Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There are ninety-two with such property. It is also known as Johnson–Zalgaller solid, and was named after two mathematicians Norman Johnson and Victor Zalgaller.

Description

The following are three examples of solids. The first solid, elongated square gyrobicupola, is Johnson solid because it has the convexity property. The second solid, stella octangula is not Johnson solid because it is not convex, meaning whenever two points are inside of it, the line connecting always be outside. The last solid is not a Johnson solid because it is not strictly convex, meaning every face is planar or the dihedral angles of two adjacent face has 180°.

A Johnson solid is a convex polyhedron in which the faces of each are regular polygon. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not uniform. This means that Johnson solid are not a Platonic solid, Archimedean solid, prism, or antiprism.^[1]^[2] Equivalently, a Johnson solid is defined as the strictly convex polyhedron, meaning that every face is not coplanar.^{[citation needed]} A convex polyhedron in which all faces are close enough to become regular but some of them are not precisely regular is known as near-miss Johnson solid.^[3]

The Johnson solid, sometimes known as Johnson–Zalgaller solid, was named after two mathematicians Norman Johnson and Victor Zalgaller.^[4] Johnson (1966) published a list including ninety-two Johnson solids—excluding the five Platonic solids, the thirteen Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms—and gave them their names and numbers. He did not prove that there were only ninety-two, but he did conjecture that there were no others.^[5] Zalgaller (1969) proved that Johnson's list was complete.^[6]

Some of the Johnson solids do have Rupert property, meaning they do have the polyhedron of the same or larger size that may pass through a hole inside of them. However, the other five Johnson solids do not have this property: gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, and paragyrate diminished rhombicosidodecahedron.^[7] From all of the Johnson solids, the elongated square gyrobicupola (also called the pseudorhombicuboctahedron),^[8] is unique in being locally vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle. However, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid.^[9]^[10]^[11]

Naming and enumeration

The naming of Johnson solids follows a flexible and precise descriptive formula, with many solids can therefore be named in different ways without compromising their accuracy as a description. Most Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundas), 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:^[12]

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.

Examples of para- and meta- can be found in parabiaugmented hexagonal prism and metabiaugmented hexagonal prism

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.^[12]

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:^[12]

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.

In general, the enumeration of Johnson solids may be denoted as $J_{n}$ , where $n$ denoted the list's enumeration (an example is $J_{1}$ denoted the first Johnson solid, the equilateral square pyramid). The following is the list of Johnson solids, with the enumeration followed according to the list of Johnson (1966):

References

^ 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.
^ 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.
^ 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.
^ 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.
^ Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
^ Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.
^ Fredriksson, Albin (2024). "Optimizing for the Rupert property". The American Mathematical Monthly. 131 (3): 255–261. arXiv:2210.00601. doi:10.1080/00029890.2023.2285200.
^ GWH. "Pseudo Rhombicuboctahedra". www.georgehart.com. Retrieved 17 April 2018.
^ Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 91. ISBN 978-0-521-55432-9.
^ Grünbaum, Branko (2009). "An enduring error" (PDF). Elemente der Mathematik. 64 (3): 89–101. doi:10.4171/EM/120. MR 2520469. Reprinted in Pitici, Mircea, ed. (2011). The Best Writing on Mathematics 2010. Princeton University Press. pp. 18–31.
^ Lando, Sergei K.; Zvonkin, Alexander K. (2004). Graphs on Surfaces and Their Applications. Springer. p. 114. doi:10.1007/978-3-540-38361-1. ISBN 978-3-540-38361-1.
^ ^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.
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.

[todesco-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.

[williams-2] 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.

[kaplan-hart-3] 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.

[uehara-4] 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.

[johnson-5] Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.

[zalgaller-6] Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.

[fred-7] Fredriksson, Albin (2024). "Optimizing for the Rupert property". The American Mathematical Monthly. 131 (3): 255–261. arXiv:2210.00601. doi:10.1080/00029890.2023.2285200.

[8] GWH. "Pseudo Rhombicuboctahedra". www.georgehart.com. Retrieved 17 April 2018.

[cromwell-9] Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 91. ISBN 978-0-521-55432-9.

[grunbaum-10] Grünbaum, Branko (2009). "An enduring error" (PDF). Elemente der Mathematik. 64 (3): 89–101. doi:10.4171/EM/120. MR 2520469. Reprinted in Pitici, Mircea, ed. (2011). The Best Writing on Mathematics 2010. Princeton University Press. pp. 18–31.

[lz-11] Lando, Sergei K.; Zvonkin, Alexander K. (2004). Graphs on Surfaces and Their Applications. Springer. p. 114. doi:10.1007/978-3-540-38361-1. ISBN 978-3-540-38361-1.

[berman-12] 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.

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v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)