Johnson solid
A Johnson solid is a strictly convex regular-faced polyhedron that is not uniform. They are named after Norman W. Johnson, who in 1966 first listed all 92 such polyhedra.[1] Before him, Duncan Sommerville discovered the subset of them that are circumscribable.[2] In 1969, Victor Zalgaller proved that the list was complete.[3]
Even though they are not allowed to be uniform, Johnson solids can have just one type of polygon for their faces, as the triangular bipyramid does, or have only one vertex figure, as the elongated square gyrobicupola does.
Johnson solids only have triangles, squares, pentagons, hexagons, octagons, or decagons as faces. All have at least some degree of symmetry.
Near-miss Johnson solids[edit | edit source]
A large amount of near-miss Johnson solids may also be constructed. These polyhedra are convex and all of their faces are either regular or almost regular. They may also use polygons unavailable to the proper Johnson solids, such as the heptagon, enneagon, hendecagon, or dodecagon.
List of the 92 Johnson solids[edit | edit source]
Pyramids, cupolas, and the pentagonal rotunda[edit | edit source]
A pyramid is formed by connecting a point and an n-gon with a band of triangles that all meet at the point. The triangular pyramid is a tetrahedron, and is too symmetric to be a Johnson solid. A hexagonal pyramid would be planar.
A cupola is formed by connecting an n-gon and a 2n-gon with a band of alternating triangles and squares. A hexagonal cupola would be planar.
The pentagonal rotunda is unique. It is formed by a pentagon and a decagon connected by two sets of triangles and one set of pentagons. It has no CRF analogues based on other polygons.
Each of these polyhedra is related to a uniform polyhedron: the square pyramid with the octahedron, the pentagonal pyramid with the icosahedron, the triangular cupola with the cuboctahedron, the square cupola with the small rhombicuboctahedron, the pentagonal cupola with the small rhombicosidodecahedron and the pentagonal rotunda with the icosidodecahedron.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Square pyramid
(J1, squippy) |
|
4 triangles
1 square |
4x 3-3
4x 3-4 |
1 (34)
4 (32.4) |
B2×I, order 8 |
| Pentagonal pyramid
(J2, peppy) |
|
5 triangles
1 pentagon |
5x 3-3
5x 3-5 |
1 (35)
5 (32.5) |
H2×I, order 10 |
| Triangular cupola
(J3, tricu) |
|
1+3 triangles
3 squares 1 hexagon |
9x 3-4
3x 3-6 3x 4-6 |
3 (3.4.3.4)
6 (3.4.6) |
A2×I, order 6 |
| Square cupola
(J4, squacu) |
|
4 triangles
1+4 squares 1 octagon |
4x 4-4
8x 3-4 4x 3-8 4x 4-8 |
4 (3.43)
8 (3.4.8) |
B2×I, order 8 |
| Pentagonal cupola
(J5, pecu) |
|
5 triangles
5 squares 1 pentagon 1 decagon |
5x 4-5
10x 3-4 5x 3-10 5x 4-10 |
5 (3.4.5.4)
10 (3.4.10) |
H2×I, order 10 |
| Pentagonal rotunda
(J6, pero) |
|
5+5 triangles
1+5 pentagons 1 decagon |
25x 3-5
5x 3-10 5x 5-10 |
10x (3.5.3.5)
10x (3.5.10) |
H2×I, order 10 |
Elongations[edit | edit source]
Some solids can be "elongated" by attaching a prism to one of their faces, usually the largest face. The elongated square cupola can also be constructed as a diminished rhombicuboctahedron.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Elongated triangular pyramid
(J7, etripy) |
|
1+3 triangles
3 squares |
3x 3-3
3x 3-4 3x 4-4 3x 3-4 |
1 (33)
3 (32.42) 3 (3.42) |
A2×I, order 6 |
| Elongated square pyramid
(J8, esquipy) |
|
4 triangles
1+4 squares |
4x 3-3
4x 3-4 8x 4-4 |
1 (34)
4 (32.42) 4 (43) |
B2×I, order 8 |
| Elongated pentagonal pyramid
(J9, epeppy) |
|
5 triangles
5 squares 1 pentagon |
5x 3-3
5x 3-4 5x 4-4 5x 4-5 |
1 (35)
5 (32.42) 5 (42.5) |
H2×I, order 10 |
| Elongated triangular cupola
(J18, etcu) |
|
1+3 triangles
3+3+3 squares 1 hexagon |
9x 3-4
3x 4-4 3x 3-4 6x 4-4 6x 4-6 |
3 (3.4.3.4)
6 (3.43) 6 (42.6) |
A2×I, order 6 |
| Elongated square cupola
(J19, escu) |
|
4 triangles
1+4+4+4 squares 1 octagon |
16x 4-4
12x 3-4 8x 4-8 |
12 (3.43)
8 (42.8) |
B2×I, order 8 |
| Elongated pentagonal cupola
(J20, epcu) |
|
5 triangles
5+5+5 squares 1 pentagon 1 decagon |
5x 4-5
10x 3-4 5x 4-4 5x 3-4 10x 4-4 10x 4-10 |
5 (3.4.5.4)
10 (3.43) 10 (42.10) |
H2×I, order 10 |
| Elongated pentagonal rotunda
(J21, epro) |
|
5+5 triangles
5+5 squares 1+5 pentagons 1 decagon |
25x 3-5
5x 3-4 5x 4-5 10x 4-4 10x 4-10 |
10 (3.5.3.5)
10 (3.42.5) 10(42.10) |
H2×I, order 10 |
Gyroelongations[edit | edit source]
"Gyroelongation" adds an antiprism to said face instead of a prism.
The triangular pyramid cannot be gyroelongated because its faces would be coplanar to those of the added antiprism, and the resulting polyhedron would not be strictly convex.
The gyroelongated pentagonal pyramid can also be constructed as a diminished icosahedron.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Gyroelongated square pyramid
(J10, gyesp) |
|
4+4+4 triangles
1 square |
4x 3-3
4x 3-3 8x 3-3 4x 3-4 |
1 (34)
4 (35) 4 (33.4) |
B2×I, order 8 |
| Gyroelongated pentagonal pyramid
(J11, gyepip) |
|
5+5+5 triangles
1 pentagon |
20x 3-3
5x 3-5 |
6 (35)
5 (33.5) |
H2×I, order 10 |
| Gyroelongated triangular cupola
(J22, gyetcu) |
|
1+3+3+3+6 triangles
3 squares 1 hexagon |
9x 3-4
3x 3-3 3x 3-4 12x 3-3 6x 3-6 |
3 (3.4.3.4)
6 (34.4) 6 (33.6) |
A2×I, order 6 |
| Gyroelongated square cupola
(J23, gyescu) |
|
4+4+4+8 triangles
1+4 squares 1 octagon |
4x 4-4
8x 3-4 4x 3-3 4x 3-4 16x 3-3 8x 3-8 |
4 (3.43)
8 (34.4) 8 (33.8) |
B2×I, order 8 |
| Gyroelongated pentagonal cupola
(J24, gyepcu) |
|
5+5+5+10 triangles
5 squares 1 pentagon 1 decagon |
5x 4-5
10x 3-4 5x 3-3 5x 3-4 20x 3-3 10x 3-10 |
5 (3.4.5.4)
10 (34.4) 10 (33.10) |
H2×I, order 10 |
| Gyroelongated pentagonal rotunda
(J25, gyepro) |
|
5+5+5+5+10 triangles
1+5 pentagons 1 decagon |
25x 3-5
5x 3-3 5x 3-5 20x 3-3 10x 3-10 |
10 (3.5.3.5)
10 (34.5) 10 (33.10) |
H2×I, order 10 |
Bipyramids, bicupolas, birotundas, and cupolarotundas[edit | edit source]
Two pyramids, cupolas, or rotundas can be joined together, typically by their largest face. (Joining them by another face would create a nonconvex polyhedron.) The pentagonal cupola can join with the pentagonal rotunda due to both being decagon-based.
Bicupolas, birotundas, and the pentagonal cupola-rotunda compound can be aligned in two different ways, with the "top" faces (the ones parallel to the "base") either aligned with one another or 180° out of alignment. When aligned, the compound is called "ortho-," and when out of alignment, the compound is called "gyro-."
The gyrobifastigium is a "digonal gyrobicupola." The "digonal cupola" is just a triangular prism, thought of as turned on its side; a band of triangles and squares joining a 2-gon and a 4-gon.
The square bipyramid is not included because it is an octahedron, which is regular and therefore not a Johnson solid.
The triangular gyrobicupola and the pentagonal gyrobirotunda are not included because they are the uniform cuboctahedron and icosidodecahedron, respectively. Their "ortho" forms can be constructed as gyrations of those polyhedra.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Triangular bipyramid
(J12, tridpy) |
|
6 triangles | 6x 3-3
3x 3-3 |
2 (33)
3 (34) |
A2×A1, order 12 |
| Pentagonal bipyramid
(J13, pedpy) |
|
10 triangles | 10x 3-3
5x 3-3 |
2 (35)
5 (34) |
H2×A1, order 20 |
| Gyrobifastigium
(J26, gybef) |
|
4 triangles
4 squares |
2x 4-4
8x 3-4 4x 3-4 |
4 (3.42)
4 (3.4.3.4) |
B2×A1/2, order 8 |
| Triangular orthobicupola
(J27, tobcu) |
|
2+6 triangles
6 squares |
18x 3-4
3x 4-4 3x 3-3 |
6 (3.4.3.4)
6 (32.42) |
A2×A1, order 12 |
| Square orthobicupola
(J28, squobcu) |
|
8 triangles
2+8 squares |
8x 4-4
16x 3-4 4x 4-4 4x 3-3 |
8 (3.43)
8 (32.42) |
B2×A1, order 16 |
| Square gyrobicupola
(J29, squigybcu) |
|
8 triangles
2+8 squares |
8x 4-4
16x 3-4 8x 3-4 |
8 (3.43)
8 (3.4.3.4) |
I2(8)×A1/2, order 16 |
| Pentagonal orthobicupola
(J30, pobcu) |
|
10 triangles
10 squares 2 pentagons |
10x 4-5
20x 3-4 5x 3-3 5x 4-4 |
10 (3.4.5.4)
10 (32.42) |
H2×A1, order 20 |
| Pentagonal gyrobicupola
(J31, pegybcu) |
|
10 triangles
10 squares 2 pentagons |
10x 4-5
20x 3-4 10x 3-4 |
10 (3.4.5.4)
10 (3.4.3.4) |
I2(10)×A1/2, order 20 |
| Pentagonal orthobirotunda
(J34, pobro) |
|
10+10 triangles
2+10 pentagons |
50x 3-5
5x 3-3 5x 5-5 |
20 (3.5.3.5)
10 (32.52) |
H2×A1, order 20 |
| Pentagonal orthocupolarotunda
(J32, pocuro) |
|
5+5+5 triangles
5 squares 1+1+5 pentagons |
25x 3-5
5x 3-4 5x 3-5 10x 3-4 5x 4-5 |
10 (3.5.3.5)
10 (3.4.3.5) 5 (3.4.5.4) |
H2×I, order 10 |
| Pentagonal gyrocupolarotunda
(J33, pegycuro) |
|
5+5+5 triangles
5 squares 1+1+5 pentagons |
25x 3-5
5x 3-3 5x 4-5 10x 3-4 5x 4-5 |
10 (3.5.3.5)
10 (32.4.5) 5 (3.4.5.4) |
H2×I, order 10 |
Elongations and gyroelongations of the pairs[edit | edit source]
Elongations and gyroelongations of the above pairs place a prism or antiprism in between the two parts.
The elongated square gyrobicupola can be constructed as a gyrate rhombicuboctahedron.
The elongated square orthobicupola is not included because it is a uniform small rhombicuboctahedron. The gyroelongated pentagonal bipyramid is not included because it is a regular icosahedron.
The gyrobifastigium cannot be elongated, nor can the triangular bipyramid be gyroelongated, because some adjacent faces would be coplanar.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Elongated triangular bipyramid
(J14, etidpy) |
|
6 triangles
3 squares |
6x 3-3
6x 3-4 3x 4-4 |
2 (33)
6 (32.42) |
A2×A1, order 12 |
| Elongated square bipyramid
(J15, esquidpy) |
|
8 triangles
4 squares |
8x 3-3
8x 3-4 4x 4-4 |
2 (34)
8 (32.42) |
B2×A1, order 16 |
| Elongated pentagonal bipyramid
(J16, epedpy) |
|
10 triangles
5 squares |
10x 3-3
10x 3-4 5x 4-4 |
2 (35)
10 (32.42) |
H2×A1, order 20 |
| Elongated triangular orthobicupola
(J35, etobcu) |
|
2+6 triangles
6+3+3 squares |
18x 3-4
6x 4-4 6x 3-4 6x 4-4 |
6 (3.4.3.4)
12 (3.43) |
A2×A1, order 12 |
| Elongated triangular gyrobicupola
(J36, etigybcu) |
|
2+6 triangles
6+6 squares |
18x 3-4
6x 4-4 6x 3-4 6x 4-4 |
6 (3.4.3.4)
12 (3.43) |
G2×A1/2, order 12 |
| Elongated square gyrobicupola
(J37, esquigybcu) |
|
8 triangles
2+8+8 squares |
24x 4-4
24x 3-4 |
24 (3.43) | I2(8)×A1/2, order 16 |
| Elongated pentagonal orthobicupola
(J38, epobcu) |
|
10 triangles
10+5+5 squares 2 pentagons |
10x 4-5
20x 3-4 10x 4-4 10x 3-4 10x 4-4 |
10 (3.4.5.4)
20 (3.43) |
H2×A1, order 20 |
| Elongated pentagonal gyrobicupola
(J39, epigybcu) |
|
10 triangles
10+10 squares 2 pentagons |
10x 4-5
20x 3-4 10x 4-4 10x 3-4 10x 4-4 |
10 (3.4.5.4)
20 (3.43) |
I2(10)×A1/2, order 20 |
| Elongated pentagonal orthobirotunda
(J42, epobro) |
|
10+10 triangles
5+5 squares 2+10 pentagons |
50x 3-5
10x 3-4 10x 4-5 10x 4-4 |
20 (3.5.3.5)
20 (3.42.5) |
H2×A1, order 20 |
| Elongated pentagonal gyrobirotunda
(J43, epgybro) |
|
10+10 triangles
10 squares 2+10 pentagons |
50x 3-5
10x 3-4 10x 4-5 10x 4-4 |
20 (3.5.3.5)
20 (3.42.5) |
I2(10)×A1/2, order 20 |
| Elongated pentagonal orthocupolarotunda
(J40, epocuro) |
|
5+5+5 triangles
5+5+5 squares 1+5+1 pentagons |
25x 3-5
5x 3-4 5x 4-5 10x 4-4 5x 3-4 5x 4-4 10x 3-4 5x 4-5 |
10 (3.5.3.5)
10 (3.42.5) 10 (3.43) 5 (3.4.5.4) |
H2×I, order 10 |
| Elongated pentagonal gyrocupolarotunda
(J41, epgycuro) |
|
5+5+5 triangles
5+5+5 squares 1+5+1 pentagons |
25x 3-5
5x 3-4 5x 4-5 10x 4-4 5x 3-4 5x 4-4 10x 3-4 5x 4-5 |
10 (3.5.3.5)
10 (3.42.5) 10 (3.43) 5 (3.4.5.4) |
H2×I, order 10 |
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Gyroelongated square bipyramid
(J17, gyesqidpy) |
|
8+8 triangles | 8x 3-3
8x 3-3 8x 3-3 |
2 (34)
8 (35) |
I2(8)×A1/2, order 16 |
| Gyroelongated triangular bicupola
(J44, gyetibcu) |
|
2+6+6+6 triangles
6 squares |
18x 3-4
6x 3-3 6x 3-4 12x 3-3 |
6 (3.4.3.4)
12 (34.4) |
A2×A1+, order 6 |
| Gyroelongated square bicupola
(J45, gyesquibcu) |
|
8+8+8 triangles
2+8 squares |
8x 4-4
16x 3-4 8x 3-3 8x 3-4 16x 3-3 |
8 (3.43)
16 (34.4) |
B2×A1+, order 8 |
| Gyroelongated pentagonal bicupola
(J46, gyepibcu) |
|
10+10+10 triangles
10 squares 2 pentagons |
10x 4-5
20x 3-4 10x 3-3 10x 3-4 20x 3-3 |
10 (3.4.5.4)
20 (34.4) |
H2×A1+, order 10 |
| Gyroelongated pentagonal birotunda
(J48, gyepabro) |
|
10+10+10+10 triangles
2+10 pentagons |
50x 3-5
10x 3-3 10x 3-5 20x 3-3 |
20 (3.5.3.5)
20 (34.5) |
H2×A1+, order 10 |
| Gyroelongated pentagonal cupolarotunda
(J47, gyepcuro) |
|
5+5+10+10+5 triangles
5 squares 1+5+1 pentagons |
25x 3-5
5x 3-3 5x 3-5 20x 3-3 5x 3-4 5x 3-3 10x 3-4 5x 4-5 |
10 (3.5.3.5)
10 (34.5) 10 (34.4) 5 (3.4.5.4) |
H2+×I, order 5 |
Augmentations[edit | edit source]
Polyhedra can be "augmented" by adding pyramids or cupolae to their faces. However, too many augmentations will make the solid nonconvex, giving each polyhedron an upper limit of possible augmentations. Most polyhedra do not admit augmentations (while maintaining strict convexity).
A pair of augmentations is referred to as "para-" if on parallel faces of a polyhedron, and as "meta-" if not (following the conventions used in similar situations in organic chemistry - "ortho" augmentations would be non-convex). If multiple augmentations on a polyhedron can only be arranged in one way (while maintaining convexity of the resulting polyhedron), like in the biaugmented truncated cube, no such clarification is needed.
The various possible "augmented square prisms" are not shown here because they would be the same as elongated square (bi)pyramids, which have already been covered.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Augmented triangular prism
(J49, autip) |
|
2+2+2 triangles
2 squares |
4x 3-3
2x 3-4 2x 3-3 4x 3-4 1x 4-4 |
1 (34)
4 (33.4) 2 (3.42) |
K2×I, order 4 |
| Biaugmented triangular prism
(J50, bautip) |
|
2+2+2+4 triangles
1 square |
8x 3-3
2x 3-4 4x 3-3 1x 3-3 2x 3-4 |
2 (34)
4 (33.4) 2 (35) |
K2×I, order 4 |
| Triaugmented triangular prism
(J51, tautip) |
|
2+6+6 triangles | 12x 3-3
6x 3-3 3x 3-3 |
3 (34)
6 (35) |
A2×A1, order 12 |
| Augmented pentagonal prism
(J52, aupip) |
|
2+2 triangles
2+2 squares 2 pentagons |
4x 3-3
2x 3-4 2x 3-5 3x 4-4 8x 4-5 |
1 (34)
4 (32.4.5) 6 (42.5) |
K2×I, order 4 |
| Biaugmented pentagonal prism
(J53, baupip) |
|
2+2+4 triangles
1+2 squares 2 pentagons |
8x 3-3
4x 3-4 4x 3-5 1x 4-4 6x 4-5 |
2 (34)
8 (32.4.5) 2 (42.5) |
K2×I, order 4 |
| Augmented hexagonal prism
(J54, auhip) |
|
2+2 triangles
1+2+2 squares 2 hexagons |
4x 3-3
2x 3-4 2x 3-6 4x 4-4 10x 4-6 |
1 (34)
4 (32.4.6) 8 (42.6) |
K2×I, order 4 |
| Parabiaugmented hexagonal prism
(J55, pabauhip) |
|
4+4 triangles
4 squares 2 hexagons |
8x 3-3
4x 3-4 4x 3-6 2x 4-4 8x 4-6 |
2 (34)
8 (32.4.6) 4 (42.6) |
K3, order 8 |
| Metabiaugmented hexagonal prism
(J56, mabauhip) |
|
2+2+4 triangles
1+1+2 squares 2 hexagons |
8x 3-3
4x 3-4 4x 3-6 2x 4-4 8x 4-6 |
2 (34)
8 (32.4.6) 4 (42.6) |
K2×I, order 4 |
| Triaugmented hexagonal prism
(J57, tauhip) |
|
6+6 triangles
3 squares 2 hexagons |
12x 3-3
6x 3-4 6x 3-6 6x 4-6 |
3 (34)
12 (32.4.6) |
A2×A1, order 12 |
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Augmented dodecahedron
(J58, aud) |
|
5 triangles
1+5+5 pentagons |
5x 3-3
5x 3-5 25x 5-5 |
1 (35)
5 (32.52) 15 (53) |
H2×I, order 10 |
| Parabiaugmented dodecahedron
(J59, pabaud) |
|
10 triangles
10 pentagons |
10x 3-3
10x 3-5 20x 5-5 |
2 (35)
10 (32.52) 10 (53) |
I2(10)×A1/2, order 20 |
| Metabiaugmented dodecahedron
(J60, mabaud) |
|
2+4+4 triangles
2+2+2+4 pentagons |
10x 3-3
10x 3-5 20x 5-5 |
2 (35)
10 (32.52) 10 (53) |
K2×I, order 4 |
| Triaugmented dodecahedron
(J61, taud) |
|
3+6+6 triangles
3+3+3 pentagons |
15x 3-3
15x 3-5 15x 5-5 |
3 (35)
15 (32.52) 5 (53) |
A2×I, order 6 |
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Augmented truncated tetrahedron
(J65, autut) |
|
1+3+3+1 triangles
3 squares 3 hexagons |
9x 3-4
3x 3-6 3x 3-4 3x 6-6 3x 3-6 |
3 (3.4.3.4)
6 (3.4.3.6) 6 (3.62) |
A2×I, order 6 |
| Augmented truncated cube
(J66, autic) |
|
4+4+4 triangles
1+4 squares 1+4 octagons |
4x 4-4
8x 3-4 4x 3-8 4x 3-4 8x 8-8 20x 3-8 |
4 (3.43)
8 (3.4.3.8) 16 (3.82) |
B2×I, order 8 |
| Biaugmented truncated cube
(J67, bautic) |
|
8+8 triangles
2+8 squares 4 octagons |
8x 4-4
16x 3-4 8x 3-8 8x 3-4 4x 8-8 16x 3-8 |
8 (3.43)
16 (3.4.3.8) 8 (3.82) |
B2×A1, order 16 |
| Augmented truncated dodecahedron
(J68, autid) |
|
5+20 triangles
5 squares 1 pentagon 1+5+5 decagons |
5x 4-5
10x 3-4 5x 3-10 5x 3-4 25x 10-10 55x 3-10 |
5 (3.4.5.4)
10 (3.4.3.10) 50 (3.102) |
H2×I, order 10 |
| Parabiaugmented truncated dodecahedron
(J69, pabautid) |
|
10+20 triangles
10 squares 2 pentagons 10 decagons |
10x 4-5
20x 3-4 10x 3-10 10x 3-4 20x 10-10 50x 3-10 |
10 (3.4.5.4)
20 (3.4.3.10) 40 (3.102) |
I2(10)×A1/2, order 20 |
| Metabiaugmented truncated dodecahedron
(J70, mabautid) |
|
2+2+2+2+2+4+4+4+4+4 triangles
2+4+4 squares 2 pentagons 2+2+2+4 decagons |
10x 4-5
20x 3-4 10x 3-10 10x 3-4 20x 10-10 50x 3-10 |
10 (3.4.5.4)
20 (3.4.3.10) 40 (3.102) |
K2×I, order 4 |
| Triaugmented truncated dodecahedron
(J71, tautid) |
|
1+1+3+3+3+6+6+6+6 triangles
3+6+6 squares 3 pentagons 3+3+3 decagons |
15x 4-5
30x 3-4 15x 3-10 15x 3-4 15x 10-10 45x 3-10 |
10 (3.4.5.4)
20 (3.4.3.10) 40 (3.102) |
A2×I, order 6 |
Diminishings[edit | edit source]
"Diminishing" is the opposite of augmenting: it removes one or more pyramids or cupolae from a solid, leaving behind regular polygonal face(s). A "diminished icosahedron" would be a gyroelongated pentagonal pyramid, and a "parabidiminished icosahedron" would be a pentagonal antiprism.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Metabidiminished icosahedron
(J62, mibdi) |
|
2+2+2+4 triangles
2 pentagons |
10x 3-3
8x 3-5 1x 5-5 |
2 (35)
6 (33.5) 2 (3.52) |
K2×I, order 6 |
| Tridiminished icosahedron
(J63, teddi) |
|
1+1+3 triangles
3 pentagons |
3x 3-3
9x 3-5 3x 5-5 |
6 (3.52)
3 (33.5) |
A2×I, order 6 |
| Augmented tridiminished icosahedron
(J64, auteddi) |
|
1+3+3 triangles
3 pentagons |
3x 3-3
3x 3-5 3x 5-5 6x 3-5 3x 3-3 |
1 (33)
3 (32.52) 3 (3.52) 3 (33.5) |
A2×I, order 6 |
Gyrations and diminishings of small rhombicosidodecahedron[edit | edit source]
A solid that can be diminished can be "gyrated" instead if the diminishing would remove a cupola (or rotunda). The gyration rotates this cupola.
Gyrating a pyramid would not change the solid. Gyrating a cuboctahedron or icosidodecahedron would produce a triangular orthobicupola or pentagonal orthobirotunda, respectively. Gyrating a small rhombicuboctahedron would produce an elongated square gyrobicupola, and diminishing it would produce an elongated square cupola.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Gyrate rhombicosidodecahedron
(J72, gyrid) |
|
5+5+5+5 triangles
5+5+5+5+10 squares 1+1+5+5 pentagons |
55x 4-5
5x 3-5 55x 3-4 5x 4-4 |
45+5 (3.4.5.4)
10 (3.42.5) |
H2×I, order 10 |
| Parabigyrate rhombicosidodecahedron
(J73, pabgyrid) |
|
10+10 triangles
10+10+10 squares 2+10 pentagons |
50x 4-5
10x 3-5 50x 3-4 10x 4-4 |
30+10 (3.4.5.4)
20 (3.42.5) |
I2(10)×A1/2, order 20 |
| Metabigyrate rhombicosidodecahedron
(J74, mabgyrid) |
|
2+2+2+2+4+4+4 triangles
1+1+2+2+4+4+4+4+4+4 squares 2+2+2+2+4 pentagons |
50x 4-5
10x 3-5 50x 3-4 10x 4-4 |
30+10 (3.4.5.4)
20 (3.42.5) |
K2×I, order 4 |
| Trigyrate rhombicosidodecahedron
(J75, tagyrid) |
|
1+1+3+3+6+6 triangles
3+3+3+3+6+6+6 squares 3+3+3+3 pentagons |
45x 4-5
15x 3-5 45x 3-4 15x 4-4 |
15+15 (3.4.5.4)
30 (3.42.5) |
A2×I, order 6 |
| Diminished rhombicosidodecahedron
(J76, dirid) |
|
5+5+5 triangles
5+5+5+10 squares 1+5+5 pentagons 1 decagon |
50x 4-5
5x 5-10 45x 3-4 5x 4-10 |
45 (3.4.5.4)
10 (4.5.10) |
H2×I, order 10 |
| Paragyrate diminished rhombicosidodecahedron
(J77, pagydrid) |
|
5+5+5 triangles
5+5+5+10 squares 1+5+5 pentagons 1 decagon |
45x 4-5
5x 3-5 5x 5-10 40x 3-4 5x 4-4 5x 4-10 |
30+5 (3.4.5.4)
10 (3.42.5) 10 (4.5.10) |
H2×I, order 10 |
| Metagyrate diminished rhombicosidodecahedron
(J78, magydrid) |
|
1+1+1+2+2+2+2+2+2 triangles
1+1+1+2+2+2+2+2+2+2+2+2+2+2 squares 1+1+1+2+2+2+2 pentagons 1 decagon |
45x 4-5
5x 3-5 5x 5-10 40x 3-4 5x 4-4 5x 4-10 |
30+5 (3.4.5.4)
10 (3.42.5) 10 (4.5.10) |
A1×I×I, order 2 |
| Bigyrate diminished rhombicosidodecahedron
(J79, bagydrid) |
|
1+1+1+2+2+2+2+2+2 triangles
1+1+1+2+2+2+2+2+2+2+2+2+2+2 squares 1+1+1+2+2+2+2 pentagons 1 decagon |
40x 4-5
10x 3-5 5x 5-10 35x 3-4 10x 4-4 5x 4-10 |
15+10 (3.4.5.4)
20 (3.42.5) 10 (4.5.10) |
A1×I×I, order 2 |
| Parabidiminished rhombicosidodecahedron
(J80, pabidrid) |
|
10 triangles
10+10 squares 10 pentagons 2 decagons |
40x 4-5
10x 5-10 30x 3-4 10x 4-10
|
30 (3.4.5.4)
20 (4.5.10) |
I2(10)×A1/2, order 20 |
| Metabidiminished rhombicosidodecahedron
(J81, mabidrid) |
|
2+2+2+4 triangles
1+1+2+4+4+4+4 squares 2+2+2+4 pentagons 2 decagons |
40x 4-5
10x 5-10 30x 3-4 10x 4-10
|
30 (3.4.5.4)
20 (4.5.10) |
K2×I, order 4 |
| Gyrate bidiminished rhombicosidodecahedron
(J82, gybadrid) |
|
1+1+1+1+2+2+2 triangles
1+1+1+1+2+2+2+2+2+2+2+2 squares 1+1+1+1+2+2+2 pentagons 2 decagons |
35x 4-5
5x 3-5 10x 5-10 25x 3-4 5x 4-4 10x 4-10 |
15+5 (3.4.5.4)
10 (3.42.5) 20 (4.5.10) |
A1×I×I, order 2 |
| Tridiminished rhombicosidodecahedron
(J83, tedrid) |
|
1+1+3 triangles
3+3+3+6 squares 3+3+3 pentagons 3 decagons |
30x 4-5
15x 5-10 15x 3-4 15x 4-10 |
15 (3.4.5.4)
30 (4.5.10) |
A2×I, order 6 |
The elementary Johnson solids[edit | edit source]
All Johnson solids up to this point were made by "cutting and pasting" pieces of Platonic solids, Archimedean solids, prisms, and antiprisms. The last few Johnson solids do not result from such simple manipulations, although some of them have subtle relationships with other solids. Only the augmented sphenocorona can be constructed by augmentation.
| Name | Image | Faces | Edges | Vertices | Symmetry group |
|---|---|---|---|---|---|
| Snub disphenoid
(J84, snadow) |
|
4+8 triangles | 2x 3-3
8x 3-3 4x 3-3 4x 3-3 |
4 (34)
4 (35) |
B2×A1/2, order 8 |
| Snub square antiprism
(J85, snisquap) |
|
8+16 triangles
2 squares |
8x 3-4
16x 3-3 8x 3-3 8x 3-3 |
8 (34.4)
8 (35) |
I2(8)×A1/2, order 16 |
| Sphenocorona
(J86, waco) |
|
2+2+4+4 triangles
2 squares |
1x 3-3
4x 3-3 4x 3-3 4x 3-3 2x 3-3 2x 3-4 4x 3-4 1x 4-4 |
2 (35)
2 (35) 4 (33.4) 2 (32.42) |
K2×I, order 4 |
| Augmented sphenocorona
(J87, auwaco) |
|
1+1+1+1+2+2+2+2+2+2 triangles
1 square |
1x 3-3
4x 3-3 4x 3-3 4x 3-3 2x 3-3 1x 3-4 2x 3-4
2x 3-3 1x 3-4 4x 3-3 |
2 (35)
2 (35) 2 (33.4) 2 (34.4) 1 (34) |
A1×I×I, order 2 |
| Sphenomegacorona
(J88, wamco) |
|
2+2+4+4+4 triangles
2 squares |
1x 3-3
4x 3-3 4x 3-3 4x 3-3 2x 3-3 4x 3-3 2x 3-3 2x 3-4 4x 3-4 1x 4-4 |
2 (35)
2 (35) 2 (34) 4 (34.4) 2 (32.42) |
K2×I, order 4 |
| Hebesphenomegacorona
(J89, hawmco) |
|
1+2+2+2+2+4+4+4+4+4+4 triangles
1+2 squares |
1x 3-3
4x 3-3 4x 3-3 4x 3-3 2x 3-3 4x 3-3 4x 3-3 2x 3-4 4x 3-4 2x 3-4 2x 4-4 |
2 (35)
2 (35) 2 (35) 4 (34.4) 4 (32.42) |
K2×I, order 4 |
| Disphenocingulum
(J90, dawci) |
|
4+8+8 triangles
4 squares |
2x 4-4
4x 3-4 4x 3-3 8x 3-4 4x 3-3 8x 3-3 |
4 (32.42)
4 (35) 8 (34.4) |
B2×A1/2, order 8 |
| Bilunabirotunda
(J91, bilbiro) |
|
4+4 triangles
2 squares 4 pentagons |
8x 3-5
4x 3-4 4x 3-4 8x 3-5 2x 5-5 |
2 (3.5.3.5)
8 (3.4.3.5) 4 (3.52) |
K3, order 8 |
| Triangular hebesphenorotunda
(J92, thawro) |
|
1+3+3+6 triangles
3 squares 3 pentagons 1 hexagon |
9x 3-5
3x 3-4 6x 3-5 6x 3-3 6x 3-4 3x 4-6 3x 3-6 |
3 (3.5.3.5)
6 (3.4.3.6) 3 (33.5) 6 (32.4.6) |
A2×I, order 6 |
Johnson solids by properties[edit | edit source]
Symmetry[edit | edit source]
The following Johnson solids have reflection planes:
A1×I×I: J78, J79, J82, J87
K2×I: J49, J50, J52, J53, J54, J56, J60, J62, J70, J74, J81, J86, J88, J89
A2×I: J3, J7, J18, J22, J61, J63, J64, J65, J71, J75, J83, J92
B2×I: J1, J4, J8, J10, J19, J23, J66
H2×I: J2, J5, J6, J9, J11, J20, J21, J24, J25, J32, J33, J40, J41, J58, J68, J72, J76, J77
K3: J55, J91
B2×A1/2: J26, J84, J90
A2×A1: J12, J14, J27, J35, J51, J57
B2×A1: J15, J28, J67
H2×A1: J13, J16, J30, J34, J38, J42
G2×A1/2: J36
I2(8)×A1/2: J17, J29, J37, J85
I2(10)×A1/2: J31, J39, J43, J59, J69, J73, J80
The following Johnson solids do not have reflection planes:
H2+×I: J47
A2×A1+: J44
B2×A1+: J45
H2×A1+: J46, J48
Face types[edit | edit source]
The following Johnson solids only have triangular faces: J12, J13, J17, J51, J84
The following Johnson solids have triangular and square faces: J1, J7, J8, J10, J14, J15, J16, J26, J27, J28, J29, J35, J36, J37, J44, J45, J49, J50, J85, J86, J87, J88, J89, J90
The following Johnson solids have triangular and pentagonal faces: J2, J11, J34, J48, J58, J59, J60, J61, J62, J63, J64
The following Johnson solids have triangular, square and pentagonal faces: J9, J30, J31, J32, J33, J38, J39, J40, J41, J42, J43, J46, J47, J52, J53, J72, J73, J74, J75, J91
The following Johnson solids have triangular, square and hexagonal faces: J3, J18, J22, J54, J55, J56, J57, J65
The following Johnson solids have triangular, square and octagonal faces: J4, J19, J23, J66, J67
The following Johnson solids have triangular, pentagonal and decagonal faces: J6, J25
The following Johnson solids have triangular, square, pentagonal and decagonal faces: J5, J20, J21, J24, J68, J69, J70, J71, J76, J77, J78, J79, J80, J81, J82, J83
Only one Johnson solid has triangular, square, pentagonal and hexagonal faces: J92
Inspheres and circumspheres[edit | edit source]
The following Johnson solids can be inscribed in spheres:
Radius 0.71: J1
Radius 0.95: J2, J11, J62, J63
Radius 1: J3, J27
Radius 1.40: J4, J19, J37
Radius 1.62: J6, J34
Radius 2.23: J5, J72, J73, J74, J75, J76, J77, J78, J79, J80, J81, J82, J83
The following Johnson solids can be circumscribed around spheres:
Radius ?: J1
Radius ?: J2
Radius 0.27: J12
Radius 0.42: J13
Generalizations[edit | edit source]
Relaxing the requirement of convexity (and allowing uniforms) results in the acrohedra, which also encompass the Stewart toroids. The entire set of acrohedra is not generally studied, but it is interesting to ask if an acrohedron exists under certain constraints, such as having a certain configuration of faces around at least one vertex.
The higher-dimensional generalizations of the Johnson solids include the non-uniform CRF polytopes, where faces must be regular, and the non-uniform Blind polytopes, where facets (e.g. cells in a 4-polytope) are regular. The Blind polytopes have been completely enumerated, while the CRF polytopes are a much broader class.
While all Johnson solids are symmetrical, the Blind polytopes contain asymmetrical polytopes, all of which are special cuts, which are 4D analogues of the diminished icosahedra in the Johnson solids.
Roger Kaufman investigated the convex triamond polyhedra, which are convex polyhedra with all regular faces except for at least one "triamond," defined as a trapezoid with edge lengths 1:1:1:2.
References[edit | edit source]
- ↑ Johnson, Norman W. (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. ISSN 0008-414X. Zbl 0132.14603.
- ↑ Sommerville, D. M. Y. (1905), "Semi-regular networks of the plane in absolute geometry", Transactions of the Royal Society of Edinburgh, 41: 725–747, doi:10.1017/s0080456800035560.
- ↑ Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. Zbl 0177.24802. No ISBN. The first proof that there are only 92 Johnson solids: see also Zalgaller, Victor A. (1967). "Convex Polyhedra with Regular Faces". Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova (in Russian). 2: 1–221. ISSN 0373-2703. Zbl 0165.56302.
External links[edit | edit source]
Wikipedia contributors, "Johnson solids."