Johnson solid
A Johnson solid is a strictly convex regularfaced 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 are symmetrical.
Nearmiss Johnson solids[edit  edit source]
A large amount of nearmiss 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 ngon 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 ngon and a 2ngon 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 33
4x 34 
1 (3^{4})
4 (3^{2}.4) 
B_{2}×I, order 8  
Pentagonal pyramid
(J2, peppy) 
5 triangles
1 pentagon 
5x 33
5x 35 
1 (3^{5})
5 (3^{2}.5) 
H_{2}×I, order 10  
Triangular cupola
(J3, tricu) 
1+3 triangles
3 squares 1 hexagon 
9x 34
3x 36 3x 46 
3 (3.4.3.4)
6 (3.4.6) 
A_{2}×I, order 6  
Square cupola
(J4, squacu) 
4 triangles
1+4 squares 1 octagon 
4x 44
8x 34 4x 38 4x 48 
4 (3.4^{3})
8 (3.4.8) 
B_{2}×I, order 8  
Pentagonal cupola
(J5, pecu) 
5 triangles
5 squares 1 pentagon 1 decagon 
5x 45
10x 34 5x 310 5x 410 
5 (3.4.5.4)
10 (3.4.10) 
H_{2}×I, order 10  
Pentagonal rotunda
(J6, pero) 
5+5 triangles
1+5 pentagons 1 decagon 
25x 35
5x 310 5x 510 
10x (3.5.3.5)
10x (3.5.10) 
H_{2}×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 33
3x 34 3x 44 3x 34 
1 (3^{3})
3 (3^{2}.4^{2}) 3 (3.4^{2}) 
A_{2}×I, order 6  
Elongated square pyramid
(J8, esquipy) 
4 triangles
1+4 squares 
4x 33
4x 34 8x 44 
1 (3^{4})
4 (3^{2}.4^{2}) 4 (4^{3}) 
B_{2}×I, order 8  
Elongated pentagonal pyramid
(J9, epeppy) 
5 triangles
5 squares 1 pentagon 
5x 33
5x 34 5x 44 5x 45 
1 (3^{5})
5 (3^{2}.4^{2}) 5 (4^{2}.5) 
H_{2}×I, order 10  
Elongated triangular cupola
(J18, etcu) 
1+3 triangles
3+3+3 squares 1 hexagon 
9x 34
3x 44 3x 34 6x 44 6x 46 
3 (3.4.3.4)
6 (3.4^{3}) 6 (4^{2}.6) 
A_{2}×I, order 6  
Elongated square cupola
(J19, escu) 
4 triangles
1+4+4+4 squares 1 octagon 
16x 44
12x 34 8x 48 
12 (3.4^{3})
8 (4^{2}.8) 
B_{2}×I, order 8  
Elongated pentagonal cupola
(J20, epcu) 
5 triangles
5+5+5 squares 1 pentagon 1 decagon 
5x 45
10x 34 5x 44 5x 34 10x 44 10x 410 
5 (3.4.5.4)
10 (3.4^{3}) 10 (4^{2}.10) 
H_{2}×I, order 10  
Elongated pentagonal rotunda
(J21, epro) 
5+5 triangles
5+5 squares 1+5 pentagons 1 decagon 
25x 35
5x 34 5x 45 10x 44 10x 410 
10 (3.5.3.5)
10 (3.4^{2}.5) 10(4^{2}.10) 
H_{2}×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 33
4x 33 8x 33 4x 34 
1 (3^{4})
4 (3^{5}) 4 (3^{3}.4) 
B_{2}×I, order 8  
Gyroelongated pentagonal pyramid
(J11, gyepip) 
5+5+5 triangles
1 pentagon 
20x 33
5x 35 
6 (3^{5})
5 (3^{3}.5) 
H_{2}×I, order 10  
Gyroelongated triangular cupola
(J22, gyetcu) 
1+3+3+3+6 triangles
3 squares 1 hexagon 
9x 34
3x 33 3x 34 12x 33 6x 36 
3 (3.4.3.4)
6 (3^{4}.4) 6 (3^{3}.6) 
A_{2}×I, order 6  
Gyroelongated square cupola
(J23, gyescu) 
4+4+4+8 triangles
1+4 squares 1 octagon 
4x 44
8x 34 4x 33 4x 34 16x 33 8x 38 
4 (3.4^{3})
8 (3^{4}.4) 8 (3^{3}.8) 
B_{2}×I, order 8  
Gyroelongated pentagonal cupola
(J24, gyepcu) 
5+5+5+10 triangles
5 squares 1 pentagon 1 decagon 
5x 45
10x 34 5x 33 5x 34 20x 33 10x 310 
5 (3.4.5.4)
10 (3^{4}.4) 10 (3^{3}.10) 
H_{2}×I, order 10  
Gyroelongated pentagonal rotunda
(J25, gyepro) 
5+5+5+5+10 triangles
1+5 pentagons 1 decagon 
25x 35
5x 33 5x 35 20x 33 10x 310 
10 (3.5.3.5)
10 (3^{4}.5) 10 (3^{3}.10) 
H_{2}×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 decagonbased.
Bicupolas, birotundas, and the pentagonal cupolarotunda 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 2gon and a 4gon.
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 33
3x 33 
2 (3^{3})
3 (3^{4}) 
A_{2}×A_{1}, order 12  
Pentagonal bipyramid
(J13, pedpy) 
10 triangles  10x 33
5x 33 
2 (3^{5})
5 (3^{4}) 
H_{2}×A_{1}, order 20  
Gyrobifastigium
(J26, gybef) 
4 triangles
4 squares 
2x 44
8x 34 4x 34 
4 (3.4^{2})
4 (3.4.3.4) 
B_{2}×A_{1}/2, order 8  
Triangular orthobicupola
(J27, tobcu) 
2+6 triangles
6 squares 
18x 34
3x 44 3x 33 
6 (3.4.3.4)
6 (3^{2}.4^{2}) 
A_{2}×A_{1}, order 12  
Square orthobicupola
(J28, squobcu) 
8 triangles
2+8 squares 
8x 44
16x 34 4x 44 4x 33 
8 (3.4^{3})
8 (3^{2}.4^{2}) 
B_{2}×A_{1}, order 16  
Square gyrobicupola
(J29, squigybcu) 
8 triangles
2+8 squares 
8x 44
16x 34 8x 34 
8 (3.4^{3})
8 (3.4.3.4) 
I_{2}(8)×A_{1}/2, order 16  
Pentagonal orthobicupola
(J30, pobcu) 
10 triangles
10 squares 2 pentagons 
10x 45
20x 34 5x 33 5x 44 
10 (3.4.5.4)
10 (3^{2}.4^{2}) 
H_{2}×A_{1}, order 20  
Pentagonal gyrobicupola
(J31, pegybcu) 
10 triangles
10 squares 2 pentagons 
10x 45
20x 34 10x 34 
10 (3.4.5.4)
10 (3.4.3.4) 
I_{2}(10)×A_{1}/2, order 20  
Pentagonal orthobirotunda
(J34, pobro) 
10+10 triangles
2+10 pentagons 
50x 35
5x 33 5x 55 
20 (3.5.3.5)
10 (3^{2}.5^{2}) 
H_{2}×A_{1}, order 20  
Pentagonal orthocupolarotunda
(J32, pocuro) 
5+5+5 triangles
5 squares 1+1+5 pentagons 
25x 35
5x 34 5x 35 10x 34 5x 45 
10 (3.5.3.5)
10 (3.4.3.5) 5 (3.4.5.4) 
H_{2}×I, order 10  
Pentagonal gyrocupolarotunda
(J33, pegycuro) 
5+5+5 triangles
5 squares 1+1+5 pentagons 
25x 35
5x 33 5x 45 10x 34 5x 45 
10 (3.5.3.5)
10 (3^{2}.4.5) 5 (3.4.5.4) 
H_{2}×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 33
6x 34 3x 44 
2 (3^{3})
6 (3^{2}.4^{2}) 
A_{2}×A_{1}, order 12  
Elongated square bipyramid
(J15, esquidpy) 
8 triangles
4 squares 
8x 33
8x 34 4x 44 
2 (3^{4})
8 (3^{2}.4^{2}) 
B_{2}×A_{1}, order 16  
Elongated pentagonal bipyramid
(J16, epedpy) 
10 triangles
5 squares 
10x 33
10x 34 5x 44 
2 (3^{5})
10 (3^{2}.4^{2}) 
H_{2}×A_{1}, order 20  
Elongated triangular orthobicupola
(J35, etobcu) 
2+6 triangles
6+3+3 squares 
18x 34
6x 44 6x 34 6x 44 
6 (3.4.3.4)
12 (3.4^{3}) 
A_{2}×A_{1}, order 12  
Elongated triangular gyrobicupola
(J36, etigybcu) 
2+6 triangles
6+6 squares 
18x 34
6x 44 6x 34 6x 44 
6 (3.4.3.4)
12 (3.4^{3}) 
G_{2}×A_{1}/2, order 12  
Elongated square gyrobicupola
(J37, esquigybcu) 
8 triangles
2+8+8 squares 
24x 44
24x 34 
24 (3.4^{3})  I_{2}(8)×A_{1}/2, order 16  
Elongated pentagonal orthobicupola
(J38, epobcu) 
10 triangles
10+5+5 squares 2 pentagons 
10x 45
20x 34 10x 44 10x 34 10x 44 
10 (3.4.5.4)
20 (3.4^{3}) 
H_{2}×A_{1}, order 20  
Elongated pentagonal gyrobicupola
(J39, epigybcu) 
10 triangles
10+10 squares 2 pentagons 
10x 45
20x 34 10x 44 10x 34 10x 44 
10 (3.4.5.4)
20 (3.4^{3}) 
I_{2}(10)×A_{1}/2, order 20  
Elongated pentagonal orthobirotunda
(J42, epobro) 
10+10 triangles
5+5 squares 2+10 pentagons 
50x 35
10x 34 10x 45 10x 44 
20 (3.5.3.5)
20 (3.4^{2}.5) 
H_{2}×A_{1}, order 20  
Elongated pentagonal gyrobirotunda
(J43, epgybro) 
10+10 triangles
10 squares 2+10 pentagons 
50x 35
10x 34 10x 45 10x 44 
20 (3.5.3.5)
20 (3.4^{2}.5) 
I_{2}(10)×A_{1}/2, order 20  
Elongated pentagonal orthocupolarotunda
(J40, epocuro) 
5+5+5 triangles
5+5+5 squares 1+5+1 pentagons 
25x 35
5x 34 5x 45 10x 44 5x 34 5x 44 10x 34 5x 45 
10 (3.5.3.5)
10 (3.4^{2}.5) 10 (3.4^{3}) 5 (3.4.5.4) 
H_{2}×I, order 10  
Elongated pentagonal gyrocupolarotunda
(J41, epgycuro) 
5+5+5 triangles
5+5+5 squares 1+5+1 pentagons 
25x 35
5x 34 5x 45 10x 44 5x 34 5x 44 10x 34 5x 45 
10 (3.5.3.5)
10 (3.4^{2}.5) 10 (3.4^{3}) 5 (3.4.5.4) 
H_{2}×I, order 10 
Name  Image  Faces  Edges  Vertices  Symmetry group 

Gyroelongated square bipyramid
(J17, gyesqidpy) 
8+8 triangles  8x 33
8x 33 8x 33 
2 (3^{4})
8 (3^{5}) 
I_{2}(8)×A_{1}/2, order 16  
Gyroelongated triangular bicupola
(J44, gyetibcu) 
2+6+6+6 triangles
6 squares 
18x 34
6x 33 6x 34 12x 33 
6 (3.4.3.4)
12 (3^{4}.4) 
A_{2}×A_{1}+, order 6  
Gyroelongated square bicupola
(J45, gyesquibcu) 
8+8+8 triangles
2+8 squares 
8x 44
16x 34 8x 33 8x 34 16x 33 
8 (3.4^{3})
16 (3^{4}.4) 
B_{2}×A_{1}+, order 8  
Gyroelongated pentagonal bicupola
(J46, gyepibcu) 
10+10+10 triangles
10 squares 2 pentagons 
10x 45
20x 34 10x 33 10x 34 20x 33 
10 (3.4.5.4)
20 (3^{4}.4) 
H_{2}×A_{1}+, order 10  
Gyroelongated pentagonal birotunda
(J48, gyepabro) 
10+10+10+10 triangles
2+10 pentagons 
50x 35
10x 33 10x 35 20x 33 
20 (3.5.3.5)
20 (3^{4}.5) 
H_{2}×A_{1}+, order 10  
Gyroelongated pentagonal cupolarotunda
(J47, gyepcuro) 
5+5+10+10+5 triangles
5 squares 1+5+1 pentagons 
25x 35
5x 33 5x 35 20x 33 5x 34 5x 33 10x 34 5x 45 
10 (3.5.3.5)
10 (3^{4}.5) 10 (3^{4}.4) 5 (3.4.5.4) 
H_{2}+×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 nonconvex). 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 33
2x 34 2x 33 4x 34 1x 44 
1 (3^{4})
4 (3^{3}.4) 2 (3.4^{2}) 
K_{2}×I, order 4  
Biaugmented triangular prism
(J50, bautip) 
2+2+2+4 triangles
1 square 
8x 33
2x 34 4x 33 1x 33 2x 34 
2 (3^{4})
4 (3^{3}.4) 2 (3^{5}) 
K_{2}×I, order 4  
Triaugmented triangular prism
(J51, tautip) 
2+6+6 triangles  12x 33
6x 33 3x 33 
3 (3^{4})
6 (3^{5}) 
A_{2}×A_{1}, order 12  
Augmented pentagonal prism
(J52, aupip) 
2+2 triangles
2+2 squares 2 pentagons 
4x 33
2x 34 2x 35 3x 44 8x 45 
1 (3^{4})
4 (3^{2}.4.5) 6 (4^{2}.5) 
K_{2}×I, order 4  
Biaugmented pentagonal prism
(J53, baupip) 
2+2+4 triangles
1+2 squares 2 pentagons 
8x 33
4x 34 4x 35 1x 44 6x 45 
2 (3^{4})
8 (3^{2}.4.5) 2 (4^{2}.5) 
K_{2}×I, order 4  
Augmented hexagonal prism
(J54, auhip) 
2+2 triangles
1+2+2 squares 2 hexagons 
4x 33
2x 34 2x 36 4x 44 10x 46 
1 (3^{4})
4 (3^{2}.4.6) 8 (4^{2}.6) 
K_{2}×I, order 4  
Parabiaugmented hexagonal prism
(J55, pabauhip) 
4+4 triangles
4 squares 2 hexagons 
8x 33
4x 34 4x 36 2x 44 8x 46 
2 (3^{4})
8 (3^{2}.4.6) 4 (4^{2}.6) 
K_{3}, order 8  
Metabiaugmented hexagonal prism
(J56, mabauhip) 
2+2+4 triangles
1+1+2 squares 2 hexagons 
8x 33
4x 34 4x 36 2x 44 8x 46 
2 (3^{4})
8 (3^{2}.4.6) 4 (4^{2}.6) 
K_{2}×I, order 4  
Triaugmented hexagonal prism
(J57, tauhip) 
6+6 triangles
3 squares 2 hexagons 
12x 33
6x 34 6x 36 6x 46 
3 (3^{4})
12 (3^{2}.4.6) 
A_{2}×A_{1}, order 12 
Name  Image  Faces  Edges  Vertices  Symmetry group 

Augmented dodecahedron
(J58, aud) 
5 triangles
1+5+5 pentagons 
5x 33
5x 35 25x 55 
1 (3^{5})
5 (3^{2}.5^{2}) 15 (5^{3}) 
H_{2}×I, order 10  
Parabiaugmented dodecahedron
(J59, pabaud) 
10 triangles
10 pentagons 
10x 33
10x 35 20x 55 
2 (3^{5})
10 (3^{2}.5^{2}) 10 (5^{3}) 
I_{2}(10)×A_{1}/2, order 20  
Metabiaugmented dodecahedron
(J60, mabaud) 
2+4+4 triangles
2+2+2+4 pentagons 
10x 33
10x 35 20x 55 
2 (3^{5})
10 (3^{2}.5^{2}) 10 (5^{3}) 
K_{2}×I, order 4  
Triaugmented dodecahedron
(J61, taud) 
3+6+6 triangles
3+3+3 pentagons 
15x 33
15x 35 15x 55 
3 (3^{5})
15 (3^{2}.5^{2}) 5 (5^{3}) 
A_{2}×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 34
3x 36 3x 34 3x 66 3x 36 
3 (3.4.3.4)
6 (3.4.3.6) 6 (3.6^{2}) 
A_{2}×I, order 6  
Augmented truncated cube
(J66, autic) 
4+4+4 triangles
1+4 squares 1+4 octagons 
4x 44
8x 34 4x 38 4x 34 8x 88 20x 38 
4 (3.4^{3})
8 (3.4.3.8) 16 (3.8^{2}) 
B_{2}×I, order 8  
Biaugmented truncated cube
(J67, bautic) 
8+8 triangles
2+8 squares 4 octagons 
8x 44
16x 34 8x 38 8x 34 4x 88 16x 38 
8 (3.4^{3})
16 (3.4.3.8) 8 (3.8^{2}) 
B_{2}×A_{1}, order 16  
Augmented truncated dodecahedron
(J68, autid) 
5+20 triangles
5 squares 1 pentagon 1+5+5 decagons 
5x 45
10x 34 5x 310 5x 34 25x 1010 55x 310 
5 (3.4.5.4)
10 (3.4.3.10) 50 (3.10^{2}) 
H_{2}×I, order 10  
Parabiaugmented truncated dodecahedron
(J69, pabautid) 
10+20 triangles
10 squares 2 pentagons 10 decagons 
10x 45
20x 34 10x 310 10x 34 20x 1010 50x 310 
10 (3.4.5.4)
20 (3.4.3.10) 40 (3.10^{2}) 
I_{2}(10)×A_{1}/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 45
20x 34 10x 310 10x 34 20x 1010 50x 310 
10 (3.4.5.4)
20 (3.4.3.10) 40 (3.10^{2}) 
K_{2}×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 45
30x 34 15x 310 15x 34 15x 1010 45x 310 
10 (3.4.5.4)
20 (3.4.3.10) 40 (3.10^{2}) 
A_{2}×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 33
8x 35 1x 55 
2 (3^{5})
6 (3^{3}.5) 2 (3.5^{2}) 
K_{2}×I, order 6  
Tridiminished icosahedron
(J63, teddi) 
1+1+3 triangles
3 pentagons 
3x 33
9x 35 3x 55 
6 (3.5^{2})
3 (3^{3}.5) 
A_{2}×I, order 6  
Augmented tridiminished icosahedron
(J64, auteddi) 
1+3+3 triangles
3 pentagons 
3x 33
3x 35 3x 55 6x 35 3x 33 
1 (3^{3})
3 (3^{2}.5^{2}) 3 (3.5^{2}) 3 (3^{3}.5) 
A_{2}×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 45
5x 35 55x 34 5x 44 
45+5 (3.4.5.4)
10 (3.4^{2}.5) 
H_{2}×I, order 10  
Parabigyrate rhombicosidodecahedron
(J73, pabgyrid) 
10+10 triangles
10+10+10 squares 2+10 pentagons 
50x 45
10x 35 50x 34 10x 44 
30+10 (3.4.5.4)
20 (3.4^{2}.5) 
I_{2}(10)×A_{1}/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 45
10x 35 50x 34 10x 44 
30+10 (3.4.5.4)
20 (3.4^{2}.5) 
K_{2}×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 45
15x 35 45x 34 15x 44 
15+15 (3.4.5.4)
30 (3.4^{2}.5) 
A_{2}×I, order 6  
Diminished rhombicosidodecahedron
(J76, dirid) 
5+5+5 triangles
5+5+5+10 squares 1+5+5 pentagons 1 decagon 
50x 45
5x 510 45x 34 5x 410 
45 (3.4.5.4)
10 (4.5.10) 
H_{2}×I, order 10  
Paragyrate diminished rhombicosidodecahedron
(J77, pagydrid) 
5+5+5 triangles
5+5+5+10 squares 1+5+5 pentagons 1 decagon 
45x 45
5x 35 5x 510 40x 34 5x 44 5x 410 
30+5 (3.4.5.4)
10 (3.4^{2}.5) 10 (4.5.10) 
H_{2}×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 45
5x 35 5x 510 40x 34 5x 44 5x 410 
30+5 (3.4.5.4)
10 (3.4^{2}.5) 10 (4.5.10) 
A_{1}×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 45
10x 35 5x 510 35x 34 10x 44 5x 410 
15+10 (3.4.5.4)
20 (3.4^{2}.5) 10 (4.5.10) 
A_{1}×I×I, order 2  
Parabidiminished rhombicosidodecahedron
(J80, pabidrid) 
10 triangles
10+10 squares 10 pentagons 2 decagons 
40x 45
10x 510 30x 34 10x 410

30 (3.4.5.4)
20 (4.5.10) 
I_{2}(10)×A_{1}/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 45
10x 510 30x 34 10x 410

30 (3.4.5.4)
20 (4.5.10) 
K_{2}×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 45
5x 35 10x 510 25x 34 5x 44 10x 410 
15+5 (3.4.5.4)
10 (3.4^{2}.5) 20 (4.5.10) 
A_{1}×I×I, order 2  
Tridiminished rhombicosidodecahedron
(J83, tedrid) 
1+1+3 triangles
3+3+3+6 squares 3+3+3 pentagons 3 decagons 
30x 45
15x 510 15x 34 15x 410 
15 (3.4.5.4)
30 (4.5.10) 
A_{2}×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 33
8x 33 4x 33 4x 33 
4 (3^{4})
4 (3^{5}) 
B_{2}×A_{1}/2, order 8  
Snub square antiprism
(J85, snisquap) 
8+16 triangles
2 squares 
8x 34
16x 33 8x 33 8x 33 
8 (3^{4}.4)
8 (3^{5}) 
I_{2}(8)×A_{1}/2, order 16  
Sphenocorona
(J86, waco) 
2+2+4+4 triangles
2 squares 
1x 33
4x 33 4x 33 4x 33 2x 33 2x 34 4x 34 1x 44 
2 (3^{5})
2 (3^{5}) 4 (3^{3}.4) 2 (3^{2}.4^{2}) 
K_{2}×I, order 4  
Augmented sphenocorona
(J87, auwaco) 
1+1+1+1+2+2+2+2+2+2 triangles
1 square 
1x 33
4x 33 4x 33 4x 33 2x 33 1x 34 2x 34
2x 33 1x 34 4x 33 
2 (3^{5})
2 (3^{5}) 2 (3^{3}.4) 2 (3^{4}.4) 1 (3^{4}) 
A_{1}×I×I, order 2  
Sphenomegacorona
(J88, wamco) 
2+2+4+4+4 triangles
2 squares 
1x 33
4x 33 4x 33 4x 33 2x 33 4x 33 2x 33 2x 34 4x 34 1x 44 
2 (3^{5})
2 (3^{5}) 2 (3^{4}) 4 (3^{4}.4) 2 (3^{2}.4^{2}) 
K2×I, order 4  
Hebesphenomegacorona
(J89, hawmco) 
1+2+2+2+2+4+4+4+4+4+4 triangles
1+2 squares 
1x 33
4x 33 4x 33 4x 33 2x 33 4x 33 4x 33 2x 34 4x 34 2x 34 2x 44 
2 (3^{5})
2 (3^{5}) 2 (3^{5}) 4 (3^{4}.4) 4 (3^{2}.4^{2}) 
K_{2}×I, order 4  
Disphenocingulum
(J90, dawci) 
4+8+8 triangles
4 squares 
2x 44
4x 34 4x 33 8x 34 4x 33 8x 33 
4 (3^{2}.4^{2})
4 (3^{5}) 8 (3^{4}.4) 
B_{2}×A_{1}/2, order 8  
Bilunabirotunda
(J91, bilbiro) 
4+4 triangles
2 squares 4 pentagons 
8x 35
4x 34 4x 34 8x 35 2x 55 
2 (3.5.3.5)
8 (3.4.3.5) 4 (3.5^{2}) 
K_{3}, order 8  
Triangular hebesphenorotunda
(J92, thawro) 
1+3+3+6 triangles
3 squares 3 pentagons 1 hexagon 
9x 35
3x 34 6x 35 6x 33 6x 34 3x 46 3x 36 
3 (3.5.3.5)
6 (3.4.3.6) 3 (3^{3}.5) 6 (3^{2}.4.6) 
A_{2}×I, order 6 
Johnson solids by properties[edit  edit source]
Symmetry[edit  edit source]
The following Johnson solids have reflection planes:
A_{1}×I×I: J78, J79, J82, J87
K_{2}×I: J49, J50, J52, J53, J54, J56, J60, J62, J70, J74, J81, J86, J88, J89
A_{2}×I: J3, J7, J18, J22, J61, J63, J64, J65, J71, J75, J83, J92
B_{2}×I: J1, J4, J8, J10, J19, J23, J66
H_{2}×I: J2, J5, J6, J9, J11, J20, J21, J24, J25, J32, J33, J40, J41, J58, J68, J72, J76, J77
K_{3}: J55, J91
B_{2}×A_{1}/2: J26, J84, J90
A_{2}×A_{1}: J12, J14, J27, J35, J51, J57
B_{2}×A_{1}: J15, J28, J67
H_{2}×A_{1}: J13, J16, J30, J34, J38, J42
G_{2}×A_{1}/2: J36
I_{2}(8)×A_{1}/2: J17, J29, J37, J85
I_{2}(10)×A_{1}/2: J31, J39, J43, J59, J69, J73, J80
The following Johnson solids do not have reflection planes:
H_{2}+×I: J47
A_{2}×A_{1}+: J44
B_{2}×A_{1}+: J45
H_{2}×A_{1}+: 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 higherdimensional generalizations of the Johnson solids include the nonuniform CRF polytopes, where faces must be regular, and the nonuniform Blind polytopes, where facets (e.g. cells in a 4polytope) 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/cjm19660218. ISSN 0008414X. Zbl 0132.14603.
 ↑ Sommerville, D. M. Y. (1905), "Semiregular 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 03732703. Zbl 0165.56302.
External links[edit  edit source]
Wikipedia contributors, "Johnson solids."