Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J₁); it has 1 square face and 4 triangular faces. Some authors require that the solid is not uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”.

The elongated square gyrobicupola (J₃₇), a Johnson solid

This 24 equilateral triangle example is not a Johnson solid because it is not convex.

This 24-square example is not a Johnson solid because it is not strictly convex (has 180° dihedral angles.)

As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J₂) is an example that has a degree-5 vertex.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids which are not uniform (i.e., not a Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 Johnson solids (excluding the 5 Platonic solids , the 13 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 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

Of the Johnson solids, the elongated square gyrobicupola (J₃₇), also called the pseudorhombicuboctahedron,[1] is unique in being locally vertex-uniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 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.

Names

The naming of Johnson solids follows a flexible and precise descriptive formula, such that many solids can 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 rotunda), together with the Platonic and Archimedean solids, prisms, and antiprisms; the centre 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:

Bi- indicates that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, the solids can be joined so that either like faces (ortho-) or unlike faces (gyro-) meet. Using this nomenclature, an octahedron can be described as a square bipyramid, a cuboctahedron as a triangular gyrobicupola, and an icosidodecahedron as a pentagonal gyrobirotunda.
Elongated indicates a prism is joined to the base of the solid in question, or between the bases in the case of Bi- solids. A rhombicuboctahedron can thus be described as an elongated square orthobicupola.
Gyroelongated indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids. An icosahedron can thus be described as a gyroelongated pentagonal bipyramid.
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 2 oblique faces gyrated.

The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson[2] with the following nomenclature:

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 12 triangles.
The suffix -cingulum indicates a belt of 12 triangles.

Enumeration

Pyramids, cupolae and rotundae

The first 6 Johnson solids are pyramids, cupolae, or rotundae with at most 5 lateral faces. Pyramids and cupolae with 6 or more lateral faces are coplanar and are hence not Johnson solids.

Pyramids

The first two Johnson solids, J1 and J2, are pyramids. The triangular pyramid is the regular tetrahedron, so it is not a Johnson solid. They represent sections of regular polyhedra.

Regular	J1	J2
Triangular pyramid (Tetrahedron)	Square pyramid	Pentagonal pyramid


Related regular polyhedra
Tetrahedron	Octahedron	Icosahedron

Cupolae and rotunda

The next four Johnson solids are three cupolae and one rotunda. They represent sections of uniform polyhedra.

Cupola				Rotunda
Uniform	J3	J4	J5	J6
Fastigium (Digonal cupola) (Triangular prism)	Triangular cupola	Square cupola	Pentagonal cupola	Pentagonal rotunda


Related uniform polyhedra
	Cuboctahedron	Rhombicuboctahedron	Rhombicosidodecahedron	Icosidodecahedron

Modified pyramids

Johnson solids 7 to 17 are derived from pyramids.

Elongated and gyroelongated pyramids

In the gyroelongated triangular pyramid, three pairs of adjacent triangles are coplanar and form non-square rhombi, so it is not a Johnson solid.

Elongated pyramids			Gyroelongated pyramids
J7	J8	J9	Coplanar	J10	J11
Elongated triangular pyramid	Elongated square pyramid	Elongated pentagonal pyramid	Gyroelongated triangular pyramid (diminished trigonal trapezohedron)	Gyroelongated square pyramid	Gyroelongated pentagonal pyramid


Augmented from polyhedra
tetrahedron triangular prism	square pyramid cube	pentagonal pyramid pentagonal prism	tetrahedron octahedron	square pyramid square antiprism	pentagonal pyramid pentagonal antiprism

Bipyramids

The square bipyramid is the regular octahedron, while the gyroelongated pentagonal bipyramid is the regular icosahedron, so they are not Johnson solids. In the gyroelongated triangular bipyramid, six pairs of adjacent triangles are coplanar and form non-square rhombi, so it is also not a Johnson solid.

Bipyramids			Elongated bipyramids			Gyroelongated bipyramids
J12	Regular	J13	J14	J15	J16	Coplanar	J17	Regular
Triangular bipyramid	Square bipyramid (octahedron)	Pentagonal bipyramid	Elongated triangular bipyramid	Elongated square bipyramid	Elongated pentagonal bipyramid	Gyroelongated triangular bipyramid (trigonal trapezohedron)	Gyroelongated square bipyramid	Gyroelongated pentagonal bipyramid (icosahedron)


Augmented from polyhedra
tetrahedron	square pyramid	pentagonal pyramid	tetrahedron triangular prism	square pyramid cube	pentagonal pyramid pentagonal prism	tetrahedron Octahedron	square pyramid square antiprism	pentagonal pyramid pentagonal antiprism

Modified cupolae and rotundae

Johnson solids 18 to 48 are derived from cupolae and rotundae.

Elongated and gyroelongated cupolae and rotundae

Elongated cupola				Elongated rotunda	Gyroelongated cupola				Gyroelongated rotunda
Coplanar	J18	J19	J20	J21	Concave	J22	J23	J24	J25
Elongated fastigium	Elongated triangular cupola	Elongated square cupola	Elongated pentagonal cupola	Elongated pentagonal rotunda	Gyroelongated fastigium	Gyroelongated triangular cupola	Gyroelongated square cupola	Gyroelongated pentagonal cupola	Gyroelongated pentagonal rotunda


Augmented from polyhedra
Square prism Triangular prism	Hexagonal prism Triangular cupola	Octagonal prism Square cupola	Decagonal prism Pentagonal cupola	Decagonal prism Pentagonal rotunda	square antiprism Triangular prism	Hexagonal antiprism Triangular cupola	Octagonal antiprism Square cupola	Decagonal antiprism Pentagonal cupola	Decagonal antiprism Pentagonal rotunda

Bicupolae

The triangular gyrobicupola is an Archimedean solid (in this case the cuboctahedron), so it is not a Johnson solid.

Orthobicupola				Gyrobicupola
Coplanar	J27	J28	J30	J26	Semiregular	J29	J31
Orthobifastigium	Triangular orthobicupola	Square orthobicupola	Pentagonal orthobicupola	Gyrobifastigium	Triangular gyrobicupola (cuboctahedron)	Square gyrobicupola	Pentagonal gyrobicupola


Augmented from polyhedron
Triangular prism	Triangular cupola	Square cupola	Pentagonal cupola	Triangular prism	Triangular cupola	Square cupola	Pentagonal cupola

Cupola-rotundae and birotunda

The pentagonal gyrobirotunda is an Archimedean solid (in this case the icosidodecahedron), so it is not a Johnson solid.

Cupola-rotunda		Birotunda
J32	J33	J34	Semiregular
Pentagonal orthocupolarotunda	Pentagonal gyrocupolarotunda	Pentagonal orthobirotunda	Pentagonal gyrobirotunda (icosidodecahedron)


Augmented from polyhedra
Pentagonal cupola Pentagonal rotunda		Pentagonal rotunda

Elongated bicupolae

The elongated square orthobicupola is an Archimedean solid (in this case the rhombicuboctahedron), so it is not a Johnson solid.

Elongated orthobicupola				Elongated gyrobicupola
Coplanar	J35	Semiregular	J38	Coplanar	J36	J37	J39
Elongated orthobifastigium	Elongated triangular orthobicupola	Elongated square orthobicupola (rhombicuboctahedron)	Elongated pentagonal orthobicupola	Elongated gyrobifastigium	Elongated triangular gyrobicupola	Elongated square gyrobicupola	Elongated pentagonal gyrobicupola


Augmented from polyhedra
Square prism Triangular prism	Hexagonal prism Triangular cupola	Octagonal prism Square cupola	Decagonal prism Pentagonal cupola	Square prism Triangular prism	Hexagonal prism Triangular cupola	Octagonal prism Square cupola	Decagonal prism Pentagonal cupola

Elongated cupola-rotundae and birotundae

Elongated cupola-rotunda		Elongated birotunda
J40	J41	J42	J43
Elongated pentagonal orthocupolarotunda	Elongated pentagonal gyrocupolarotunda	Elongated pentagonal orthobirotunda	Elongated pentagonal gyrobirotunda


Augmented from polyhedra
Decagonal prism Pentagonal cupola Pentagonal rotunda		Decagonal prism Pentagonal rotunda

Gyroelongated bicupolae, cupola-rotunda, and birotunda

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola				Gyroelongated cupola-rotunda	Gyroelongated birotunda
Concave	J44	J45	J46	J47	J48
Gyroelongated bifastigium	Gyroelongated triangular bicupola	Gyroelongated square bicupola	Gyroelongated pentagonal bicupola	Gyroelongated pentagonal cupolarotunda	Gyroelongated pentagonal birotunda


Augmented from polyhedra
Triangular prism Square antiprism	Triangular cupola Hexagonal antiprism	Square cupola Octagonal antiprism	Pentagonal cupola Decagonal antiprism	Pentagonal cupola Pentagonal rotunda Decagonal antiprism	Pentagonal rotunda Decagonal antiprism

Augmented prisms

Johnson solids 49 to 57 are built by augmenting the sides of prisms with square pyramids.

Augmented triangular prisms			Augmented pentagonal prisms		Augmented hexagonal prisms
J49	J50	J51	J52	J53	J54	J55	J56	J57
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


Augmented from polyhedra
Triangular prism Square pyramid			Pentagonal prism Square pyramid		Hexagonal prism Square pyramid

Modified Platonic solids

Johnson solids 58 to 64 are built by augmenting or diminishing Platonic solids.

Augmented dodecahedra

J58	J59	J60	J61
Augmented dodecahedron	Parabiaugmented dodecahedron	Metabiaugmented dodecahedron	Triaugmented dodecahedron


Augmented from polyhedra
Dodecahedron and pentagonal pyramid

Diminished and augmented diminished icosahedra

Diminished icosahedron				Augmented tridiminished icosahedron
J11 (Repeated)	Uniform	J62	J63	J64
Diminished icosahedron (Gyroelongated pentagonal pyramid)	Parabidiminished icosahedron (Pentagonal antiprism)	Metabidiminished icosahedron	Tridiminished icosahedron	Augmented tridiminished icosahedron

Modified Archimedean solids

Johnson solids 65 to 83 are built by augmenting, diminishing or gyrating Archimedean solids.

Augmented Archimedean solids

Augmented truncated tetrahedron	Augmented truncated cubes		Augmented truncated dodecahedra
J65	J66	J67	J68	J69	J70	J71
Augmented truncated tetrahedron	Augmented truncated cube	Biaugmented truncated cube	Augmented truncated dodecahedron	Parabiaugmented truncated dodecahedron	Metabiaugmented truncated dodecahedron	Triaugmented truncated dodecahedron


Augmented from polyhedra
truncated tetrahedron triangular cupola	truncated cube square cupola		truncated dodecahedron pentagonal cupola

Gyrate and diminished rhombicosidodecahedra

Gyrate rhombicosidodecahedra
J72	J73	J74	J75
Gyrate rhombicosidodecahedron	Parabigyrate rhombicosidodecahedron	Metabigyrate rhombicosidodecahedron	Trigyrate rhombicosidodecahedron


Diminished rhombicosidodecahedra
J76	J80	J81	J83
Diminished rhombicosidodecahedron	Parabidiminished rhombicosidodecahedron	Metabidiminished rhombicosidodecahedron	Tridiminished rhombicosidodecahedron


Gyrate diminished rhombicosidodecahedra
J77	J78	J79	J82
Paragyrate diminished rhombicosidodecahedron	Metagyrate diminished rhombicosidodecahedron	Bigyrate diminished rhombicosidodecahedron	Gyrate bidiminished rhombicosidodecahedron

J37 would also appear here as a duplicate (it is a gyrate rhombicuboctahedron).

Other gyrate and diminished archimedean solids

Other archimedean solids can be gyrated and diminished, but they all result in previously counted solids.

J27	J3	J34	J6	J37	J19	Uniform
Gyrate cuboctahedron (triangular orthobicupola)	Diminished cuboctahedron (triangular cupola)	Gyrate icosidodecahedron (pentagonal orthobirotunda)	Diminished icosidodecahedron (pentagonal rotunda)	Gyrate rhombicuboctahedron (elongated square gyrobicupola)	Diminished rhombicuboctahedron (elongated square cupola)	Bidiminished rhombicuboctahedron (octagonal prism)


Gyrated or diminished from polyhedra
Cuboctahedron		Icosidodecahedron		Rhombicuboctahedron

Elementary solids

Johnson solids 84 to 92 are not derived from "cut-and-paste" manipulations of uniform solids.

Snub antiprisms

The snub antiprisms can be constructed as an alternation of a truncated antiprism. The gyrobianticupolae are another construction for the snub antiprisms. Only snub antiprisms with at most 4 sides can be constructed from regular polygons. The snub triangular antiprism is the regular icosahedron, so it is not a Johnson solid.

J84	Regular	J85
Snub disphenoid ss{2,4}	Icosahedron ss{2,6}	Snub square antiprism ss{2,8}
Digonal gyrobianticupola	Triangular gyrobianticupola	Square gyrobianticupola

Others

J86	J87	J88
Sphenocorona	Augmented sphenocorona	Sphenomegacorona


J89	J90	J91	J92
Hebesphenomegacorona	Disphenocingulum	Bilunabirotunda	Triangular hebesphenorotunda

Classification by types of faces

Triangle-faced Johnson solids

Five Johnson solids are deltahedra, with all equilateral triangle faces:

J12 Triangular bipyramid J13 Pentagonal bipyramid J17 Gyroelongated square bipyramid	J51 Triaugmented triangular prism J84 Snub disphenoid

Triangle and square-faced Johnson solids

Twenty four Johnson solids have only triangle or square faces:

Triangle and pentagonal-faced Johnson solids

Eleven Johnson solids have only triangle and pentagonal faces:

J2 Pentagonal pyramid J11 Gyroelongated pentagonal pyramid J34 Pentagonal orthobirotunda J48 Gyroelongated pentagonal birotunda J58 Augmented dodecahedron J59 Parabiaugmented dodecahedron	J60 Metabiaugmented dodecahedron J61 Triaugmented dodecahedron J62 Metabidiminished icosahedron J63 Tridiminished icosahedron J64 Augmented tridiminished icosahedron

Triangle, square, and pentagonal-faced Johnson solids

Twenty Johnson solids have only triangle, square and pentagonal faces:

Triangle, square, and hexagonal-faced Johnson solids

Eight Johnson solids have only triangle, square and hexagonal faces:

J3 Triangular cupola J18 Elongated triangular cupola J22 Gyroelongated triangular cupola J54 Augmented hexagonal prism	J55 Parabiaugmented hexagonal prism J56 Metabiaugmented hexagonal prism J57 Triaugmented hexagonal prism J65 Augmented truncated tetrahedron

Triangle, square, and octagonal-faced Johnson solids

Five Johnson solids have only triangle, square and octagonal faces:

J4 Square cupola J19 Elongated square cupola J23 Gyroelongated square cupola	J66 Augmented truncated cube J67 Biaugmented truncated cube

Triangle, pentagon, and decagonal-faced Johnson solids

Two Johnson solids have only triangle, pentagon and decagonal faces:

J06 Pentagonal rotunda J25 Gyroelongated pentagonal rotunda

Triangle, square, pentagon, and hexagonal-faced Johnson solids

Only one Johnson solid has triangle, square, pentagon and hexagonal faces:

J92 Triangular hebesphenorotunda

Triangle, square, pentagon, and decagonal-faced Johnson solids

Sixteen Johnson solids have only triangle, square, pentagon and decagonal faces:

J05 Pentagonal cupola J20 Elongated pentagonal cupola J21 Elongated pentagonal rotunda J24 Gyroelongated pentagonal cupola J68 Augmented truncated dodecahedron J69 Parabiaugmented truncated dodecahedron J70 Metabiaugmented truncated dodecahedron J71 Triaugmented truncated dodecahedron	J76 Diminished rhombicosidodecahedron J77 Paragyrate diminished rhombicosidodecahedron J78 Metagyrate diminished rhombicosidodecahedron J79 Bigyrate diminished rhombicosidodecahedron J80 Parabidiminished rhombicosidodecahedron J81 Metabidiminished rhombicosidodecahedron J82 Gyrate bidiminished rhombicosidodecahedron J83 Tridiminished rhombicosidodecahedron

Circumscribable Johnson solids

25 of the Johnson solids have vertices that exist on the surface of a sphere: 1–6,11,19,27,34,37,62,63,72–83. All of them can be seen to be related to a regular or uniform polyhedron by gyration, diminishment, or dissection.[3]

Octahedron	Cuboctahedron		Rhombicuboctahedron
J1	J3	J27	J4	J19	J37

Icosahedron				Icosidodecahedron
J2	J11	J62	J63	J6	J34

Rhombicosidodecahedron
J5	J72	J73	J74	J75	J76	J77
J5	J78	J79	J80	J81	J82	J83

References

Johnson, Norman W. (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. ISSN 0008-414X. Zbl 0132.14603. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
Zalgaller, Victor A. (1967). "Convex Polyhedra with Regular Faces". Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (in Russian). 2: 1–221. ISSN 0373-2703. Zbl 0165.56302. The first proof that there are only 92 Johnson solids. English translation: Zalgaller, Victor A. (1969). "Convex Polyhedra with Regular Faces". Seminars in Mathematics, V. A. Steklov Math. Inst., Leningrad. Consultants Bureau. 2. ISSN 0080-8873. Zbl 0177.24802.
Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 3 Further Convex polyhedra

GWH. "Pseudo Rhombicuboctahedra". www.georgehart.com. Retrieved 17 April 2018.
George Hart (quoting Johnson) (1996). "Johnson Solids". Virtual Polyhedra. Retrieved 5 February 2014.
Klitzing, Dr. Richard. "Johnson solids et al". bendwavy.org. Retrieved 17 April 2018.

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 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, 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.

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

[2] George Hart (quoting Johnson) (1996). "Johnson Solids". Virtual Polyhedra. Retrieved 5 February 2014.

[3] Klitzing, Dr. Richard. "Johnson solids et al". bendwavy.org. Retrieved 17 April 2018.