Johnson solid

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.

Definition and background

Among these three polyhedra, only the first, the elongated square gyrobicupola, is a Johnson solid. The second, the stella octangula, is not convex, as some of its diagonals lie outside the shape. The third presents coplanar faces.

A Johnson solid is a convex polyhedron whose faces are all regular polygons.^[1] Here, a polyhedron is said to be convex if the shortest path between any two of its vertices lies either within its interior or on its boundary, none of its faces are coplanar (meaning they do not share the same plane, and do not "lie flat"), and none of its edges are colinear (meaning they are not segments of the same line).^[2][3] 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 a Johnson solid is not a Platonic solid, Archimedean solid, prism, or antiprism.^[4][5] A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a near-miss Johnson solid.^[6]

The Johnson solid, sometimes known as Johnson–Zalgaller solid, was named after two mathematicians Norman Johnson and Victor Zalgaller.^[7] 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.^[8] Zalgaller (1969) proved that Johnson's list was complete.^[9]

Naming and enumeration

Main article: List of Johnson solids

An example is triaugmented triangular prism. Here, it is constructed from triangular prism by joining three equilateral square pyramids onto each of its squares (tri-). The process of this construction known as "augmentation", making its first name is "triaugmented".

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:^[10]

• 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.^[10]

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:^[10]

• 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.

The enumeration of Johnson solids may be denoted as ${\displaystyle J_{n}}$, where ${\displaystyle n}$ denoted the list's enumeration (an example is ${\displaystyle J_{1}}$ denoted the first Johnson solid, the equilateral square pyramid).^[7] The following is the list of ninety-two Johnson solids, with the enumeration followed according to the list of Johnson (1966):

1. Equilateral square pyramid
2. Pentagonal pyramid
3. Triangular cupola
4. Square cupola
5. Pentagonal cupola
6. Pentagonal rotunda
7. Elongated triangular pyramid
8. Elongated square pyramid
9. Elongated pentagonal pyramid
10. Gyroelongated square pyramid
11. Gyroelongated pentagonal pyramid
12. Triangular bipyramid
13. Pentagonal bipyramid
14. Elongated triangular bipyramid
15. Elongated square bipyramid
16. Elongated pentagonal bipyramid
17. Gyroelongated square bipyramid
18. Elongated triangular cupola
19. Elongated square cupola
20. Elongated pentagonal cupola
21. Elongated pentagonal rotunda
22. Gyroelongated triangular cupola
23. Gyroelongated square cupola
24. Gyroelongated pentagonal cupola
25. Gyroelongated pentagonal rotunda
26. Gyrobifastigium
27. Triangular orthobicupola
28. Square orthobicupola
29. Square gyrobicupola
30. Pentagonal orthobicupola
31. Pentagonal gyrobicupola
32. Pentagonal orthocupolarotunda
33. Pentagonal gyrocupolarotunda
34. Pentagonal orthobirotunda
35. Elongated triangular orthobicupola
36. Elongated triangular gyrobicupola
37. Elongated square gyrobicupola
38. Elongated pentagonal orthobicupola
39. Elongated pentagonal gyrobicupola
40. Elongated pentagonal orthocupolarotunda
41. Elongated pentagonal gyrocupolarotunda
42. Elongated pentagonal orthobirotunda
43. Elongated pentagonal gyrobirotunda
44. Gyroelongated triangular bicupola
45. Gyroelongated square bicupola
46. Gyroelongated pentagonal bicupola
47. Gyroelongated pentagonal cupolarotunda
48. Gyroelongated pentagonal birotunda
49. Augmented triangular prism
50. Biaugmented triangular prism
51. Triaugmented triangular prism
52. Augmented pentagonal prism
53. Biaugmented pentagonal prism
54. Augmented hexagonal prism
55. Parabiaugmented hexagonal prism
56. Metabiaugmented hexagonal prism
57. Triaugmented hexagonal prism
58. Augmented dodecahedron
59. Parabiaugmented dodecahedron
60. Metabiaugmented dodecahedron
61. Triaugmented dodecahedron
62. Metabidiminished icosahedron
63. Tridiminished icosahedron
64. Augmented tridiminished icosahedron
65. Augmented truncated tetrahedron
66. Augmented truncated cube
67. Biaugmented truncated cube
68. Augmented truncated dodecahedron
69. Parabiaugmented truncated dodecahedron
70. Metabiaugmented truncated dodecahedron
71. Triaugmented truncated dodecahedron
72. Gyrate rhombicosidodecahedron
73. Parabigyrate rhombicosidodecahedron
74. Metabigyrate rhombicosidodecahedron
75. Trigyrate rhombicosidodecahedron
76. Diminished rhombicosidodecahedron
77. Paragyrate diminished rhombicosidodecahedron
78. Metagyrate diminished rhombicosidodecahedron
79. Bigyrate diminished rhombicosidodecahedron
80. Parabidiminished rhombicosidodecahedron
81. Metabidiminished rhombicosidodecahedron
82. Gyrate bidiminished rhombicosidodecahedron
83. Tridiminished rhombicosidodecahedron
84. Snub disphenoid
85. Snub square antiprism
86. Sphenocorona
87. Augmented sphenocorona
88. Sphenomegacorona
89. Hebesphenomegacorona
90. Disphenocingulum
91. Bilunabirotunda
92. Triangular hebesphenorotunda

Some of the Johnson solids may be categorized as elementary polyhedra. This means the polyhedron cannot be separated by a plane to create two small convex polyhedra with regular faces; examples of Johnson solids are the first six Johnson solids—square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda—tridiminished icosahedron, parabidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron, snub disphenoid, snub square antiprism, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum, bilunabirotunda, and triangular hebesphenorotunda.^[8][11] The other Johnson solids are composite polyhedron because they are constructed by attaching some elementary polyhedra.^[12]

Properties

As the definition above, a Johnson solid is a convex polyhedron with regular polygons as their faces. However, there are several properties possessed by each of them.

• All but five of the 92 Johnson solids are known to have the Rupert property, meaning that it is possible for a larger copy of themselves to pass through a hole inside of them. The five which are not known to have this property are: gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, and paragyrate diminished rhombicosidodecahedron.^[13]
• From all of the Johnson solids, the elongated square gyrobicupola (also called the pseudorhombicuboctahedron) 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.^[14][15][16]

See also

• List of Johnson solids
• Near-miss Johnson solid
• Blind polytope

References

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.
2. Litchenberg, Dorovan R. (1988). "Pyramids, Prisms, Antiprisms, and Deltahedra". The Mathematics Teacher. 81 (4): 261–265. doi:10.5951/MT.81.4.0261. JSTOR 27965792.
3. Boissonnat, J. D.; Yvinec, M. (June 1989). "Probing a scene of non convex polyhedra". Proceedings of the Fifth Annual Symposium on Computational Geometry. pp. 237–246. doi:10.1145/73833.73860. ISBN 0-89791-318-3.
4. 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.
5. 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.
6. 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.
7. 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.
8. Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
9. Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.
10. 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.
11. Hartshorne, Robin (2000). Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer-Verlag. p. 464. ISBN 9780387986500.
12. Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.
13. 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.
14. Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 91. ISBN 978-0-521-55432-9.
15. 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.
16. 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.

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.