and antiprisms;
- 35 formed by modifying uniform polyhedra, by augmenting with primitives, diminishing, or gyrating; and
- 9 which are not derived from "cut-and-paste" manipulations of uniform solids.
Definition and background
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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 and construction of solids
[edit | edit source]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.
Elementary combinations
[edit | edit source]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]
- 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.
- invalid, - Platonic, - Archimedean.
| Fastigium | |
|---|---|
| gyrobi- | 26 Gyrobifastigium File:Gyrobifastigium.png |
Modified uniform polyhedra
[edit | edit source]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]
- 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]
Non cut-and-paste
[edit | edit source]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]
- 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.
| Snub polyhedra | |||
|---|---|---|---|
| 84 Snub disphenoid File:Snub disphenoid.png |
85 Snub square antiprism File:Snub square antiprism.png | ||
| Rotundoids | |
|---|---|
| 91 Bilunabirotunda File:Bilunabirotunda.png |
92 Triangular hebesphenorotunda File:Triangular hebesphenorotunda.png |
See also
[edit | edit source]References
[edit | edit source]- ↑ 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. 8973. Cham: Springer International Publishing. pp. 294–305. doi:10.1007/978-3-319-15612-5_26. ISBN 978-3-319-15612-5.
- ↑ 2.0 2.1 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.
- ↑ 3.0 3.1 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.
- ↑ 4.0 4.1 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. 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.
- ↑ 11.0 11.1 11.2 11.3 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
[edit | edit source]- 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.