Johnson solid

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The elongated square gyrobicupola (J37), a Johnson solid Elongated square gyrobicupola.png
The elongated square gyrobicupola (J37), a Johnson solid
This 24 equilateral triangle example is not a Johnson solid because it is not convex Stella octangula.png
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 180deg dihedral angles) Partial cubic honeycomb.png
This 24-square example is not a Johnson solid because it is not strictly convex (has 180° dihedral angles)

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

Contents

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

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

In 1966, Norman Johnson published a list which included all 92 Johnson solids (excluding the 5 Platonic solids, the 13 Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms), and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

Of the Johnson solids, the elongated square gyrobicupola (J37), also called the pseudorhombicuboctahedron, [1] is unique in being locally vertex-uniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 triangle. However, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid.

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. [2]

Names

The naming of Johnson solids follows a flexible and precise descriptive formula, such that many solids can be named in different ways without compromising their accuracy as a description. Most Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundas), together with the Platonic and Archimedean solids, prisms, and antiprisms; the centre of a particular solid's name will reflect these ingredients. From there, a series of prefixes are attached to the word to indicate additions, rotations, and transformations:

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 [3] with the following nomenclature:

Enumeration

Pyramids, cupolae, and rotundas

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

Pyramids

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

Regular 3> TJ1 4>J2 5>
Triangular pyramid
(Tetrahedron)		 Square pyramid Pentagonal pyramid
Tetrahedron.png Square pyramid.png Pentagonal pyramid.png
Tetrahedron flat.svg Johnson solid 1 net.png Johnson solid 2 net.png
Related regular polyhedra
Tetrahedron Octahedron Icosahedron
Tetrahedron.png Octahedron.png Icosahedron.png

Cupolae and rotunda

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

Cupola Rotunda
UniformJ3 3c aC-J4 4cJ5 5cJ6 5r aD-
Fastigium
(Digonal cupola)
(Triangular prism)		 Triangular cupola Square cupola Pentagonal cupola Pentagonal rotunda
Triangular prism wedge.png Triangular cupola.png Square cupola.png Pentagonal cupola.png Pentagonal rotunda.png
Triangular prism net.png Johnson solid 3 net.png Johnson solid 4 net.png Johnson solid 5 net.png Johnson solid 6 net.png
Related uniform polyhedra
Rhombohedron Cuboctahedron Rhombicuboctahedron Rhombicosidodecahedron Icosidodecahedron
Rhombohedron.svg Cuboctahedron.png Small rhombicuboctahedron.png Small rhombicosidodecahedron.png Icosidodecahedron.png

Modified pyramids

Johnson solids 7 to 17 are derived from pyramids.

Elongated and gyroelongated pyramids

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

Elongated pyramids Gyroelongated pyramids
J7 3=>J8 4=>J9 5=>CoplanarJ10 4z>J11 5z> I-
Elongated triangular pyramid Elongated square pyramid Elongated pentagonal pyramid Gyroelongated triangular pyramid
(diminished trigonal trapezohedron)		 Gyroelongated square pyramid Gyroelongated pentagonal pyramid
Elongated triangular pyramid.png Elongated square pyramid.png Elongated pentagonal pyramid.png Augmented octahedron.png Gyroelongated square pyramid.png Gyroelongated pentagonal pyramid.png
Johnson solid 7 net.png Johnson solid 8 net.png Johnson solid 9 net.png Augmented octahedgon net.png Johnson solid 10 net.png Johnson solid 11 net.png
Augmented from polyhedra
tetrahedron
triangular prism 		square pyramid
cube 		pentagonal pyramid
pentagonal prism 		tetrahedron
octahedron 		square pyramid
square antiprism 		pentagonal pyramid
pentagonal antiprism
Tetrahedron.png Triangular prism.png Square pyramid.png Hexahedron.png Pentagonal pyramid.png Pentagonal prism.png Tetrahedron.png Octahedron.png Square pyramid.png Square antiprism.png Pentagonal pyramid.png Pentagonal antiprism.png

Bipyramids

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

Bipyramids Elongated bipyramids Gyroelongated bipyramids
J12 3<>RegularJ13 5<>J14 3<=>J15 4<=>J16 5<=>CoplanarJ17 4<z>Regular
Triangular bipyramid Square bipyramid
(octahedron)		 Pentagonal bipyramid Elongated triangular bipyramid Elongated square bipyramid Elongated pentagonal bipyramid Gyroelongated triangular bipyramid
(trigonal trapezohedron)		 Gyroelongated square bipyramid Gyroelongated pentagonal bipyramid
(icosahedron)
Triangular dipyramid.png Octahedron.png Pentagonal dipyramid.png Elongated triangular dipyramid.png Elongated square dipyramid.png Elongated pentagonal dipyramid.png Gyroelongated triangular bipyramid.png Gyroelongated square dipyramid.png Pentagonal gyroelongated bipyramid.png
Johnson solid 12 net.png Octahedron flat.svg Johnson solid 13 net.png Johnson solid 14 net.png Johnson solid 15 net.png Johnson solid 16 net.png Johnson solid 17 net.png Icosahedron flat.svg
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
Tetrahedron.png Square pyramid.png Pentagonal pyramid.png Tetrahedron.png Triangular prism.png Square pyramid.png Hexahedron.png Pentagonal pyramid.png Pentagonal prism.png Tetrahedron.png Octahedron.png Square pyramid.png Square antiprism.png Pentagonal pyramid.png Pentagonal antiprism.png

Modified cupolae and rotundas

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

Elongated and gyroelongated cupolae and rotundas

Elongated cupola Elongated rotunda Gyroelongated cupola Gyroelongated rotunda
CoplanarJ18 3c=J19 4c= eC-J20 5c=J21 5r=ConcaveJ22 3czJ23 4czJ24 5czJ25 5rz
Elongated fastigium Elongated triangular cupola Elongated square cupola Elongated pentagonal cupola Elongated pentagonal rotunda Gyroelongated fastigium Gyroelongated triangular cupola Gyroelongated square cupola Gyroelongated pentagonal cupola Gyroelongated pentagonal rotunda
Elongated digonal cupola.png Elongated triangular cupola.png Elongated square cupola.png Elongated pentagonal cupola.svg Elongated pentagonal rotunda.png Gyroelongated digonal cupola.png Gyroelongated triangular cupola.png Gyroelongated square cupola.png Gyroelongated pentagonal cupola.png Gyroelongated pentagonal rotunda.png
Johnson solid 18 net.png Johnson solid 19 net.png Johnson solid 20 net.png Johnson solid 21 net.png Johnson solid 22 net.png Johnson solid 23 net.png Johnson solid 24 net.png Johnson solid 25 net.png
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
Tetragonal prism.png Triangular prism.png Hexagonal prism.png Triangular cupola.png Octagonal prism.png Square cupola.png Decagonal prism.png Pentagonal cupola.png Decagonal prism.png Pentagonal rotunda.png Square antiprism.png Triangular prism.png Hexagonal antiprism.png Triangular cupola.png Octagonal antiprism.png Square cupola.png Decagonal antiprism.png Pentagonal cupola.png Decagonal antiprism.png Pentagonal rotunda.png

Bicupolae

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

Orthobicupola Gyrobicupola
CoplanarJ27 3ccJ28 4ccJ30 5ccJ26 2cc*SemiregularJ29 4cc*J31 5cc*
Orthobifastigium Triangular orthobicupola Square orthobicupola Pentagonal orthobicupola Gyrobifastigium Triangular gyrobicupola
(cuboctahedron)		 Square gyrobicupola Pentagonal gyrobicupola
Digonal orthobicupola.png Triangular orthobicupola.png Square orthobicupola.png Pentagonal orthobicupola.png Gyrobifastigium.png Cuboctahedron.png Square gyrobicupola.png Pentagonal gyrobicupola.png
Johnson solid 27 net.png Johnson solid 28 net.png Johnson solid 30 net.png Johnson solid 26 net.png Cuboctahedron flat.svg Johnson solid 29 net.png Johnson solid 31 net.png
Augmented from polyhedron
Triangular prism Triangular cupola Square cupola Pentagonal cupola Triangular prism Triangular cupola Square cupola Pentagonal cupola
Triangular prism.png Triangular cupola.png Square cupola.png Pentagonal cupola.png Triangular prism.png Triangular cupola.png Square cupola.png Pentagonal cupola.png

Cupola-rotundas and birotundas

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

Cupola-rotundaBirotunda
J32 5crJ33 5cr*J34 5rr aD*Semiregular
Pentagonal orthocupolarotunda Pentagonal gyrocupolarotunda Pentagonal orthobirotunda Pentagonal gyrobirotunda
(icosidodecahedron)
Pentagonal orthocupolarotunda.png Pentagonal gyrocupolarotunda.png Pentagonal orthobirotunda.png Icosidodecahedron.png
Johnson solid 32 net.png Johnson solid 33 net.png Johnson solid 34 net.png Icosidodecahedron flat.svg
Augmented from polyhedra
Pentagonal cupola
Pentagonal rotunda 		Pentagonal rotunda
Pentagonal cupola.png Pentagonal rotunda.png Pentagonal rotunda.png

Elongated bicupolae

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

Elongated orthobicupola Elongated gyrobicupola
CoplanarJ35 3c=cSemiregularJ38 5c=cCoplanarJ36 3c=c*J37 4c=c* eC*J39 5c=c*
Elongated orthobifastigium Elongated triangular orthobicupola Elongated square orthobicupola
(rhombicuboctahedron)		 Elongated pentagonal orthobicupola Elongated gyrobifastigium Elongated triangular gyrobicupola Elongated square gyrobicupola Elongated pentagonal gyrobicupola
Elongated digonal orthobicupola.png Elongated triangular orthobicupola.png Small rhombicuboctahedron.png Elongated pentagonal orthobicupola.png Elongated digonal gyrobicupola.png Elongated triangular gyrobicupola.png Elongated square gyrobicupola.png Elongated pentagonal gyrobicupola.png
Johnson solid 35 net.png Rhombicuboctahedron flat.png Johnson solid 38 net.png Johnson solid 36 net.png Johnson solid 37 net.png Johnson solid 39 net.png
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
Tetragonal prism.png Triangular prism.png Hexagonal prism.png Triangular cupola.png Octagonal prism.png Square cupola.png Decagonal prism.png Pentagonal cupola.png Tetragonal prism.png Triangular prism.png Hexagonal prism.png Triangular cupola.png Octagonal prism.png Square cupola.png Decagonal prism.png Pentagonal cupola.png

Elongated cupola-rotundas and birotundas

Elongated cupola-rotundaElongated birotunda
J40 5c=rJ41 5c=r*J42 5r=rJ43 5r=r*
Elongated pentagonal orthocupolarotunda Elongated pentagonal gyrocupolarotunda Elongated pentagonal orthobirotunda Elongated pentagonal gyrobirotunda
Elongated pentagonal orthocupolarotunda.png Elongated pentagonal gyrocupolarotunda.png Elongated pentagonal orthobirotunda.png Elongated pentagonal gyrobirotunda.png
Johnson solid 40 net.png Johnson solid 41 net.png Johnson solid 42 net.png Johnson solid 43 net.png
Augmented from polyhedra
Decagonal prism
Pentagonal cupola
Pentagonal rotunda 		Decagonal prism
Pentagonal rotunda
Decagonal prism.png Pentagonal cupola.png Pentagonal rotunda.png Decagonal prism.png Pentagonal rotunda.png

Gyroelongated bicupolae, cupola-rotundas, and birotundas

These Johnson solids have 2 chiral forms.

Gyroelongated bicupola Gyroelongated cupola-rotundaGyroelongated birotunda
ConcaveJ44 3czcJ45 4czcJ46 5czcJ47 5czrJ48 5rzr
Gyroelongated bifastigium Gyroelongated triangular bicupola Gyroelongated square bicupola Gyroelongated pentagonal bicupola Gyroelongated pentagonal cupolarotunda Gyroelongated pentagonal birotunda
Gyroelongated digonal bicupola.png Gyroelongated triangular bicupola.png Gyroelongated square bicupola.png Gyroelongated pentagonal bicupola.png Gyroelongated pentagonal cupolarotunda.png Gyroelongated pentagonal birotunda.png
Johnson solid 44 net.png Johnson solid 45 net.png Johnson solid 46 net.png Johnson solid 47 net.png Johnson solid 48 net.png
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
Triangular prism.png Square antiprism.png Triangular cupola.png Hexagonal antiprism.png Square cupola.png Octagonal antiprism.png Pentagonal cupola.png Decagonal antiprism.png Pentagonal cupola.png Pentagonal rotunda.png Decagonal antiprism.png Pentagonal rotunda.png Decagonal antiprism.png

Augmented prisms

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

Augmented triangular prismsAugmented pentagonal prismsAugmented hexagonal prisms
J49 3=+J50 3=++J51 3=+++J52 5=+J53 5=++J54 6=+J55 6=++J56 6=+xJ57 6=+++
Augmented triangular prism Biaugmented triangular prism Triaugmented triangular prism Augmented pentagonal prism Biaugmented pentagonal prism Augmented hexagonal prism Parabiaugmented hexagonal prism Metabiaugmented hexagonal prism Triaugmented hexagonal prism
Augmented triangular prism.png Biaugmented triangular prism.png Triaugmented triangular prism.png Augmented pentagonal prism.png Biaugmented pentagonal prism.png Augmented hexagonal prism.png Parabiaugmented hexagonal prism.png Metabiaugmented hexagonal prism.png Triaugmented hexagonal prism.png
Johnson solid 49 net.png Johnson solid 50 net.png Johnson solid 51 net.png Johnson solid 52 net.png Johnson solid 53 net.png Johnson solid 54 net.png Johnson solid 55 net.png Johnson solid 56 net.png Johnson solid 57 net.png
Augmented from polyhedra
Triangular prism
Square pyramid 		Pentagonal prism
Square pyramid 		Hexagonal prism
Square pyramid
Triangular prism.png Square pyramid.png Pentagonal prism.png Square pyramid.png Hexagonal prism.png Square pyramid.png

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

Modified Platonic solids

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

Augmented dodecahedra

J58 D+J59 D++J60 D+xJ61 D+++
Augmented dodecahedron Parabiaugmented dodecahedron Metabiaugmented dodecahedron Triaugmented dodecahedron
Augmented dodecahedron.png Parabiaugmented dodecahedron.png Metabiaugmented dodecahedron.png Triaugmented dodecahedron.png
Johnson solid 58 net.png Johnson solid 59 net.png Johnson solid 60 net.png Johnson solid 61 net.png
Augmented from polyhedra
Dodecahedron and pentagonal pyramid
Dodecahedron.png Pentagonal pyramid.png

Diminished and augmented diminished icosahedra

Diminished icosahedronAugmented tridiminished icosahedron
J11
(Repeated)		UniformJ62 I-/J63 I---J64 I---+
Diminished icosahedron
(Gyroelongated pentagonal pyramid)		Parabidiminished icosahedron
(Pentagonal antiprism)		 Metabidiminished icosahedron Tridiminished icosahedron Augmented tridiminished icosahedron
Gyroelongated pentagonal pyramid.png Pentagonal antiprism.png Metabidiminished icosahedron.png Tridiminished icosahedron.png Augmented tridiminished icosahedron.png
Johnson solid 11 net.png Johnson solid 62 net.png Johnson solid 63 net.png Johnson solid 64 net.png

Modified Archimedean solids

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

Augmented Archimedean solids

Augmented truncated tetrahedronAugmented truncated cubesAugmented truncated dodecahedra
J65 tT+J66 tC+J67 tC++J68 tD+J69 tD++J70 tD+xJ71 tD+++
Augmented truncated tetrahedron Augmented truncated cube Biaugmented truncated cube Augmented truncated dodecahedron Parabiaugmented truncated dodecahedron Metabiaugmented truncated dodecahedron Triaugmented truncated dodecahedron
Augmented truncated tetrahedron.png Augmented truncated cube.png Biaugmented truncated cube.png Augmented truncated dodecahedron.png Parabiaugmented truncated dodecahedron.png Metabiaugmented truncated dodecahedron.png Triaugmented truncated dodecahedron.png
Johnson solid 65 net.png Johnson solid 66 net.png Johnson solid 67 net.png Johnson solid 68 net.png Johnson solid 69 net.png Johnson solid 70 net.png Johnson solid 71 net.png
Augmented from polyhedra
truncated tetrahedron
triangular cupola 		truncated cube
square cupola 		truncated dodecahedron
pentagonal cupola
Truncated tetrahedron.png Triangular cupola.png Truncated hexahedron.png Square cupola.png Truncated dodecahedron.png Pentagonal cupola.png

Gyrate and diminished rhombicosidodecahedra

Gyrate rhombicosidodecahedra
J72 eD*J73 eD**J74 eD*'J75 eD***
Gyrate rhombicosidodecahedron Parabigyrate rhombicosidodecahedron Metabigyrate rhombicosidodecahedron Trigyrate rhombicosidodecahedron
Gyrate rhombicosidodecahedron.png Parabigyrate rhombicosidodecahedron.png Metabigyrate rhombicosidodecahedron.png Trigyrate rhombicosidodecahedron.png
Johnson solid 72 net.png Johnson solid 73 net.png Johnson solid 74 net.png Johnson solid 75 net.png
Diminished rhombicosidodecahedra
J76 eD-J80 eD--J81 eD-/J83 eD---
Diminished rhombicosidodecahedron Parabidiminished rhombicosidodecahedron Metabidiminished rhombicosidodecahedron Tridiminished rhombicosidodecahedron
Diminished rhombicosidodecahedron.png Parabidiminished rhombicosidodecahedron.png Metabidiminished rhombicosidodecahedron.png Tridiminished rhombicosidodecahedron.png
Johnson solid 76 net.png Johnson solid 80 net.png Johnson solid 81 net.png Johnson solid 83 net.png
Gyrate diminished rhombicosidodecahedra
J77 -*J78 -'J79 -**J82 --*
Paragyrate diminished rhombicosidodecahedron Metagyrate diminished rhombicosidodecahedron Bigyrate diminished rhombicosidodecahedron Gyrate bidiminished rhombicosidodecahedron
Paragyrate diminished rhombicosidodecahedron.png Metagyrate diminished rhombicosidodecahedron.png Bigyrate diminished rhombicosidodecahedron.png Gyrate bidiminished rhombicosidodecahedron.png
Johnson solid 77 net.png Johnson solid 78 net.png Johnson solid 79 net.png Johnson solid 82 net.png

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.

J27J3J34J6J37J19Uniform
Gyrate cuboctahedron
(triangular orthobicupola)		Diminished cuboctahedron
(triangular cupola)		Gyrate icosidodecahedron
(pentagonal orthobirotunda)		Diminished icosidodecahedron
(pentagonal rotunda)		Gyrate rhombicuboctahedron
(elongated square gyrobicupola)		Diminished rhombicuboctahedron
(elongated square cupola)		Bidiminished rhombicuboctahedron
(octagonal prism)
Triangular orthobicupola.png Triangular cupola.png Pentagonal orthobirotunda.png Pentagonal rotunda.png Elongated square gyrobicupola.png Elongated square cupola.png Octagonal prism.png
Johnson solid 27 net.png Johnson solid 3 net.png Johnson solid 34 net.png Johnson solid 6 net.png Johnson solid 37 net.png Johnson solid 19 net.png
Gyrated or diminished from polyhedra
CuboctahedronIcosidodecahedronRhombicuboctahedron
Cuboctahedron.png Icosidodecahedron.png Small rhombicuboctahedron.png

Elementary solids

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

Snub antiprisms

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

J84RegularJ85
Snub disphenoid
ss{2,4}		 Icosahedron
ss{2,6}		 Snub square antiprism
ss{2,8}
Digonal gyrobianticupolaTriangular gyrobianticupolaSquare gyrobianticupola
Snub digonal antiprism.png Snub triangular antiprism.png Snub square antiprism colored.png
Snub disphenoid net snubcoloring.png Snub triangular antiprism net.png Snub square antiprism net snubcoloring.png

Others

J86J87J88
Sphenocorona Augmented sphenocorona Sphenomegacorona
Sphenocorona.png Augmented sphenocorona.png Sphenomegacorona.png
Johnson solid 86 net.png Johnson solid 87 net.png Johnson solid 88 net.png
J89J90J91J92
Hebesphenomegacorona Disphenocingulum Bilunabirotunda Triangular hebesphenorotunda
Hebesphenomegacorona.png Disphenocingulum.png Bilunabirotunda.png Triangular hebesphenorotunda.png
Johnson solid 89 net.png Johnson solid 90 net.png Johnson solid 91 net.png Johnson solid 92 net.png

Classification by types of faces

Triangle-faced Johnson solids

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

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

Triangle and square-faced Johnson solids

Twenty four Johnson solids have only triangle or square faces:

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

Triangle and pentagon-faced Johnson solids

Eleven Johnson solids have only triangle and pentagon faces:

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

Triangle, square, and pentagon-faced Johnson solids

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

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

Triangle, square, and hexagon-faced Johnson solids

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

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

Triangle, square, and octagon-faced Johnson solids

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

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

Triangle, pentagon, and decagon-faced Johnson solids

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

J06 Pentagonal rotunda Pentagonal rotunda.png
J25 Gyroelongated pentagonal rotunda Gyroelongated pentagonal rotunda.png

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

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

J92 Triangular hebesphenorotunda Triangular hebesphenorotunda.png

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

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

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

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. [4]

Octahedron Cuboctahedron Rhombicuboctahedron
J1
Square pyramid.png 		J3
Triangular cupola.png 		J27
Triangular orthobicupola.png 		J4
Square cupola.png 		J19
Elongated square cupola.png 		J37
Elongated square gyrobicupola.png
Icosahedron Icosidodecahedron
J2
Pentagonal pyramid.png 		J11
Gyroelongated pentagonal pyramid.png 		J62
Metabidiminished icosahedron.png 		J63
Tridiminished icosahedron.png 		J6
Pentagonal rotunda.png 		J34
Pentagonal orthobirotunda.png
Rhombicosidodecahedron
J5
Pentagonal cupola.png 		J72
Gyrate rhombicosidodecahedron.png 		J73
Parabigyrate rhombicosidodecahedron.png 		J74
Metabigyrate rhombicosidodecahedron.png 		J75
Trigyrate rhombicosidodecahedron.png 		J76
Diminished rhombicosidodecahedron.png 		J77
Paragyrate diminished rhombicosidodecahedron.png
J78
Metagyrate diminished rhombicosidodecahedron.png 		J79
Bigyrate diminished rhombicosidodecahedron.png 		J80
Parabidiminished rhombicosidodecahedron.png 		J81
Metabidiminished rhombicosidodecahedron.png 		J82
Gyrate bidiminished rhombicosidodecahedron.png 		J83
Tridiminished rhombicosidodecahedron.png

See also

Related Research Articles

<span class="mw-page-title-main">Gyroelongated pentagonal pyramid</span> 11th Johnson solid (16 faces)

In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids. As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base.

<span class="mw-page-title-main">Elongated square gyrobicupola</span> 37th Johnson solid

In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids. It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that map every vertex to every other vertex. It strongly resembles, but should not be mistaken for, the rhombicuboctahedron, which is an Archimedean solid. It is also a canonical polyhedron.

<span class="mw-page-title-main">Gyroelongated pentagonal rotunda</span>

In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (J25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (J6) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal birotunda (J48) with one pentagonal rotunda removed.

<span class="mw-page-title-main">Square orthobicupola</span> 28th Johnson solid; 2 square cupolae joined base-to-base

In geometry, the square orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola.

<span class="mw-page-title-main">Square gyrobicupola</span> 29th Johnson solid; 2 square cupolae joined base-to-base

In geometry, the square gyrobicupola is one of the Johnson solids. Like the square orthobicupola, it can be obtained by joining two square cupolae along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.

<span class="mw-page-title-main">Gyrate rhombicosidodecahedron</span> 72nd Johnson solid

In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids. It is also a canonical polyhedron.

<span class="mw-page-title-main">Metabidiminished rhombicosidodecahedron</span> 81st Johnson solid

In geometry, the metabidiminished rhombicosidodecahedron is one of the Johnson solids.

<span class="mw-page-title-main">Trigyrate rhombicosidodecahedron</span> 75th Johnson solid

In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids. It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron.

<span class="mw-page-title-main">Elongated pentagonal bipyramid</span> 16th Johnson solid; pentagonal prism capped by pyramids

In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal bipyramid by inserting a pentagonal prism between its congruent halves.

<span class="mw-page-title-main">Augmented pentagonal prism</span> 52nd Johnson solid

In geometry, the augmented pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a pentagonal prism by attaching a square pyramid to one of its equatorial faces.

<span class="mw-page-title-main">Parabiaugmented dodecahedron</span> 59th Johnson solid

In geometry, the parabiaugmented dodecahedron is one of the Johnson solids. It can be seen as a dodecahedron with two pentagonal pyramids attached to opposite faces. When pyramids are attached to a dodecahedron in other ways, they may result in an augmented dodecahedron, a metabiaugmented dodecahedron, a triaugmented dodecahedron, or even a pentakis dodecahedron if the faces are made to be irregular.

<span class="mw-page-title-main">Parabigyrate rhombicosidodecahedron</span> 73rd Johnson solid

In geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.

<span class="mw-page-title-main">Gyrate bidiminished rhombicosidodecahedron</span> 82nd Johnson solid

In geometry, the gyrate bidiminished rhombicosidodecahedron is one of the Johnson solids.

<span class="mw-page-title-main">Metabigyrate rhombicosidodecahedron</span> 74th Johnson solid

In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids. It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.

<span class="mw-page-title-main">Gyroelongated triangular cupola</span>

In geometry, the gyroelongated triangular cupola is one of the Johnson solids (J22). It can be constructed by attaching a hexagonal antiprism to the base of a triangular cupola (J3). This is called "gyroelongation", which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

<span class="mw-page-title-main">Bicupola (geometry)</span> Solid made from 2 cupolae joined base-to-base

In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.

In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.

References

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

  1. ↑ GWH. "Pseudo Rhombicuboctahedra". www.georgehart.com. Retrieved 17 April 2018.
  2. ↑ 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
  3. ↑ George Hart (quoting Johnson) (1996). "Johnson Solids". Virtual Polyhedra. Retrieved 5 February 2014.
  4. ↑ Klitzing, Dr. Richard. "Johnson solids et al". bendwavy.org. Retrieved 17 April 2018.
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