Johnson solid

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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.)

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 not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a "Johnson solid".

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 rotundas), 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 rotundas, 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[4<>], a cuboctahedron as a triangular gyrobicupola[3cc*], and an icosidodecahedron as a pentagonal gyrobirotunda[5rr*].
  • 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[z] 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 rotundas

The first 6 Johnson solids are pyramids, cupolae, or rotundas 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 3> T J1 4> J2 5>
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 3c aC- J4 4c J5 5c J6 5r aD-
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 3=> J8 4=> J9 5=> Coplanar J10 4z> J11 5z> I-
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 3<> Regular J13 5<> J14 3<=> J15 4<=> J16 5<=> Coplanar J17 4<z> 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 rotundas

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

Elongated and gyroelongated cupolae and rotundas

Elongated cupola Elongated rotunda Gyroelongated cupola Gyroelongated rotunda
Coplanar J18 3c= J19 4c= eC- J20 5c= J21 5r= Concave J22 3cz J23 4cz J24 5cz J25 5rz
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 3cc J28 4cc J30 5cc J26 2cc* Semiregular J29 4cc* J31 5cc*
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-rotundas and birotundas

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

Cupola-rotunda Birotunda
J32 5cr J33 5cr* J34 5rr aD* 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 3c=c Semiregular J38 5c=c Coplanar J36 3c=c* J37 4c=c* eC* J39 5c=c*
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-rotundas and birotundas

Elongated cupola-rotunda Elongated birotunda
J40 5c=r J41 5c=r* J42 5r=r J43 5r=r*
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-rotundas, and birotundas

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola Gyroelongated cupola-rotunda Gyroelongated birotunda
Concave J44 3czc J45 4czc J46 5czc J47 5czr J48 5rzr
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 3=+ J50 3=++ J51 3=+++ J52 5=+ J53 5=++ J54 6=+ J55 6=++ J56 6=+x J57 6=+++
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

J8 and J15 would also fit here, as an augmented square prism and biaugmented square prism.

Modified Platonic solids

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

Augmented dodecahedra

J58 D+ J59 D++ J60 D+x J61 D+++
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 I-/ J63 I--- J64 I---+
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 tT+ J66 tC+ J67 tC++ J68 tD+ J69 tD++ J70 tD+x J71 tD+++
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 eD* J73 eD** J74 eD*' J75 eD***
Gyrate rhombicosidodecahedron Parabigyrate rhombicosidodecahedron Metabigyrate rhombicosidodecahedron Trigyrate rhombicosidodecahedron
Diminished rhombicosidodecahedra
J76 eD- J80 eD-- J81 eD-/ J83 eD---
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:

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 pentagon-faced Johnson solids

Eleven Johnson solids have only triangle and pentagon 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 pentagon-faced Johnson solids

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

J09 Elongated pentagonal pyramid
J30 Pentagonal orthobicupola
J31 Pentagonal gyrobicupola
J32 Pentagonal orthocupolarotunda
J33 Pentagonal gyrocupolarotunda
J38 Elongated pentagonal orthobicupola
J39 Elongated pentagonal gyrobicupola
J40 Elongated pentagonal orthocupolarotunda
J41 Elongated pentagonal gyrocupolarotunda
J42 Elongated pentagonal orthobirotunda
J43 Elongated pentagonal gyrobirotunda
J46 Gyroelongated pentagonal bicupola
J47 Gyroelongated pentagonal cupolarotunda
J52 Augmented pentagonal prism
J53 Biaugmented pentagonal prism
J72 Gyrate rhombicosidodecahedron
J73 Parabigyrate rhombicosidodecahedron
J74 Metabigyrate rhombicosidodecahedron
J75 Trigyrate rhombicosidodecahedron
J91 Bilunabirotunda

Triangle, square, and hexagon-faced Johnson solids

Eight Johnson solids have only triangle, square, and hexagon 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 octagon-faced Johnson solids

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

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

Triangle, pentagon, and decagon-faced Johnson solids

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

J06 Pentagonal rotunda
J25 Gyroelongated pentagonal rotunda

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

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

J92 Triangular hebesphenorotunda

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

Sixteen Johnson solids have only triangle, square, pentagon, and decagon 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 polyhedra 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
J78
J79
J80
J81
J82
J83

See also

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
  • Timofeenko, A.V. (2009). "The Non-Platonic and Non-Archimedean Noncomposite Polyhedra". J. Math. Sci. 162 (5): 710–729. doi:10.1007/s10958-009-9655-0. S2CID 120114341. [1]

olyhedra." J. Math. Sci. 162, 710-729, 2009.

  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.

External links