# Johnson solid

(Redirected from Johnson solids)

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 are symmetrical.

## Near-miss Johnson solids

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

### Pyramids, cupolas, and the pentagonal rotunda

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.

Pyramids, cupolas, and rotundas
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

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.

Elongations of pyramids, cupolas, and rotundas
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

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.

Gyroelongations of pyramids, cupolas, and rotundas
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

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.

Bipyramids, bicupolas, birotundas, and cupolarotundas
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

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.

Elongations of the pairs
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
Gyroelongations of the pairs
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

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.

Augmented prisms
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
Augmented Platonic solids
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
Augmented Archimedean solids
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

"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.

Diminished Platonic solids
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

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.

Diminished Archimedean solids
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

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

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.

Elementary Johnson solids
Name Image Faces Edges Vertices Symmetry group
Snub disphenoid

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

1x 3-3

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

### Symmetry

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

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

The following Johnson solids can be inscribed in spheres:

Radius 0.95: J2, J11, J62, J63

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:

## Generalizations

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

1. 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.
2. 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.
3. 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.