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{{short description|92 non-uniform convex polyhedra, with each face a regular polygon}}
{{short description|92 non-uniform convex polyhedra, with each face a regular polygon}}
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In [[geometry]], a '''Johnson solid''' is a strictly [[convex polyhedron]] each face of which is a [[regular polygon]]. It is also '''Johnson–Zalgaller solid''', was named after two mathematicians [[Norman Johnson (mathematician)|Norman Johnson]] and [[Victor Zalgaller]].
[[Image:Elongated square gyrobicupola.png|thumb|right|The [[elongated square gyrobicupola]] ({{math|''J''{{sub|37}}}}), a Johnson solid]]
[[Image:Stella octangula.png|160px|thumb|This [[Stella octangula|24 equilateral triangle]] example is not a Johnson solid because it is not convex]]
[[Image:Partial cubic honeycomb.png|160px|thumb|This 24-square example is not a Johnson solid because it is not strictly convex (has 180° [[dihedral angle]]s)]]


== Description ==
In [[geometry]], a '''Johnson solid''' is a strictly [[convex polyhedron]] each face of which is a [[regular polygon]]. There is no requirement that [[isohedral|each face must be the same polygon]], or that the same polygons join around each [[Vertex (geometry)|vertex]]. An example of a Johnson solid is the square-based [[Pyramid (geometry)|pyramid]] with [[equilateral]] sides ([[square pyramid|{{math|''J''{{sub|1}}}}]]); it has 1 square face and 4 triangular faces. Some authors require that the solid not be [[uniform polyhedron|uniform]] (i.e., not [[Platonic solid]], [[Archimedean solid]], [[prism (geometry)|uniform prism]], or uniform [[antiprism]]) before they refer to it as a "Johnson solid".
{{multiple image
| image1 = Elongated square gyrobicupola.png
| image2 = Stella octangula.png
| image3 = Partial cubic honeycomb.png
| total_width = 500
| align = right
| footer = The following are three examples of solids. The first solid, [[elongated square gyrobicupola]], is Johnson solid because it has the convexity property. The second solid, [[stella octangula]] is not Johnson solid because it is not convex, meaning whenever two points are inside of it, the line connecting always be outside. The last solid is not a Johnson solid because it is not strictly convex, meaning every face is planar or the dihedral angles of two adjacent face has 180°.
}}
A Johnson solid is a convex polyhedron in which the faces of each are [[regular polygon]]. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not [[Uniform polyhedron|uniform]]. This means that Johnson solid are not a [[Platonic solid]], [[Archimedean solid]], [[Prism (geometry)|prism]], or [[antiprism]].{{r|todesco|williams}} Equivalently, a Johnson solid is defined as the ''strictly'' convex polyhedron, meaning that every face is not [[Coplanarity|coplanar]]. A convex polyhedron in which all faces are close enough to become regular but some of them are not precisely regular is known as [[near-miss Johnson solid]].


The Johnson solid, sometimes known as '''Johnson–Zalgaller solid''', was named after two mathematicians [[Norman Johnson (mathematician)|Norman Johnson]] and [[Victor Zalgaller]].{{r|uehara}} {{harvtxt|Johnson|1966}} published a list including 92 Johnson solids—excluding the five Platonic solids, the thirteen 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.{{r|johnson}} {{harvtxt|Zalgaller|1969}} proved that Johnson's list was complete.{{r|zalgaller}}
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]] ({{math|''J''{{sub|2}}}}) is an example that has a degree-5 vertex.


Some of the Johnson solids do have [[Rupert property]], meaning they do have the polyhedron of the same or larger size that may pass through a hole inside of them. However, the other five Johnson solids do not have this property: [[gyrate rhombicosidodecahedron]], [[parabigyrate rhombicosidodecahedron]], [[metabigyrate rhombicosidodecahedron]], [[trigyrate rhombicosidodecahedron]], and [[paragyrate diminished rhombicosidodecahedron]].{{r|fred}} From all of the Johnson solids, the [[elongated square gyrobicupola]] (also called the pseudorhombicuboctahedron),<ref>{{cite web|url=http://www.georgehart.com/virtual-polyhedra/pseudo-rhombicuboctahedra.html|title=Pseudo Rhombicuboctahedra|last=GWH|website=www.georgehart.com|access-date=17 April 2018}}</ref> is unique in being locally vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one 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]].
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 polyhedron|uniform]] (i.e., not a [[Platonic solid]], [[Archimedean solid]], [[prism (geometry)|uniform prism]], or uniform [[antiprism]]) always have 3, 4, 5, 6, 8, or 10 sides.


== Naming and enumeration ==
In 1966, [[Norman Johnson (mathematician)|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.
{{main article|List of Johnson solids}}
[[File:Triaugmented triangular prism (symmetric view).svg|thumb|An example is [[triaugmented triangular prism]]. Here, it is constructed from triangular prism by joining three equilateral square pyramids into each of its squares (tri-). The process of this construction known as "augmentation", making its first name is "triaugmented".]]
The naming of Johnson solids follows a flexible and precise descriptive formula, with many solids can therefore be named in different ways without compromising their accuracy as a description. Most Johnson solids can be constructed from the first few ([[Pyramid (geometry)|pyramids]], [[cupola (geometry)|cupolae]], and [[rotunda (geometry)|rotundas]]), together with the [[Platonic solid|Platonic]] and [[Archimedean solid|Archimedean]] solids, [[prism (geometry)|prisms]], and [[antiprism]]s; the center 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:{{r|berman}}
* ''Bi-'' indicates that two copies of the solid 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, 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.
* ''Elongated'' indicates a [[prism (geometry)|prism]] is joined to the base of the solid, or between the bases; ''gyroelongated'' indicates an [[antiprism]]. ''Augmented'' indicates another polyhedron, namely a [[pyramid (geometry)|pyramid]] or [[Cupola (geometry)|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.
*''[[gyration|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.{{r|berman}}


The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson with the following nomenclature:{{r|berman}}
Of the Johnson solids, the [[elongated square gyrobicupola]] ({{math|''J''{{sub|37}}}}), also called the pseudorhombicuboctahedron,<ref>{{cite web|url=http://www.georgehart.com/virtual-polyhedra/pseudo-rhombicuboctahedra.html|title=Pseudo Rhombicuboctahedra|last=GWH|website=www.georgehart.com|access-date=17 April 2018}}</ref> 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]].
*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.

The following is the list of Johnson solids, with the enumeration followed according to the list of {{harvtxt|Johnson|1966}}:
# [[Equilateral square pyramid]]
# [[Pentagonal pyramid]]
# [[Triangular cupola]]
# [[Square cupola]]
# [[Pentagonal cupola]]
# [[Pentagonal rotunda]]
# [[Elongated triangular pyramid]]
# [[Elongated square pyramid]]
# [[Elongated pentagonal pyramid]]
# [[Gyroelongated square pyramid]]
# [[Gyroelongated pentagonal pyramid]]
# [[Triangular bipyramid]]
# [[Pentagonal bipyramid]]
# [[Elongated triangular bipyramid]]
# [[Elongated square bipyramid]]
# [[Elongated pentagonal bipyramid]]
# [[Gyroelongated square bipyramid]]
# [[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 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 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 dodecahedron]]
# [[Parabiaugmented dodecahedron]]
# [[Metabiaugmented dodecahedron]]
# [[Triaugmented dodecahedron]]
# [[Metabidiminished icosahedron]]
# [[Tridiminished icosahedron]]
# [[Augmented tridiminished icosahedron]]
# [[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]]
# [[Snub disphenoid]]
# [[Snub square antiprism]]
# [[Sphenocorona]]
# [[Augmented sphenocorona]]
# [[Sphenomegacorona]]
# [[Hebesphenomegacorona]]
# [[Disphenocingulum]]
# [[Bilunabirotunda]]
# [[Triangular hebesphenorotunda]]


== References ==
87 of the 92 Johnson solids have the [[Rupert property]]: a copy of the solid, of the same or larger shape, can be passed through a hole in the solid.
{{reflist|refs =
<ref name=fred>{{citation

<ref name="berman">{{cite journal
| last = Berman | first = Martin
| doi = 10.1016/0016-0032(71)90071-8
| journal = Journal of the Franklin Institute
| mr = 290245
| pages = 329–352
| title = Regular-faced convex polyhedra
| volume = 291
| year = 1971| issue = 5
}}</ref>

<ref name="fred">{{cite journal
| last = Fredriksson | first = Albin
| last = Fredriksson | first = Albin
| title = Optimizing for the Rupert property
| title = Optimizing for the Rupert property
Line 28: Line 155:
}}</ref>
}}</ref>


<ref name="johnson">{{cite journal
== Names ==
| last = Johnson | first = Norman | authorlink = Norman Johnson (mathematician)
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 ([[Pyramid (geometry)|pyramids]], [[cupola (geometry)|cupolae]], and [[rotunda (geometry)|rotundas]]), together with the [[Platonic solid|Platonic]] and [[Archimedean solid|Archimedean]] solids, [[prism (geometry)|prisms]], and [[antiprism]]s; 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:
| title = Convex Solids with Regular Faces
| journal = Canadian Journal of Mathematics
| volume = 18
| year = 1966
| pages = 169–200
| doi = 10.4153/CJM-1966-021-8
}}</ref>


<ref name="todesco">{{cite book
*'''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*]''.
| last = Todesco | first = Gian Marco
*'''Elongated[=]''' indicates a [[prism (geometry)|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''.
| editor-last1 = Emmer | editor-first1 = Michele
*'''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''.
| editor-last2 = Abate | editor-first2 = Marco
*'''Augmented[+]''' indicates another polyhedron, namely a [[pyramid (geometry)|pyramid]] or [[Cupola (geometry)|cupola]], is joined to one or more faces of the solid in question.
| year = 2020
*'''Diminished[-]''' indicates a pyramid or cupola is removed from one or more faces of the solid in question.
| contribution = Hyperbolic Honeycomb
*'''[[gyration|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.
| title = Imagine Math 7: Between Culture and Mathematics
| publisher = Springer
| doi = 10.1007/978-3-030-42653-8
| isbn = 978-3-030-42653-8
| page = 282
| url = https://books.google.com/books?id=wtIBEAAAQBAJ&pg=PA282
}}</ref>


<ref name="uehara">{{cite book
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.
| last = Uehara | first = Ryuhei
| year = 2020
| title = Introduction to Computational Origami: The World of New Computational Geometry
| url = https://books.google.com/books?id=51juDwAAQBAJ
| publisher = Springer
| isbn = 978-981-15-4470-5
| doi = 10.1007/978-981-15-4470-5
}}</ref>


<ref name="williams">{{cite book
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.
| last1 = Williams | first1 = Kim
| last2 = Monteleone | first2 = Cosino
| year = 2021
| title = Daniele Barbaro’s Perspective of 1568
| publisher = Springer
| isbn = 978-3-030-76687-0
| doi = 10.1007/978-3-030-76687-0
| page = 23
| url = https://books.google.com/books?id=w5RBEAAAQBAJ&pg=PA23
}}</ref>


<ref name="zalgaller">{{cite book
The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson<ref>{{cite web | url=http://www.georgehart.com/virtual-polyhedra/johnson-info.html | title = Johnson Solids | access-date = 5 February 2014 | author = George Hart (quoting Johnson) | website = Virtual Polyhedra | year = 1996}}</ref>
| last = Zalgaller | first = Victor A. | author-link = Victor Zalgaller
with the following nomenclature:
| title = Convex Polyhedra with Regular Faces

| publisher = Consultants Bureau
*A ''lune'' is a complex of two triangles attached to opposite sides of a square.
| year = 1969
*''Spheno''- indicates a wedgelike complex formed by two adjacent lunes. ''Dispheno-'' indicates two such complexes.
}}</ref>
*''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 ==
{{Further|List of Johnson solids}}
=== 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 [[Pyramid (geometry)|pyramids]]. The ''triangular pyramid'' is the regular [[tetrahedron]], so it is not a Johnson solid. They represent sections of regular polyhedra.
{| class="wikitable" width=640
|-
!Regular 3> T
!J1 4>
!J2 5>
|- align=center
| Triangular pyramid<br>([[Tetrahedron]])
| [[Square pyramid]]
| [[Pentagonal pyramid]]
|- align=center
| [[Image:Tetrahedron.png|120px]]
| [[Image:Square pyramid.png|120px]]
| [[Image:Pentagonal pyramid.png|120px]]
|- align=center
| [[File:Tetrahedron flat.svg|120px]]
| [[Image:Johnson solid 1 net.png|120px]]
| [[Image:Johnson solid 2 net.png|120px]]
|-
!colspan=3|Related regular polyhedra
|- align=center
|[[Tetrahedron]]
|[[Octahedron]]
|[[Icosahedron]]
|- align=center
| [[Image:Tetrahedron.png|80px]]
| [[Image:Octahedron.png|80px]]
| [[Image:Icosahedron.png|80px]]
|}

==== Cupolae and rotunda ====
The next four Johnson solids are three [[cupola (geometry)|cupolae]] and one [[rotunda (geometry)|rotunda]]. They represent sections of uniform polyhedra.
{| class="wikitable" width=640
|-
!colspan=4|[[Cupola (geometry)|Cupola]]
![[Rotunda (geometry)|Rotunda]]
|-
!Uniform
!J3 3c aC-
!J4 4c
!J5 5c
!J6 5r aD-
|- align=center
| Fastigium<br>(Digonal cupola)<br>([[Triangular prism]])
| [[Triangular cupola]]
| [[Square cupola]]
| [[Pentagonal cupola]]
| [[Pentagonal rotunda]]
|- align=center
|[[File:Triangular prism wedge.png|80px]]
| [[Image:Triangular cupola.png|80px]]
| [[Image:Square cupola.png|80px]]
| [[Image:Pentagonal cupola.png|80px]]
| [[Image:Pentagonal rotunda.png|80px]]
|- align=center
| [[File:Triangular prism net.png|
80px]]
| [[Image:Johnson solid 3 net.png|80px]]
| [[Image:Johnson solid 4 net.png|80px]]
| [[Image:Johnson solid 5 net.png|80px]]
| [[Image:Johnson solid 6 net.png|80px]]
|-
!colspan=5|Related uniform polyhedra
|- align=center
|[[Rhombohedron]]
|[[Cuboctahedron]]
|[[Rhombicuboctahedron]]
|[[Rhombicosidodecahedron]]
|[[Icosidodecahedron]]
|- align=center
| [[Image:Rhombohedron.svg|80px]]
| [[Image:Cuboctahedron.png|80px]]
| [[Image:Small rhombicuboctahedron.png|80px]]
| [[Image:Small rhombicosidodecahedron.png|80px]]
| [[File:Icosidodecahedron.png|80px]]
|}

=== 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.
{| class="wikitable" width=640
|-
!colspan=3|[[Elongated pyramid]]s
!colspan=3|[[Gyroelongated pyramid]]s
|-
!J7 3=>
!J8 4=>
!J9 5=>
!Coplanar
!J10 4z>
!J11 5z> I-
|- align=center
| [[Elongated triangular pyramid]]
| [[Elongated square pyramid]]
| [[Elongated pentagonal pyramid]]
| Gyroelongated triangular pyramid<br>([[diminished trapezohedron|diminished trigonal trapezohedron]])
| [[Gyroelongated square pyramid]]
| [[Gyroelongated pentagonal pyramid]]
|- align=center
| [[Image:Elongated triangular pyramid.png|80px]]
| [[Image:Elongated square pyramid.png|80px]]
| [[Image:Elongated pentagonal pyramid.png|80px]]
| [[File:Augmented octahedron.png|80px]]
| [[Image:Gyroelongated square pyramid.png|80px]]
| [[Image:Gyroelongated pentagonal pyramid.png|80px]]
|- align=center
| [[Image:Johnson solid 7 net.png|80px]]
| [[Image:Johnson solid 8 net.png|80px]]
| [[Image:Johnson solid 9 net.png|80px]]
| [[File:Augmented octahedgon net.png|80px]]
| [[Image:Johnson solid 10 net.png|80px]]
| [[Image:Johnson solid 11 net.png|80px]]
|- align=center
!colspan=6|Augmented from polyhedra
|- align=center
|[[tetrahedron]]<br>[[triangular prism]]
|[[square pyramid]]<br>[[cube]]
|[[pentagonal pyramid]]<br>[[pentagonal prism]]
|[[tetrahedron]]<br>[[octahedron]]
|[[square pyramid]]<br>[[square antiprism]]
|[[pentagonal pyramid]]<br>[[pentagonal antiprism]]
|- align=center
|[[File:tetrahedron.png|40px]] [[File:triangular prism.png|40px]]
|[[File:square pyramid.png|40px]] [[File:hexahedron.png|40px]]
|[[File:pentagonal pyramid.png|40px]] [[File:pentagonal prism.png|40px]]
|[[File:tetrahedron.png|40px]] [[File:octahedron.png|40px]]
|[[File:square pyramid.png|40px]] [[File:square antiprism.png|40px]]
|[[File:pentagonal pyramid.png|40px]] [[File:pentagonal antiprism.png|40px]]
|}

==== 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.

{| class="wikitable" width=800
|-
!colspan=3|[[Bipyramid]]s
!colspan=3|[[Elongated bipyramid]]s
!colspan=3|[[Gyroelongated bipyramid]]s
|-
!J12 3<>
!Regular
!J13 5<>
!J14 3<=>
!J15 4<=>
!J16 5<=>
!Coplanar
!J17 4<z>
!Regular
|- align=center
| [[Triangular bipyramid]]
| Square bipyramid<br>([[octahedron]])
| [[Pentagonal bipyramid]]
| [[Elongated triangular bipyramid]]
| [[Elongated square bipyramid]]
| [[Elongated pentagonal bipyramid]]
| Gyroelongated triangular bipyramid<br>([[trigonal trapezohedron]])
| [[Gyroelongated square bipyramid]]
| Gyroelongated pentagonal bipyramid<br>([[icosahedron]])
|-
| [[Image:Triangular dipyramid.png|80px]]
| [[Image:Octahedron.png|80px]]
| [[Image:Pentagonal dipyramid.png|80px]]
| [[Image:Elongated triangular dipyramid.png|80px]]
| [[Image:Elongated square dipyramid.png|80px]]
| [[Image:Elongated pentagonal dipyramid.png|80px]]
| [[File:Gyroelongated triangular bipyramid.png|80px]]
| [[Image:Gyroelongated square dipyramid.png|80px]]
| [[File:Pentagonal gyroelongated bipyramid.png|80px]]
|-
| [[Image:Johnson solid 12 net.png|80px]]
| [[File:Octahedron flat.svg|80px]]
| [[Image:Johnson solid 13 net.png|80px]]
| [[Image:Johnson solid 14 net.png|80px]]
| [[Image:Johnson solid 15 net.png|80px]]
| [[Image:Johnson solid 16 net.png|80px]]
|
| [[Image:Johnson solid 17 net.png|80px]]
| [[File:Icosahedron flat.svg|80px]]
|- align=center
!colspan=9|Augmented from polyhedra
|- align=center
|[[tetrahedron]]
|[[square pyramid]]
|[[pentagonal pyramid]]
|[[tetrahedron]]<br>[[triangular prism]]
|[[square pyramid]]<br>[[cube]]
|[[pentagonal pyramid]]<br>[[pentagonal prism]]
|[[tetrahedron]]<br>[[Octahedron]]
|[[square pyramid]]<br>[[square antiprism]]
|[[pentagonal pyramid]]<br>[[pentagonal antiprism]]
|- align=center
|[[File:tetrahedron.png|40px]]
|[[File:square pyramid.png|40px]]
|[[File:pentagonal pyramid.png|40px]]
|[[File:tetrahedron.png|40px]] [[File:triangular prism.png|40px]]
|[[File:square pyramid.png|40px]] [[File:hexahedron.png|40px]]
|[[File:pentagonal pyramid.png|40px]] [[File:pentagonal prism.png|40px]]
|[[File:tetrahedron.png|40px]] [[File:octahedron.png|40px]]
|[[File:square pyramid.png|40px]] [[File:square antiprism.png|40px]]
|[[File:pentagonal pyramid.png|40px]] [[File:pentagonal antiprism.png|40px]]
|}

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

==== Elongated and gyroelongated cupolae and rotundas ====

{| class="wikitable" width=640
|-
!colspan=4|[[Elongated cupola]]
!Elongated rotunda
!colspan=4|[[Gyroelongated cupola]]
!Gyroelongated rotunda
|-
!Coplanar
!J18 3c=
!J19 4c= eC-
!J20 5c=
!J21 5r=
!Concave
!J22 3cz
!J23 4cz
!J24 5cz
!J25 5rz
|- align=center
| 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]]
|- align=center
| [[Image:Elongated digonal cupola.png|80px]]
| [[Image:Elongated triangular cupola.png|80px]]
| [[Image:Elongated square cupola.png|80px]]
| [[Image:Elongated pentagonal cupola.svg|80px]]
| [[Image:Elongated pentagonal rotunda.png|80px]]
| [[Image:Gyroelongated digonal cupola.png|80px]]
| [[Image:Gyroelongated triangular cupola.png|80px]]
| [[Image:Gyroelongated square cupola.png|80px]]
| [[Image:Gyroelongated pentagonal cupola.png|80px]]
| [[Image:Gyroelongated pentagonal rotunda.png|80px]]
|- align=center
|
| [[Image:Johnson solid 18 net.png|80px]]
| [[Image:Johnson solid 19 net.png|80px]]
| [[Image:Johnson solid 20 net.png|80px]]
| [[Image:Johnson solid 21 net.png|80px]]
|
| [[Image:Johnson solid 22 net.png|80px]]
| [[Image:Johnson solid 23 net.png|80px]]
| [[Image:Johnson solid 24 net.png|80px]]
| [[Image:Johnson solid 25 net.png|80px]]
|- align=center
!colspan=10|Augmented from polyhedra
|- align=center
|[[Square prism]]<br>[[Triangular prism]]
|[[Hexagonal prism]]<br>[[Triangular cupola]]
|[[Octagonal prism]]<br>[[Square cupola]]
|[[Decagonal prism]]<br>[[Pentagonal cupola]]
|[[Decagonal prism]]<br>[[Pentagonal rotunda]]
|[[square antiprism]]<br>[[Triangular prism]]
|[[Hexagonal antiprism]]<br>[[Triangular cupola]]
|[[Octagonal antiprism]]<br>[[Square cupola]]
|[[Decagonal antiprism]]<br>[[Pentagonal cupola]]
|[[Decagonal antiprism]]<br>[[Pentagonal rotunda]]
|- align=center
|[[File:tetragonal prism.png|40px]] [[File:Triangular prism.png|40px]]
|[[File:Hexagonal prism.png|40px]] [[File:Triangular cupola.png|40px]]
|[[File:Octagonal prism.png|40px]] [[File:Square cupola.png|40px]]
|[[File:Decagonal prism.png|40px]] [[File:Pentagonal cupola.png|40px]]
|[[File:Decagonal prism.png|40px]] [[File:Pentagonal rotunda.png|40px]]
|[[File:Square antiprism.png|40px]] [[File:Triangular prism.png|40px]]
|[[File:Hexagonal antiprism.png|40px]] [[File:Triangular cupola.png|40px]]
|[[File:Octagonal antiprism.png|40px]] [[File:Square cupola.png|40px]]
|[[File:Decagonal antiprism.png|40px]] [[File:Pentagonal cupola.png|40px]]
|[[File:Decagonal antiprism.png|40px]] [[File:Pentagonal rotunda.png|40px]]
|}

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

{| class="wikitable" width=640
|-
!colspan=4|[[Orthobicupola]]
!colspan=4|[[Gyrobicupola]]
|-
!Coplanar
!J27 3cc
!J28 4cc
!J30 5cc
!J26 2cc*
!Semiregular
!J29 4cc*
!J31 5cc*
|- align=center
| [[Orthobifastigium]]
| [[Triangular orthobicupola]]
| [[Square orthobicupola]]
| [[Pentagonal orthobicupola]]
| [[Gyrobifastigium]]
| Triangular gyrobicupola <br>([[cuboctahedron]])
| [[Square gyrobicupola]]
| [[Pentagonal gyrobicupola]]
|- align=center
| [[File:Digonal orthobicupola.png|80px]]
| [[Image:Triangular orthobicupola.png|80px]]
| [[Image:Square orthobicupola.png|80px]]
| [[Image:Pentagonal orthobicupola.png|80px]]
| [[Image:Gyrobifastigium.png|80px]]
| [[Image:Cuboctahedron.png|80px]]
| [[Image:Square gyrobicupola.png|80px]]
| [[Image:Pentagonal gyrobicupola.png|80px]]
|- align=center
|
| [[Image:Johnson solid 27 net.png|80px]]
| [[Image:Johnson solid 28 net.png|80px]]
| [[Image:Johnson solid 30 net.png|80px]]
| [[Image:Johnson solid 26 net.png|80px]]
| [[File:Cuboctahedron flat.svg|80px]]
| [[Image:Johnson solid 29 net.png|80px]]
| [[Image:Johnson solid 31 net.png|80px]]
|-
!colspan=8|Augmented from polyhedron
|- align=center
|[[Triangular prism]]
|[[Triangular cupola]]
|[[Square cupola]]
|[[Pentagonal cupola]]
|[[Triangular prism]]
|[[Triangular cupola]]
|[[Square cupola]]
|[[Pentagonal cupola]]
|- align=center
| [[Image:Triangular prism.png|40px]]
| [[Image:Triangular cupola.png|40px]]
| [[Image:Square cupola.png|40px]]
| [[Image:Pentagonal cupola.png|40px]]
| [[Image:Triangular prism.png|40px]]
| [[Image:Triangular cupola.png|40px]]
| [[Image:Square cupola.png|40px]]
| [[Image:Pentagonal cupola.png|40px]]
|}

==== Cupola-rotundas and birotundas ====
The pentagonal gyrobirotunda is an [[Archimedean solid]] (in this case the [[icosidodecahedron]]), so it is not a Johnson solid.

{| class="wikitable" width=640
!colspan=2|Cupola-rotunda
!colspan=2|Birotunda
|-
!J32 5cr
!J33 5cr*
!J34 5rr aD*
!Semiregular
|- align=center
| [[Pentagonal orthocupolarotunda]]
| [[Pentagonal gyrocupolarotunda]]
| [[Pentagonal orthobirotunda]]
| Pentagonal gyrobirotunda<br>([[icosidodecahedron]])
|- align=center
| [[Image:Pentagonal orthocupolarotunda.png|80px]]
| [[Image:Pentagonal gyrocupolarotunda.png|80px]]
| [[Image:Pentagonal orthobirotunda.png|80px]]
| [[Image:icosidodecahedron.png|80px]]
|- align=center
| [[Image:Johnson solid 32 net.png|80px]]
| [[Image:Johnson solid 33 net.png|80px]]
| [[Image:Johnson solid 34 net.png|80px]]
| [[File:Icosidodecahedron flat.svg|80px]]
|- align=center
!colspan=4|Augmented from polyhedra
|- align=center
|colspan=2|[[Pentagonal cupola]]<br>[[Pentagonal rotunda]]
|colspan=2|[[Pentagonal rotunda]]
|- align=center
|colspan=2|[[File:Pentagonal cupola.png|40px]] [[File:Pentagonal rotunda.png|40px]]
|colspan=2|[[File:Pentagonal rotunda.png|40px]]
|}

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

{| class="wikitable" width=640
!colspan=4|[[Elongated orthobicupola]]
!colspan=4|[[Elongated gyrobicupola]]
|-
!Coplanar
!J35 3c=c
!Semiregular
!J38 5c=c
!Coplanar
!J36 3c=c*
!J37 4c=c* eC*
!J39 5c=c*
|- align=center
| Elongated orthobifastigium
| [[Elongated triangular orthobicupola]]
| Elongated square orthobicupola<br>([[rhombicuboctahedron]])
| [[Elongated pentagonal orthobicupola]]
| [[Elongated gyrobifastigium]]
| [[Elongated triangular gyrobicupola]]
| [[Elongated square gyrobicupola]]
| [[Elongated pentagonal gyrobicupola]]
|- align=center
| [[File:Elongated digonal orthobicupola.png|60px]]
| [[Image:Elongated triangular orthobicupola.png|80px]]
| [[Image:small rhombicuboctahedron.png|80px]]
| [[Image:Elongated pentagonal orthobicupola.png|80px]]
| [[File:Elongated digonal gyrobicupola.png|60px]]
| [[Image:Elongated triangular gyrobicupola.png|80px]]
| [[Image:Elongated square gyrobicupola.png|80px]]
| [[Image:Elongated pentagonal gyrobicupola.png|80px]]
|- align=center
|
| [[Image:Johnson solid 35 net.png|80px]]
| [[File:Rhombicuboctahedron flat.png|80px]]
| [[Image:Johnson solid 38 net.png|80px]]
|
| [[Image:Johnson solid 36 net.png|80px]]
| [[Image:Johnson solid 37 net.png|80px]]
| [[Image:Johnson solid 39 net.png|80px]]
|- align=center
!colspan=8|Augmented from polyhedra
|- align=center
|[[Square prism]]<br>[[Triangular prism]]
|[[Hexagonal prism]]<br>[[Triangular cupola]]
|[[Octagonal prism]]<br>[[Square cupola]]
|[[Decagonal prism]]<br>[[Pentagonal cupola]]
|[[Square prism]]<br>[[Triangular prism]]
|[[Hexagonal prism]]<br>[[Triangular cupola]]
|[[Octagonal prism]]<br>[[Square cupola]]
|[[Decagonal prism]]<br>[[Pentagonal cupola]]
|- align=center
|[[File:tetragonal prism.png|40px]] [[File:Triangular prism.png|40px]]
|[[File:Hexagonal prism.png|40px]] [[File:Triangular cupola.png|40px]]
|[[File:Octagonal prism.png|40px]] [[File:Square cupola.png|40px]]
|[[File:Decagonal prism.png|40px]] [[File:Pentagonal cupola.png|40px]]
|[[File:tetragonal prism.png|40px]] [[File:Triangular prism.png|40px]]
|[[File:Hexagonal prism.png|40px]] [[File:Triangular cupola.png|40px]]
|[[File:Octagonal prism.png|40px]] [[File:Square cupola.png|40px]]
|[[File:Decagonal prism.png|40px]] [[File:Pentagonal cupola.png|40px]]
|}

==== Elongated cupola-rotundas and birotundas ====

{| class="wikitable" width=640
!colspan=2|Elongated cupola-rotunda
!colspan=2|Elongated birotunda
|-
!J40 5c=r
!J41 5c=r*
!J42 5r=r
!J43 5r=r*
|- align=center
| [[Elongated pentagonal orthocupolarotunda]]
| [[Elongated pentagonal gyrocupolarotunda]]
| [[Elongated pentagonal orthobirotunda]]
| [[Elongated pentagonal gyrobirotunda]]
|- align=center
| [[Image:Elongated pentagonal orthocupolarotunda.png|80px]]
| [[Image:Elongated pentagonal gyrocupolarotunda.png|80px]]
| [[Image:Elongated pentagonal orthobirotunda.png|80px]]
| [[Image:Elongated pentagonal gyrobirotunda.png|80px]]
|- align=center
| [[Image:Johnson solid 40 net.png|80px]]
| [[Image:Johnson solid 41 net.png|80px]]
| [[Image:Johnson solid 42 net.png|80px]]
| [[Image:Johnson solid 43 net.png|80px]]
|- align=center
!colspan=4|Augmented from polyhedra
|- align=center
|colspan=2|[[Decagonal prism]]<br>[[Pentagonal cupola]]<br>[[Pentagonal rotunda]]
|colspan=2|[[Decagonal prism]]<br>[[Pentagonal rotunda]]
|- align=center
|colspan=2|[[File:Decagonal prism.png|40px]] [[File:Pentagonal cupola.png|40px]] [[File:Pentagonal rotunda.png|40px]]
|colspan=2|[[File:Decagonal prism.png|40px]] [[File:Pentagonal rotunda.png|40px]]
|}

==== Gyroelongated bicupolae, cupola-rotundas, and birotundas ====
These Johnson solids have 2 chiral forms.
{| class="wikitable" width=640
!colspan=4|[[Gyroelongated bicupola]]
!Gyroelongated cupola-rotunda
!Gyroelongated birotunda
|-
!Concave
!J44 3czc
!J45 4czc
!J46 5czc
!J47 5czr
!J48 5rzr
|- align=center
| Gyroelongated bifastigium
| [[Gyroelongated triangular bicupola]]
| [[Gyroelongated square bicupola]]
| [[Gyroelongated pentagonal bicupola]]
| [[Gyroelongated pentagonal cupolarotunda]]
| [[Gyroelongated pentagonal birotunda]]
|- align=center
| [[Image:Gyroelongated digonal bicupola.png|80px]]
| [[Image:Gyroelongated triangular bicupola.png|80px]]
| [[Image:Gyroelongated square bicupola.png|80px]]
| [[Image:Gyroelongated pentagonal bicupola.png|80px]]
| [[Image:Gyroelongated pentagonal cupolarotunda.png|80px]]
| [[Image:Gyroelongated pentagonal birotunda.png|80px]]
|- align=center
|
| [[Image:Johnson solid 44 net.png|80px]]
| [[Image:Johnson solid 45 net.png|80px]]
| [[Image:Johnson solid 46 net.png|80px]]
| [[Image:Johnson solid 47 net.png|80px]]
| [[Image:Johnson solid 48 net.png|80px]]
|- align=center
!colspan=6|Augmented from polyhedra
|- align=center
| [[Triangular prism]]<br>[[Square antiprism]]
| [[Triangular cupola]]<br>[[Hexagonal antiprism]]
| [[Square cupola]]<br>[[Octagonal antiprism]]
| [[Pentagonal cupola]]<br>[[Decagonal antiprism]]
| [[Pentagonal cupola]]<br>[[Pentagonal rotunda]]<br>[[Decagonal antiprism]]
| [[Pentagonal rotunda]]<br>[[Decagonal antiprism]]
|- align=center
| [[File:triangular prism.png|40px]] [[File:square antiprism.png|40px]]
| [[File:triangular cupola.png|40px]] [[File:Hexagonal antiprism.png|40px]]
| [[File:Square cupola.png|40px]] [[File:Octagonal antiprism.png|40px]]
| [[File:Pentagonal cupola.png|40px]] [[File:Decagonal antiprism.png|40px]]
| [[File:Pentagonal cupola.png|40px]] [[File:Pentagonal rotunda.png|40px]] [[File:Decagonal antiprism.png|40px]]
| [[File:Pentagonal rotunda.png|40px]] [[File:Decagonal antiprism.png|40px]]
|}

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

{| class="wikitable" width=640
!colspan=3|Augmented triangular prisms
!colspan=2|Augmented pentagonal prisms
!colspan=4|Augmented hexagonal prisms
|-
!J49 3=+
!J50 3=++
!J51 3=+++
!J52 5=+
!J53 5=++
!J54 6=+
!J55 6=++
!J56 6=+x
!J57 6=+++
|- align=center
| [[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]]
|- align=center
| [[Image:Augmented triangular prism.png|80px]]
| [[Image:Biaugmented triangular prism.png|80px]]
| [[Image:Triaugmented triangular prism.png|80px]]
| [[Image:Augmented pentagonal prism.png|80px]]
| [[Image:Biaugmented pentagonal prism.png|80px]]
| [[Image:Augmented hexagonal prism.png|80px]]
| [[Image:Parabiaugmented hexagonal prism.png|80px]]
| [[Image:Metabiaugmented hexagonal prism.png|80px]]
| [[Image:Triaugmented hexagonal prism.png|80px]]
|- align=center
| [[Image:Johnson solid 49 net.png|80px]]
| [[Image:Johnson solid 50 net.png|80px]]
| [[Image:Johnson solid 51 net.png|80px]]
| [[Image:Johnson solid 52 net.png|80px]]
| [[Image:Johnson solid 53 net.png|80px]]
| [[Image:Johnson solid 54 net.png|80px]]
| [[Image:Johnson solid 55 net.png|80px]]
| [[Image:Johnson solid 56 net.png|80px]]
| [[Image:Johnson solid 57 net.png|80px]]
|- align=center
!colspan=9|Augmented from polyhedra
|- align=center
|colspan=3|[[Triangular prism]]<br>[[Square pyramid]]
|colspan=2|[[Pentagonal prism]]<br>[[Square pyramid]]
|colspan=4|[[Hexagonal prism]]<br>[[Square pyramid]]
|- align=center
|colspan=3|[[File:triangular prism.png|40px]] [[File:square pyramid.png|40px]]
|colspan=2|[[File:Pentagonal prism.png|40px]] [[File:square pyramid.png|40px]]
|colspan=4|[[File:Hexagonal prism.png|40px]] [[File:square pyramid.png|40px]]
|}
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 ====
{| class="wikitable" width=640
|-
!J58 D+
!J59 D++
!J60 D+x
!J61 D+++
|- align=center
| [[Augmented dodecahedron]]
| [[Parabiaugmented dodecahedron]]
| [[Metabiaugmented dodecahedron]]
| [[Triaugmented dodecahedron]]
|- align=center
| [[Image:Augmented dodecahedron.png|80px]]
| [[Image:Parabiaugmented dodecahedron.png|80px]]
| [[Image:Metabiaugmented dodecahedron.png|80px]]
| [[Image:Triaugmented dodecahedron.png|80px]]
|- align=center
| [[Image:Johnson solid 58 net.png|80px]]
| [[Image:Johnson solid 59 net.png|80px]]
| [[Image:Johnson solid 60 net.png|80px]]
| [[Image:Johnson solid 61 net.png|80px]]
|- align=center
!colspan=4|Augmented from polyhedra
|- align=center
|colspan=4|[[Dodecahedron]] and [[pentagonal pyramid]]
|- align=center
|colspan=4|[[File:dodecahedron.png|40px]] [[File:Pentagonal pyramid.png|40px]]
|}

==== Diminished and augmented diminished icosahedra ====
{| class="wikitable" width=640
!colspan=4|Diminished icosahedron
!Augmented tridiminished icosahedron
|-
!J11<br>(Repeated)
!Uniform
!J62 I-/
!J63 I---
!J64 I---+
|- align=center
| Diminished icosahedron<br>([[Gyroelongated pentagonal pyramid]])
| Parabidiminished icosahedron<br>([[Pentagonal antiprism]])
| [[Metabidiminished icosahedron]]
| [[Tridiminished icosahedron]]
| [[Augmented tridiminished icosahedron]]
|- align=center
| [[Image:Gyroelongated pentagonal pyramid.png|80px]]
| [[Image:Pentagonal_antiprism.png|80px]]
| [[Image:Metabidiminished icosahedron.png|80px]]
| [[Image:Tridiminished icosahedron.png|80px]]
| [[Image:Augmented tridiminished icosahedron.png|80px]]
|- align=center
| [[Image:Johnson solid 11 net.png|80px]]
|
| [[Image:Johnson solid 62 net.png|80px]]
| [[Image:Johnson solid 63 net.png|80px]]
| [[Image:Johnson solid 64 net.png|80px]]
|}

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

==== Augmented Archimedean solids====
{| class="wikitable" width=640
!Augmented truncated tetrahedron
!colspan=2|Augmented truncated cubes
!colspan=4|Augmented truncated dodecahedra
|-
!J65 tT+
!J66 tC+
!J67 tC++
!J68 tD+
!J69 tD++
!J70 tD+x
!J71 tD+++
|- align=center
| [[Augmented truncated tetrahedron]]
| [[Augmented truncated cube]]
| [[Biaugmented truncated cube]]
| [[Augmented truncated dodecahedron]]
| [[Parabiaugmented truncated dodecahedron]]
| [[Metabiaugmented truncated dodecahedron]]
| [[Triaugmented truncated dodecahedron]]
|- align=center
| [[Image:Augmented truncated tetrahedron.png|80px]]
| [[Image:Augmented truncated cube.png|80px]]
| [[Image:Biaugmented truncated cube.png|80px]]
| [[Image:Augmented truncated dodecahedron.png|80px]]
| [[Image:Parabiaugmented truncated dodecahedron.png|80px]]
| [[Image:Metabiaugmented truncated dodecahedron.png|80px]]
| [[Image:Triaugmented truncated dodecahedron.png|80px]]
|- align=center
| [[Image:Johnson solid 65 net.png|80px]]
| [[Image:Johnson solid 66 net.png|80px]]
| [[Image:Johnson solid 67 net.png|80px]]
| [[Image:Johnson solid 68 net.png|80px]]
| [[Image:Johnson solid 69 net.png|80px]]
| [[Image:Johnson solid 70 net.png|80px]]
| [[Image:Johnson solid 71 net.png|80px]]
|- align=center
!colspan=7|Augmented from polyhedra
|- align=center
|[[truncated tetrahedron]]<br>[[triangular cupola]]
|colspan=2|[[truncated cube]]<br>[[square cupola]]
|colspan=4|[[truncated dodecahedron]]<br>[[pentagonal cupola]]
|- align=center
|[[File:truncated tetrahedron.png|40px]] [[File:triangular cupola.png|40px]]
|colspan=2|[[File:truncated hexahedron.png|40px]] [[File:square cupola.png|40px]]
|colspan=4|[[File:truncated dodecahedron.png|40px]] [[File:pentagonal cupola.png|40px]]
|}

==== Gyrate and diminished rhombicosidodecahedra ====
{| class="wikitable" width=640
!colspan=4|Gyrate rhombicosidodecahedra
|-
!J72 eD*
!J73 eD**
!J74 eD*'
!J75 eD***
|- align=center
| [[Gyrate rhombicosidodecahedron]]
| [[Parabigyrate rhombicosidodecahedron]]
| [[Metabigyrate rhombicosidodecahedron]]
| [[Trigyrate rhombicosidodecahedron]]
|- align=center
| [[Image:Gyrate rhombicosidodecahedron.png|80px]]
| [[Image:Parabigyrate rhombicosidodecahedron.png|80px]]
| [[Image:Metabigyrate rhombicosidodecahedron.png|80px]]
| [[Image:Trigyrate rhombicosidodecahedron.png|80px]]
|- align=center
| [[Image:Johnson solid 72 net.png|80px]]
| [[Image:Johnson solid 73 net.png|80px]]
| [[Image:Johnson solid 74 net.png|80px]]
| [[Image:Johnson solid 75 net.png|80px]]
|-
!colspan=4|Diminished rhombicosidodecahedra
|-
!J76 eD-
!J80 eD--
!J81 eD-/
!J83 eD---
|- align=center
| [[Diminished rhombicosidodecahedron]]
| [[Parabidiminished rhombicosidodecahedron]]
| [[Metabidiminished rhombicosidodecahedron]]
| [[Tridiminished rhombicosidodecahedron]]
|- align=center
| [[Image:Diminished rhombicosidodecahedron.png|80px]]
| [[Image:Parabidiminished rhombicosidodecahedron.png|80px]]
| [[Image:Metabidiminished rhombicosidodecahedron.png|80px]]
| [[Image:Tridiminished rhombicosidodecahedron.png|80px]]
|- align=center
| [[Image:Johnson solid 76 net.png|80px]]
| [[Image:Johnson solid 80 net.png|80px]]
| [[Image:Johnson solid 81 net.png|80px]]
| [[Image:Johnson solid 83 net.png|80px]]
|-
!colspan=4|Gyrate diminished rhombicosidodecahedra
|-
!J77 -*
!J78 -'
!J79 -**
!J82 --*
|- align=center
| [[Paragyrate diminished rhombicosidodecahedron]]
| [[Metagyrate diminished rhombicosidodecahedron]]
| [[Bigyrate diminished rhombicosidodecahedron]]
| [[Gyrate bidiminished rhombicosidodecahedron]]
|- align=center
| [[Image:Paragyrate diminished rhombicosidodecahedron.png|80px]]
| [[Image:Metagyrate diminished rhombicosidodecahedron.png|80px]]
| [[Image:Bigyrate diminished rhombicosidodecahedron.png|80px]]
| [[Image:Gyrate bidiminished rhombicosidodecahedron.png|80px]]
|- align=center
| [[Image:Johnson solid 77 net.png|80px]]
| [[Image:Johnson solid 78 net.png|80px]]
| [[Image:Johnson solid 79 net.png|80px]]
| [[Image:Johnson solid 82 net.png|80px]]
|}

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.
{| class="wikitable" width=640
!J27
!J3
!J34
!J6
!J37
!J19
!Uniform
|- align=center
| Gyrate cuboctahedron<br>([[triangular orthobicupola]])
| Diminished cuboctahedron<br>([[triangular cupola]])
| Gyrate icosidodecahedron<br>([[pentagonal orthobirotunda]])
| Diminished icosidodecahedron<br>([[pentagonal rotunda]])
| Gyrate rhombicuboctahedron<br>([[elongated square gyrobicupola]])
| Diminished rhombicuboctahedron<br>([[elongated square cupola]])
| Bidiminished rhombicuboctahedron<br>([[octagonal prism]])
|- align=center
| [[Image:Triangular orthobicupola.png|80px]]
| [[Image:Triangular cupola.png|80px]]
| [[Image:Pentagonal orthobirotunda.png|80px]]
| [[Image:Pentagonal rotunda.png|80px]]
| [[Image:Elongated square gyrobicupola.png|80px]]
| [[Image:Elongated square cupola.png|80px]]
| [[Image:Octagonal prism.png|80px]]
|- align=center
| [[Image:Johnson solid 27 net.png|80px]]
| [[Image:Johnson solid 3 net.png|80px]]
| [[Image:Johnson solid 34 net.png|80px]]
| [[Image:Johnson solid 6 net.png|80px]]
| [[Image:Johnson solid 37 net.png|80px]]
| [[Image:Johnson solid 19 net.png|80px]]
|
|- align=center
!colspan=7|Gyrated or diminished from polyhedra
|- align=center
|colspan=2|Cuboctahedron
|colspan=2|Icosidodecahedron
|colspan=3|Rhombicuboctahedron
|- align=center
|colspan=2|[[Image:Cuboctahedron.png|80px]]
|colspan=2|[[Image:Icosidodecahedron.png|80px]]
|colspan=3|[[Image:Small rhombicuboctahedron.png|80px]]
|}

=== Elementary solids ===
Johnson solids 84 to 92 are not derived from "cut-and-paste" manipulations of [[uniform polyhedron|uniform]] solids.

==== Snub antiprisms ====
The [[snub (geometry)|snub]] [[antiprism]]s 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.

{| class=wikitable
!J84
!Regular
!J85
|- align=center valign=top
|[[Snub disphenoid]]<br>ss{2,4}
|[[Icosahedron]]<br>ss{2,6}
|[[Snub square antiprism]]<br>ss{2,8}
|- align=center
|Digonal gyrobianticupola
|Triangular gyrobianticupola
|Square gyrobianticupola
|- align=center
|[[File:Snub digonal antiprism.png|50px]]
|[[File:snub triangular antiprism.png|80px]]
|[[File:Snub square antiprism colored.png|80px]]<!--file misnamed!-->
|- align=center
| [[File:Snub disphenoid net snubcoloring.png|80px]]
| [[File:Snub triangular antiprism net.png|100px]]
| [[File:Snub square antiprism net snubcoloring.png|120px]]
|}

==== Others ====
{| class="wikitable" width=640
|-
!J86
!J87
!J88
|- align=center
| [[Sphenocorona]]
| [[Augmented sphenocorona]]
| [[Sphenomegacorona]]
|- align=center
| [[Image:Sphenocorona.png|80px]]
| [[Image:Augmented sphenocorona.png|80px]]
| [[Image:Sphenomegacorona.png|80px]]

|- align=center
| [[Image:Johnson solid 86 net.png|80px]]
| [[Image:Johnson solid 87 net.png|80px]]
| [[Image:Johnson solid 88 net.png|80px]]
|-
!J89
!J90
!J91
!J92
|- align=center
| [[Hebesphenomegacorona]]
| [[Disphenocingulum]]
| [[Bilunabirotunda]]
| [[Triangular hebesphenorotunda]]
|- align=center
| [[Image:Hebesphenomegacorona.png|80px]]
| [[Image:Disphenocingulum.png|80px]]
| [[Image:Bilunabirotunda.png|80px]]
| [[Image:Triangular hebesphenorotunda.png|80px]]
|- align=center
| [[Image:Johnson solid 89 net.png|100px]]
| [[Image:Johnson solid 90 net.png|100px]]
| [[Image:Johnson solid 91 net.png|80px]]
| [[Image:Johnson solid 92 net.png|80px]]
|}

== Classification by types of faces ==

=== Triangle-faced Johnson solids ===
Five Johnson solids are [[deltahedra]], with all equilateral triangle faces:
{| class=wikitable
|
:J12 [[Triangular bipyramid]] [[File:Triangular dipyramid.png|40px]]
:J13 [[Pentagonal bipyramid]] [[File:Pentagonal dipyramid.png|40px]]
:J17 [[Gyroelongated square bipyramid]] [[File:Gyroelongated square dipyramid.png|40px]]
|
:J51 [[Triaugmented triangular prism]] [[File:Triaugmented triangular prism.png|40px]]
:J84 [[Snub disphenoid]] [[File:Snub disphenoid.png|40px]]
|}

=== Triangle and square-faced Johnson solids ===
Twenty four Johnson solids have only triangle or square faces:
{| class=wikitable
|
:J1 [[Square pyramid]] [[File:Square pyramid.png|40px]]
:J7 [[Elongated triangular pyramid]] [[File:Elongated triangular pyramid.png|40px]]
:J8 [[Elongated square pyramid]] [[File:Elongated square pyramid.png|40px]]
:J10 [[Gyroelongated square pyramid]] [[File:Gyroelongated square pyramid.png|40px]]
:J14 [[Elongated triangular bipyramid]] [[File:Elongated triangular dipyramid.png|40px]]
:J15 [[Elongated square bipyramid]] [[File:Elongated square dipyramid.png|40px]]
:J16 [[Elongated pentagonal bipyramid]] [[File:Elongated pentagonal dipyramid.png|40px]]
:J26 [[Gyrobifastigium]] [[File:Gyrobifastigium.png|40px]]
|
:J27 [[Triangular orthobicupola]] [[File:Triangular orthobicupola.png|40px]]
:J28 [[Square orthobicupola]] [[File:Square orthobicupola.png|40px]]
:J29 [[Square gyrobicupola]] [[File:Square gyrobicupola.png|40px]]
:J35 [[Elongated triangular orthobicupola]] [[File:Elongated triangular orthobicupola.png|40px]]
:J36 [[Elongated triangular gyrobicupola]] [[File:Elongated triangular gyrobicupola.png|40px]]
:J37 [[Elongated square gyrobicupola]] [[File:Elongated square gyrobicupola.png|40px]]
:J44 [[Gyroelongated triangular bicupola]] [[File:Gyroelongated triangular bicupola.png|40px]]
:J45 [[Gyroelongated square bicupola]] [[File:Gyroelongated square bicupola.png|40px]]
|
:J49 [[Augmented triangular prism]] [[File:Augmented triangular prism.png|40px]]
:J50 [[Biaugmented triangular prism]] [[File:Biaugmented triangular prism.png|40px]]
:J85 [[Snub square antiprism]] [[File:Snub square antiprism.png|40px]]
:J86 [[Sphenocorona]] [[File:Sphenocorona.png|40px]]
:J87 [[Augmented sphenocorona]] [[File:Augmented sphenocorona.png|40px]]
:J88 [[Sphenomegacorona]] [[File:Sphenomegacorona.png|40px]]
:J89 [[Hebesphenomegacorona]] [[File:Hebesphenomegacorona.png|40px]]
:J90 [[Disphenocingulum]] [[File:Disphenocingulum.png|40px]]
|}

=== Triangle and pentagon-faced Johnson solids ===
Eleven Johnson solids have only triangle and pentagon faces:
{| class=wikitable
|
:J2 [[Pentagonal pyramid]] [[File:Pentagonal pyramid.png|40px]]
:J11 [[Gyroelongated pentagonal pyramid]] [[File:Gyroelongated pentagonal pyramid.png|40px]]
:J34 [[Pentagonal orthobirotunda]] [[File:Pentagonal orthobirotunda.png|40px]]
:J48 [[Gyroelongated pentagonal birotunda]] [[File:Gyroelongated pentagonal birotunda.png|40px]]
:J58 [[Augmented dodecahedron]] [[File:Augmented dodecahedron.png|40px]]
:J59 [[Parabiaugmented dodecahedron]] [[File:Parabiaugmented dodecahedron.png|40px]]
|
:J60 [[Metabiaugmented dodecahedron]] [[File:Metabiaugmented dodecahedron.png|40px]]
:J61 [[Triaugmented dodecahedron]] [[File:Triaugmented dodecahedron.png|40px]]
:J62 [[Metabidiminished icosahedron]] [[File:Metabidiminished icosahedron.png|40px]]
:J63 [[Tridiminished icosahedron]] [[File:Tridiminished icosahedron.png|40px]]
:J64 [[Augmented tridiminished icosahedron]] [[File:Augmented tridiminished icosahedron.png|40px]]
|}

=== Triangle, square, and pentagon-faced Johnson solids ===
Twenty Johnson solids have only triangle, square, and pentagon faces:
{| class=wikitable
|
:J09 [[Elongated pentagonal pyramid]] [[File:Elongated pentagonal pyramid.png|40px]]
:J30 [[Pentagonal orthobicupola]] [[File:Pentagonal orthobicupola.png|40px]]
:J31 [[Pentagonal gyrobicupola]] [[File:Pentagonal gyrobicupola.png|40px]]
:J32 [[Pentagonal orthocupolarotunda]] [[File:Pentagonal orthocupolarotunda.png|40px]]
:J33 [[Pentagonal gyrocupolarotunda]] [[File:Pentagonal gyrocupolarotunda.png|40px]]
:J38 [[Elongated pentagonal orthobicupola]] [[File:Elongated pentagonal orthobicupola.png|40px]]
:J39 [[Elongated pentagonal gyrobicupola]] [[File:Elongated pentagonal gyrobicupola.png|40px]]
:J40 [[Elongated pentagonal orthocupolarotunda]] [[File:Elongated pentagonal orthocupolarotunda.png|40px]]
:J41 [[Elongated pentagonal gyrocupolarotunda]] [[File:Elongated pentagonal gyrocupolarotunda.png|40px]]
:J42 [[Elongated pentagonal orthobirotunda]] [[File:Elongated pentagonal orthobirotunda.png|40px]]
|
:J43 [[Elongated pentagonal gyrobirotunda]] [[File:Elongated pentagonal gyrobirotunda.png|40px]]
:J46 [[Gyroelongated pentagonal bicupola]] [[File:Gyroelongated pentagonal bicupola.png|40px]]
:J47 [[Gyroelongated pentagonal cupolarotunda ]] [[File:Gyroelongated pentagonal cupolarotunda.png|40px]]
:J52 [[Augmented pentagonal prism]] [[File:Augmented pentagonal prism.png|40px]]
:J53 [[Biaugmented pentagonal prism]] [[File:Biaugmented pentagonal prism.png|40px]]
:J72 [[Gyrate rhombicosidodecahedron]] [[File:Gyrate rhombicosidodecahedron.png|40px]]
:J73 [[Parabigyrate rhombicosidodecahedron]] [[File:Parabigyrate rhombicosidodecahedron.png|40px]]
:J74 [[Metabigyrate rhombicosidodecahedron]] [[File:Metabigyrate rhombicosidodecahedron.png|40px]]
:J75 [[Trigyrate rhombicosidodecahedron]] [[File:Trigyrate rhombicosidodecahedron.png|40px]]
:J91 [[Bilunabirotunda]] [[File:Bilunabirotunda.png|40px]]
|}

=== Triangle, square, and hexagon-faced Johnson solids ===
Eight Johnson solids have only triangle, square, and hexagon faces:
{| class=wikitable
|
:J3 [[Triangular cupola]] [[File:Triangular cupola.png|40px]]
:J18 [[Elongated triangular cupola]] [[File:Elongated triangular cupola.png|40px]]
:J22 [[Gyroelongated triangular cupola]] [[File:Gyroelongated triangular cupola.png|40px]]
:J54 [[Augmented hexagonal prism]] [[File:Augmented hexagonal prism.png|40px]]
|
:J55 [[Parabiaugmented hexagonal prism]] [[File:Parabiaugmented hexagonal prism.png|40px]]
:J56 [[Metabiaugmented hexagonal prism]] [[File:Metabiaugmented hexagonal prism.png|40px]]
:J57 [[Triaugmented hexagonal prism]] [[File:Triaugmented hexagonal prism.png|40px]]
:J65 [[Augmented truncated tetrahedron]] [[File:Augmented truncated tetrahedron.png|40px]]
|}

=== Triangle, square, and octagon-faced Johnson solids ===
Five Johnson solids have only triangle, square, and octagon faces:
{| class=wikitable
|
:J4 [[Square cupola]] [[File:Square cupola.png|40px]]
:J19 [[Elongated square cupola]] [[File:Elongated square cupola.png|40px]]
:J23 [[Gyroelongated square cupola]] [[File:Gyroelongated square cupola.png|40px]]
|
:J66 [[Augmented truncated cube]] [[File:Augmented truncated cube.png|40px]]
:J67 [[Biaugmented truncated cube]] [[File:Biaugmented truncated cube.png|40px]]
|}

=== Triangle, pentagon, and decagon-faced Johnson solids ===
Two Johnson solids have only triangle, pentagon, and decagon faces:
{| class=wikitable
|
:J06 [[Pentagonal rotunda]] [[File:Pentagonal rotunda.png|40px]]
:J25 [[Gyroelongated pentagonal rotunda]] [[File:Gyroelongated pentagonal rotunda.png|40px]]
|}

=== Triangle, square, pentagon, and hexagon-faced Johnson solids ===
Only one Johnson solid has triangle, square, pentagon, and hexagon faces:
{| class=wikitable
|
:J92 [[Triangular hebesphenorotunda]] [[File:Triangular hebesphenorotunda.png|40px]]
|}

=== Triangle, square, pentagon, and decagon-faced Johnson solids ===
Sixteen Johnson solids have only triangle, square, pentagon, and decagon faces:
{| class=wikitable
|
:J05 [[Pentagonal cupola]] [[File:Pentagonal cupola.png|40px]]
:J20 [[Elongated pentagonal cupola]] [[File:Elongated pentagonal cupola.svg|40px]]
:J21 [[Elongated pentagonal rotunda]] [[File:Elongated pentagonal rotunda.png|40px]]
:J24 [[Gyroelongated pentagonal cupola]] [[File:Gyroelongated pentagonal cupola.png|40px]]
:J68 [[Augmented truncated dodecahedron]] [[File:Augmented truncated dodecahedron.png|40px]]
:J69 [[Parabiaugmented truncated dodecahedron]] [[File:Parabiaugmented truncated dodecahedron.png|40px]]
:J70 [[Metabiaugmented truncated dodecahedron]] [[File:Metabiaugmented truncated dodecahedron.png|40px]]
:J71 [[Triaugmented truncated dodecahedron]] [[File:Triaugmented truncated dodecahedron.png|40px]]
|
:J76 [[Diminished rhombicosidodecahedron]] [[File:Diminished rhombicosidodecahedron.png|40px]]
:J77 [[Paragyrate diminished rhombicosidodecahedron]] [[File:Paragyrate diminished rhombicosidodecahedron.png|40px]]
:J78 [[Metagyrate diminished rhombicosidodecahedron]] [[File:Metagyrate diminished rhombicosidodecahedron.png|40px]]
:J79 [[Bigyrate diminished rhombicosidodecahedron]] [[File:Bigyrate diminished rhombicosidodecahedron.png|40px]]
:J80 [[Parabidiminished rhombicosidodecahedron]] [[File:Parabidiminished rhombicosidodecahedron.png|40px]]
:J81 [[Metabidiminished rhombicosidodecahedron]] [[File:Metabidiminished rhombicosidodecahedron.png|40px]]
:J82 [[Gyrate bidiminished rhombicosidodecahedron]] [[File:Gyrate bidiminished rhombicosidodecahedron.png|40px]]
:J83 [[Tridiminished rhombicosidodecahedron]] [[File:Tridiminished rhombicosidodecahedron.png|40px]]
|}

== 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.<ref>{{cite web|url=http://bendwavy.org/klitzing/explain/johnson.htm|title=Johnson solids et al.|first=Dr. Richard|last=Klitzing|website=bendwavy.org|access-date=17 April 2018}}</ref>
{| class="wikitable sortable"
! [[Octahedron]]
!colspan=2|[[Cuboctahedron]]
!colspan=3|[[Rhombicuboctahedron]]
|- valign=top
! [[Square pyramid|J1]]<br>[[Image:Square pyramid.png|80px]]
! [[Triangular cupola|J3]]<br>[[Image:Triangular cupola.png|80px]]
! [[Triangular orthobicupola|J27]]<br>[[Image:Triangular orthobicupola.png|80px]]
! [[Square cupola|J4]]<br>[[Image:Square cupola.png|80px]]
! [[Elongated square cupola|J19]]<br>[[Image:Elongated square cupola.png|80px]]
! [[Elongated square gyrobicupola|J37]]<br>[[Image:Elongated square gyrobicupola.png|80px]]
|}

{| class="wikitable sortable"
!colspan=4|[[Icosahedron]]
!colspan=2|[[Icosidodecahedron]]
|- valign=top
! [[Pentagonal pyramid|J2]]<br>[[Image:Pentagonal pyramid.png|80px]]
! [[Gyroelongated pentagonal pyramid|J11]]<br>[[Image:Gyroelongated pentagonal pyramid.png|80px]]
! [[Metabidiminished icosahedron|J62]]<br>[[Image:Metabidiminished icosahedron.png|80px]]
! [[Tridiminished icosahedron|J63]]<br>[[Image:Tridiminished icosahedron.png|80px]]
! [[Pentagonal rotunda|J6]]<br>[[Image:Pentagonal rotunda.png|80px]]
! [[Pentagonal orthobirotunda|J34]]<br>[[Image:Pentagonal orthobirotunda.png|80px]]
|}

{| class="wikitable sortable"
!colspan=7|[[Rhombicosidodecahedron]]
|- valign=top
!rowspan=2| [[Pentagonal cupola|J5]]<br>[[Image:Pentagonal cupola.png|80px]]
! [[Gyrate rhombicosidodecahedron|J72]]<br>[[Image:Gyrate rhombicosidodecahedron.png|80px]]
! [[Parabigyrate rhombicosidodecahedron|J73]]<br>[[Image:Parabigyrate rhombicosidodecahedron.png|80px]]
! [[Metabigyrate rhombicosidodecahedron|J74]]<br>[[Image:Metabigyrate rhombicosidodecahedron.png|80px]]
! [[Trigyrate rhombicosidodecahedron|J75]]<br>[[Image:Trigyrate rhombicosidodecahedron.png|80px]]
! [[Diminished rhombicosidodecahedron|J76]]<br>[[Image:Diminished rhombicosidodecahedron.png|80px]]
! [[Paragyrate diminished rhombicosidodecahedron|J77]]<br>[[Image:Paragyrate diminished rhombicosidodecahedron.png|80px]]
|- valign=top
! [[Metagyrate diminished rhombicosidodecahedron|J78]]<br>[[Image:Metagyrate diminished rhombicosidodecahedron.png|80px]]
! [[Bigyrate diminished rhombicosidodecahedron|J79]]<br>[[Image:Bigyrate diminished rhombicosidodecahedron.png|80px]]
! [[Parabidiminished rhombicosidodecahedron|J80]]<br>[[Image:Parabidiminished rhombicosidodecahedron.png|80px]]
! [[Metabidiminished rhombicosidodecahedron|J81]]<br>[[Image:Metabidiminished rhombicosidodecahedron.png|80px]]
! [[Gyrate bidiminished rhombicosidodecahedron|J82]]<br>[[Image:Gyrate bidiminished rhombicosidodecahedron.png|80px]]
! [[Tridiminished rhombicosidodecahedron|J83]]<br>[[Image:Tridiminished rhombicosidodecahedron.png|80px]]
|}

== See also ==
* [[Near-miss Johnson solid]]
* [[Catalan solid]]
* [[Toroidal polyhedron]]


}}
==References==
*{{cite journal | last=Johnson | first=Norman W. | author-link=Norman Johnson (mathematician) | title=Convex Solids with Regular Faces | journal=Canadian Journal of Mathematics | volume=18 | year=1966 | pages=169–200 | zbl=0132.14603 | issn=0008-414X | doi=10.4153/cjm-1966-021-8| doi-access=free }} Contains the original enumeration of the 92 solids and the conjecture that there are no others.
* {{cite journal | last=Zalgaller | first=Victor A. | author-link=Victor Zalgaller | title=Convex Polyhedra with Regular Faces | language=ru | journal=Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova | volume=2 | pages=1–221 | year=1967 | issn=0373-2703 | zbl=0165.56302 }} The first proof that there are only 92 Johnson solids. English translation: {{cite journal | last=Zalgaller | first=Victor A. | title=Convex Polyhedra with Regular Faces| journal=Seminars in Mathematics, V. A. Steklov Math. Inst., Leningrad | volume=2 | issn=0080-8873 | publisher=Consultants Bureau|year=1969 | zbl=0177.24802 }}
* {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7 }} Chapter 3 Further Convex polyhedra
* {{cite journal | last=Timofeenko | first=A.V. | author-link=A. V. Timofeenko | title=The Non-Platonic and Non-Archimedean Noncomposite Polyhedra | language=en | journal=J. Math. Sci.| volume=162 | pages=710–729 | year=2009 | issue=5 | doi=10.1007/s10958-009-9655-0 | s2cid=120114341 }} [https://link.springer.com/article/10.1007/s10958-009-9655-0]
olyhedra." J. Math. Sci. 162, 710-729, 2009.
{{reflist}}


==External links==
==External links==

Revision as of 06:26, 6 June 2024

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. It is also Johnson–Zalgaller solid, was named after two mathematicians Norman Johnson and Victor Zalgaller.

Description

The following are three examples of solids. The first solid, elongated square gyrobicupola, is Johnson solid because it has the convexity property. The second solid, stella octangula is not Johnson solid because it is not convex, meaning whenever two points are inside of it, the line connecting always be outside. The last solid is not a Johnson solid because it is not strictly convex, meaning every face is planar or the dihedral angles of two adjacent face has 180°.

A Johnson solid is a convex polyhedron in which the faces of each are regular polygon. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not uniform. This means that Johnson solid are not a Platonic solid, Archimedean solid, prism, or antiprism.[1][2] Equivalently, a Johnson solid is defined as the strictly convex polyhedron, meaning that every face is not coplanar. A convex polyhedron in which all faces are close enough to become regular but some of them are not precisely regular is known as near-miss Johnson solid.

The Johnson solid, sometimes known as Johnson–Zalgaller solid, was named after two mathematicians Norman Johnson and Victor Zalgaller.[3] Johnson (1966) published a list including 92 Johnson solids—excluding the five Platonic solids, the thirteen 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.[4] Zalgaller (1969) proved that Johnson's list was complete.[5]

Some of the Johnson solids do have Rupert property, meaning they do have the polyhedron of the same or larger size that may pass through a hole inside of them. However, the other five Johnson solids do not have this property: gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, and paragyrate diminished rhombicosidodecahedron.[6] From all of the Johnson solids, the elongated square gyrobicupola (also called the pseudorhombicuboctahedron),[7] is unique in being locally vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one 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.

Naming and enumeration

Main article: List of Johnson solids
An example is triaugmented triangular prism. Here, it is constructed from triangular prism by joining three equilateral square pyramids into each of its squares (tri-). The process of this construction known as "augmentation", making its first name is "triaugmented".

The naming of Johnson solids follows a flexible and precise descriptive formula, with many solids can therefore 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 center 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:[8]

  • Bi- indicates that two copies of the solid 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, 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.
  • Elongated indicates a prism is joined to the base of the solid, or between the bases; gyroelongated indicates an antiprism. 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.[8]

The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson with the following nomenclature:[8]

  • 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.

The following is the list of Johnson solids, with the enumeration followed according to the list of Johnson (1966):

  1. Equilateral square pyramid
  2. Pentagonal pyramid
  3. Triangular cupola
  4. Square cupola
  5. Pentagonal cupola
  6. Pentagonal rotunda
  7. Elongated triangular pyramid
  8. Elongated square pyramid
  9. Elongated pentagonal pyramid
  10. Gyroelongated square pyramid
  11. Gyroelongated pentagonal pyramid
  12. Triangular bipyramid
  13. Pentagonal bipyramid
  14. Elongated triangular bipyramid
  15. Elongated square bipyramid
  16. Elongated pentagonal bipyramid
  17. Gyroelongated square bipyramid
  18. Elongated triangular cupola
  19. Elongated square cupola
  20. Elongated pentagonal cupola
  21. Elongated pentagonal rotunda
  22. Gyroelongated triangular cupola
  23. Gyroelongated square cupola
  24. Gyroelongated pentagonal cupola
  25. Gyroelongated pentagonal rotunda
  26. Gyrobifastigium
  27. Triangular orthobicupola
  28. Square orthobicupola
  29. Square gyrobicupola
  30. Pentagonal gyrobicupola
  31. Pentagonal orthocupolarotunda
  32. Pentagonal gyrocupolarotunda
  33. Pentagonal orthobirotunda
  34. Elongated triangular orthobicupola
  35. Elongated triangular gyrobicupola
  36. Elongated square gyrobicupola
  37. Elongated pentagonal orthobicupola
  38. Elongated pentagonal gyrobicupola
  39. Elongated pentagonal orthocupolarotunda
  40. Elongated pentagonal gyrocupolarotunda
  41. Elongated pentagonal orthobirotunda
  42. Elongated pentagonal gyrobirotunda
  43. Gyroelongated triangular bicupola
  44. Gyroelongated square bicupola
  45. Gyroelongated pentagonal bicupola
  46. Gyroelongated pentagonal cupolarotunda
  47. Gyroelongated pentagonal birotunda
  48. Augmented triangular prism
  49. Biaugmented triangular prism
  50. Triaugmented triangular prism
  51. Augmented pentagonal prism
  52. Biaugmented pentagonal prism
  53. Augmented hexagonal prism
  54. Parabiaugmented hexagonal prism
  55. Metabiaugmented hexagonal prism
  56. Triaugmented hexagonal prism
  57. Augmented dodecahedron
  58. Parabiaugmented dodecahedron
  59. Metabiaugmented dodecahedron
  60. Triaugmented dodecahedron
  61. Metabidiminished icosahedron
  62. Tridiminished icosahedron
  63. Augmented tridiminished icosahedron
  64. Augmented truncated tetrahedron
  65. Augmented truncated cube
  66. Biaugmented truncated cube
  67. Augmented truncated dodecahedron
  68. Parabiaugmented truncated dodecahedron
  69. Metabiaugmented truncated dodecahedron
  70. Triaugmented truncated dodecahedron
  71. Gyrate rhombicosidodecahedron
  72. Parabigyrate rhombicosidodecahedron
  73. Metabigyrate rhombicosidodecahedron
  74. Trigyrate rhombicosidodecahedron
  75. Diminished rhombicosidodecahedron
  76. Paragyrate diminished rhombicosidodecahedron
  77. Metagyrate diminished rhombicosidodecahedron
  78. Bigyrate diminished rhombicosidodecahedron
  79. Parabidiminished rhombicosidodecahedron
  80. Metabidiminished rhombicosidodecahedron
  81. Gyrate bidiminished rhombicosidodecahedron
  82. Tridiminished rhombicosidodecahedron
  83. Snub disphenoid
  84. Snub square antiprism
  85. Sphenocorona
  86. Augmented sphenocorona
  87. Sphenomegacorona
  88. Hebesphenomegacorona
  89. Disphenocingulum
  90. Bilunabirotunda
  91. Triangular hebesphenorotunda

References

  1. ^ 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.
  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.
  3. ^ Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5.
  4. ^ Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
  5. ^ Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.
  6. ^ Fredriksson, Albin (2024). "Optimizing for the Rupert property". The American Mathematical Monthly. 131 (3): 255–261. arXiv:2210.00601. doi:10.1080/00029890.2023.2285200.
  7. ^ GWH. "Pseudo Rhombicuboctahedra". www.georgehart.com. Retrieved 17 April 2018.
  8. ^ a b c 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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