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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, 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.

NamesEdit

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.

EnumerationEdit

Pyramids, cupolae and rotundaeEdit

The first 6 Johnson solids are pyramids, cupolae, or rotundae with at most 5 sides. 6 sided pyramids and cupolae are coplanar and are hence not Johnson solids.

PyramidsEdit

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

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

Cupolae and rotundaEdit

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
(Triangular prism)
Triangular cupola Square cupola Pentagonal cupola Pentagonal rotunda
         
       
Related uniform polyhedra
Cuboctahedron Rhombicuboctahedron Rhombicosidodecahedron Icosidodecahedron
       

Modified pyramids, cupolae and rotundaeEdit

Johnson solids 7 to 48 are derived from pyramids, cupolae and rotundae.

Elongated and gyroelongated pyramidsEdit

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
                       

BipyramidsEdit

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(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 cupolae and rotundaeEdit

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
                                       

BicupolaeEdit

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
Bifastigium 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 birotundaEdit

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)
       
       
Augumented from polyhedra
Pentagonal cupola
Pentagonal rotunda
Pentagonal rotunda
     

Elongated bicupolaeEdit

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 bifastigium 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 birotundaeEdit

Elongated cupolarotunda 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 birotundaEdit

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola Gyroelongated cupolarotunda 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 prismsEdit

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

Modified Platonic and Archimedean solidsEdit

Johnson solids 58 to 83 are built by augmenting, diminishing or gyrating Platonic or Archimedean solids.

Augmented dodecahedraEdit

J58 J59 J60 J61
Augmented dodecahedron Parabiaugmented dodecahedron Metabiaugmented dodecahedron Triaugmented dodecahedron
       
       
Augumented from polyhedra
Dodecahedron and pentagonal pyramid
   

Diminished icosahedraEdit

Diminished icosahedra 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
         
       

Augmented Archimedean solidsEdit

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

Gyrate and diminished rhombicosidodecahedraEdit

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

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
     
     

OthersEdit

J86 J87 J88
Sphenocorona Augmented sphenocorona Sphenomegacorona
     
     
J89 J90 J91 J92
Hebesphenomegacorona Disphenocingulum Bilunabirotunda Triangular hebesphenorotunda
       
       

Classification by types of facesEdit

Triangle-faced Johnson solidsEdit

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 solidsEdit

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 solidsEdit

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 solidsEdit

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 solidsEdit

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 solidsEdit

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
 

Dual PolyhedronsEdit

As of now, the website https://levskaya.github.io/polyhedronisme/ supports Conway operations applied to Johnson solids. For example, see https://levskaya.github.io/polyhedronisme/?recipe=C1000dJ73&palette=%2300ffff%20%23ff7f7f for the dual polyhedron of J73.

See alsoEdit

ReferencesEdit

  • 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
  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 linksEdit