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 is actually a stellation, the only one possible for the octahedron.)

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

In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform (i.e., not a Platonic solid, Archimedean solid, prism or antiprism), and 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.

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 actually 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 always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 solids, 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 names are listed below and are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few (pyramids, cupolae, and rotunda), together with the Platonic and Archimedean solids, prisms, and antiprisms.

Bi- means that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, they can be joined so that like faces (ortho-) or unlike faces (gyro-) meet. In this nomenclature, an octahedron would be a square bipyramid, a cuboctahedron would be a triangular gyrobicupola, and an icosidodecahedron would be a pentagonal gyrobirotunda.
Elongated means that a prism has been joined to the base of the solid in question or between the bases of the solids in question. A rhombicuboctahedron would be an elongated square orthobicupola.
Gyroelongated means that an antiprism has been joined to the base of the solid in question or between the bases of the solids in question. An icosahedron would be a gyroelongated pentagonal bipyramid.
Augmented means that a pyramid or cupola has been joined to a face of the solid in question.
Diminished means that a pyramid or cupola has been removed from the solid in question.
Gyrate means that a cupola on the solid in question has been rotated so that different edges match up, as in the difference between ortho- and gyrobicupolae.

The last three operations — augmentation, diminution, and gyration — can be performed more than once on a large enough solid. We add bi- to the name of the operation to indicate that it has been performed twice. (A bigyrate solid has had two of its cupolae rotated.) We add tri- to indicate that it has been performed three times. (A tridiminished solid has had three of its pyramids or cupolae removed.)

Sometimes, bi- alone is not specific enough. We must distinguish between a solid that has had two parallel faces altered and one that has had two oblique faces altered. When the faces altered are parallel, we add para- to the name of the operation. (A parabiaugmented solid has had two parallel faces augmented.) When they are not, we add meta- to the name of the operation. (A metabiaugmented solid has had 2 oblique faces augmented.)

The last few Johnson solids have names based on certain polygon complexes that they are assembled from. These names are defined by Johnson as follows:^[2]

If we define a lune as a complex of two triangles attached to opposite sides of a square, the prefix spheno- refers to a wedgelike complex formed by two adjacent lunes. The prefix dispheno- denotes two such complexes, while hebespheno- indicates a blunter complex of two lunes separated by a third lune. The suffix -corona refers to a crownlike complex of eight triangles, and -megacorona, to a larger such complex of 12 triangles. The suffix -cingulum indicates a belt of 12 triangles.

Enumeration

Further information: List of 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.

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

Cupolæ 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
Triangular prism	Triangular cupola	Square cupola	Pentagonal cupola	Pentagonal rotunda


Related uniform polyhedra
	Cuboctahedron	Rhombicuboctahedron	Rhombicosidodecahedron	Icosidodecahedron

Elongated and gyroelongated pyramids

The next five Johnson solids are elongated and gyroelongated pyramids. These represent the composite or augmentation of two polyhedra. 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 (or augmented prisms)			Gyroelongated pyramids (or augmented antiprisms)
J7	J8	J9	Coplanar	J10	J11
Elongated triangular pyramid	Elongated square pyramid	Elongated pentagonal pyramid	Gyroelongated triangular pyramid	Gyroelongated square pyramid	Gyroelongated pentagonal pyramid
Augmented triangular prism	Augmented cube	Augmented pentagonal prism	Augmented octahedron	Augmented square antiprism	Augmented pentagonal antiprism


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 next six Johnson solids are bipyramids, elongated bipyramids, and gyroelongated bipyramids:

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 (rhombohedron)	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

Elongated cupolæ and rotundæ

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

Bicupolæ

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

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


Augmented from polyhedron

Cupola-rotundæ and birotunda

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


Augumented from polyhedra
Pentagonal cupola Pentagonal rotunda		Pentagonal rotunda

Elongated bicupolæ

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

Elongated cupola-rotundæ and birotundæ

Elongated cupolarotunda		Elongated birotunda
J40	J41	J42	J43
Elongated pentagonal orthocupolarotunda	Elongated pentagonal gyrocupolarotunda	Elongated pentagonal orthobirotunda	Elongated pentagonal gyrobirotunda

Gyroelongated bicupolæ, cupola-rotunda, and birotunda

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola				Gyroelongated cupolarotunda	Gyroelongated birotunda
Concave	J44	J45	J46	J47	J48
Gyroelongated digonal bicupola	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 triangular prisms

J7 (repeat)	J49	J50	J51
Elongated triangular pyramid	Augmented triangular prism	Biaugmented triangular prism	Triaugmented triangular prism


Augumented from polyhedra
Triangular prism tetrahedron	Triangular prism Square pyramid

Augmented pentagonal and hexagonal prisms

Augmented pentagonal prisms		Augmented hexagonal prisms
J52	J53	J54	J55	J56	J57
Augmented pentagonal prism	Biaugmented pentagonal prism	Augmented hexagonal prism	Parabiaugmented hexagonal prism	Metabiaugmented hexagonal prism	Triaugmented hexagonal prism


Augumented from polyhedra
Pentagonal prism Square pyramid		Hexagonal prism Square pyramid

Augmented dodecahedra

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


Augumented from polyhedra
Dodecahedron and pentagonal pyramid

Diminished icosahedra

J63	J62	J11 (Repeated)	Regular	J64
Tridiminished icosahedron	Metabidiminished icosahedron	Diminished icosahedron (Gyroelongated pentagonal pyramid)	Icosahedron	Augmented tridiminished icosahedron


Augumented from polyhedra
Tridiminished icosahedron, pentagonal pyramid and tetrahedron

Augmented truncated tetrahedron and truncated cubes

J65	J66	J67
Augmented truncated tetrahedron	Augmented truncated cube	Biaugmented truncated cube


Augumented from polyhedra
truncated tetrahedron triangular cupola	truncated cube square cupola

Augmented truncated dodecahedra

Semiregular	J68	J69	J70	J71
Truncated dodecahedron	Augmented truncated dodecahedron	Parabiaugmented truncated dodecahedron	Metabiaugmented truncated dodecahedron	Triaugmented truncated dodecahedron

Gyrate rhombicosidodecahedra

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

Diminished rhombicosidodecahedra

J76	J77	J78	J79
Diminished rhombicosidodecahedron	Paragyrate diminished rhombicosidodecahedron	Metagyrate diminished rhombicosidodecahedron	Bigyrate diminished rhombicosidodecahedron



J80	J81	J82	J83
Parabidiminished rhombicosidodecahedron	Metabidiminished rhombicosidodecahedron	Gyrate bidiminished rhombicosidodecahedron	Tridiminished rhombicosidodecahedron

Snub antiprisms

The snub antiprisms can be constructed as an alternation of a truncated antiprism. Two are Johnson solids, one is a regular, and the rest can not be constructed with regular triangles.

J84	Regular	J85	Irregular
Johnson solid	Regular	Johnson solid	Concave...
Snub disphenoid ss{2,4}	icosahedron ss{2,6}	snub square antiprism ss{2,8}	ss{2,10}...

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:

J1 Square pyramid J7 Elongated triangular pyramid J8 Elongated square pyramid J10 Gyroelongated square pyramid J14 Elongated triangular bipyramid J15 Elongated square bipyramid J16 Elongated pentagonal bipyramid J26 Gyrobifastigium	J27 Triangular orthobicupola J28 Square orthobicupola J29 Square gyrobicupola J35 Elongated triangular orthobicupola J36 Elongated triangular gyrobicupola J37 Elongated square gyrobicupola J44 Gyroelongated triangular bicupola J45 Gyroelongated square bicupola	J49 Augmented triangular prism J50 Biaugmented triangular prism J85 Snub square antiprism J86 Sphenocorona J87 Augmented sphenocorona J88 Sphenomegacorona J89 Hebesphenomegacorona J90 Disphenocingulum

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 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

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	J63	J62	J11	J6	J34

Rhombicosidodecahedron (diminished)
J5	J76	J80	J81	J83

Rhombicosidodecahedron (+gyration)
J72	J73	J74	J75	J77	J78	J79	J82

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. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. Zbl 0177.24802. No ISBN. The first proof that there are only 92 Johnson solids: see also 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.
Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 3 Further Convex polyhedra

↑ http://www.georgehart.com/virtual-polyhedra/pseudo-rhombicuboctahedra.html
↑ George Hart (quoting Johnson) (1996). "Johnson Solids". Virtual Polyhedra. Retrieved 5 February 2014.
↑ http://bendwavy.org/klitzing/explain/johnson.htm

External links

Sylvain Gagnon, "Convex polyhedra with regular faces", Structural Topology, No. 6, 1982, 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

Johnson solids navigator

Prismatoids and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda

Modified pyramids and dipyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular dipyramid square dipyramid pentagonal dipyramid elongated triangular dipyramid elongated square dipyramid elongated pentagonal dipyramid gyroelongated square dipyramid

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

Miscellaneous	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)

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