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:

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

The polyhedron on the left, the elongated square gyrobicupola, is a Johnson solid. The polyhedron on the right, the stella octangula, is not a Johnson solid: it has regular faces, but is not convex, since some of its diagonals lie outside the polyhedron.

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

The 92 Johnson Solids and some related shapes. (see an animated version here).

- invalid, - Platonic, - Archimedean, - Gyrated sections.

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

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.

	Pyramids			Cupolas			Cupola-Rotunda	Rotundas
	Tetrahedron "triangular pyramid"	1 Square pyramid	2 Pentagonal pyramid	3 Triangular cupola	4 Square cupola	5 Pentagonal cupola		6 Pentagonal rotunda
Elongated	7 Elongated triangular pyramid	8 Elongated square pyramid	9 Elongated pentagonal pyramid	18 Elongated triangular cupola	19 Elongated square cupola	20 Elongated pentagonal cupola		21 Elongated pentagonal rotunda
Gyroelongated	Augmented octahedron "Gyroelongated triangular pyramid"	10 Gyroelongated square pyramid	11 Gyroelongated pentagonal pyramid	22 Gyroelongated triangular cupola	23 Gyroelongated square cupola	24 Gyroelongated pentagonal cupola		25 Gyroelongated pentagonal rotunda
orthobi-	12 Triangular bipyramid	Octahedron "Square bipyramid"	13 Pentagonal bipyramid	27 Triangular orthobicupola	28 Square orthobicupola	30 Pentagonal orthobicupola	32 Pentagonal orthocupolarotunda	34 Pentagonal orthobirotunda
gyrobi-	12 Triangular bipyramid	Octahedron "Square bipyramid"	13 Pentagonal bipyramid	Cuboctahedron "Triangular gyrobicupola"	29 Square gyrobicupola	31 Pentagonal gyrobicupola	33 Pentagonal gyrocupolarotunda	Icosidodecahedron "pentagonal gyrobirotunda"
Elongated orthobi-	14 Elongated triangular bipyramid	15 Elongated square bipyramid	16 Elongated pentagonal bipyramid	35 Elongated triangular orthobicupola	Rhombicuboctahedron "Elongated square orthobicupola"	38 Elongated pentagonal orthobicupola	40 Elongated pentagonal orthocupolarotunda	42 Elongated pentagonal orthobirotunda
Elongated gyrobi-	14 Elongated triangular bipyramid	15 Elongated square bipyramid	16 Elongated pentagonal bipyramid	36 Elongated triangular gyrobicupola	37 Elongated square gyrobicupola	39 Elongated pentagonal gyrobicupola	41 Elongated pentagonal gyrocupolarotunda	43 Elongated pentagonal gyrobirotunda
Gyroelongated bi-	Trigonal trapezohedron "Gyroelongated triangular bipyramid"	17 Gyroelongated square bipyramid	Icosahedron "Gyroelongated pentagonal bipyramid"	44 Gyroelongated triangular bicupola	45 Gyroelongated square bicupola	46 Gyroelongated pentagonal bicupola	47 Gyroelongated pentagonal cupolarotunda	48 Gyroelongated pentagonal birotunda

	Fastigium
gyrobi-	26 Gyrobifastigium

Modified uniform polyhedra

A triangular prism is augmented by three square pyramids, becoming a triaugmented triangular prism.

A rhombicosidodecahedron being diminished.

A rhombicosidodecahedron being gyrated

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]

Augmented Prisms
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

Modified Platonics
58 Augmented dodecahedron	59 Parabiaugmented dodecahedron	60 Metabiaugmented dodecahedron	61 Triaugmented dodecahedron
62 Metabidiminished icosahedron	63 Tridiminished icosahedron	64 Augmented tridiminished icosahedron

Modified Archimedeans
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

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]

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	85 Snub square antiprism

Others
86 Sphenocorona	87 Augmented sphenocorona
88 Sphenomegacorona	89 Hebesphenomegacorona	90 Disphenocingulum

Rotundoids
91 Bilunabirotunda	92 Triangular hebesphenorotunda

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.
1 2 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.
1 2 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.
1 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.
↑ 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.
1 2 3 4 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.

[1] 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.

[strictly_convex-2] 1 2 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.

[todesco-3] 1 2 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-4] 1 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.

[bk-5] 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-6] 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-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.

[zalgaller-8] Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.

[johnson-9] Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.

[kaplan-hart-10] 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.

[berman-11] 1 2 3 4 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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