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Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. 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 one square face and four triangular faces.

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 actually 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 always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 solids, 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) is unique in being locally vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle. However, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid.

Names

The names are listed below and are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundae), together with the Platonic and Archimedean solids, prisms, and antiprisms.

Bi- means that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, they can be joined so that like faces (ortho-) or unlike faces (gyro-) meet. In this nomenclature, an octahedron would be a square bipyramid, a cuboctahedron would be a triangular gyrobicupola, and an icosidodecahedron would be a pentagonal gyrobirotunda.
Elongated means that a prism has been joined to the base of the solid in question or between the bases of the solids in question. A rhombicuboctahedron would be an elongated square orthobicupola.
Gyroelongated means that an antiprism has been joined to the base of the solid in question or between the bases of the solids in question. An icosahedron would be a gyroelongated pentagonal bipyramid.
Augmented means that a pyramid or cupola has been joined to a face of the solid in question.
Diminished means that a pyramid or cupola has been removed from the solid in question.
Gyrate means that a cupola on the solid in question has been rotated so that different edges match up, as in the difference between ortho- and gyrobicupolae.

The last three operations — augmentation, diminution, and gyration — can be performed more than once on a large enough solid. We add bi- to the name of the operation to indicate that it has been performed twice. (A bigyrate solid has had two of its cupolae rotated.) We add tri- to indicate that it has been performed three times. (A tridiminished solid has had three of its pyramids or cupolae removed.)

Sometimes, bi- alone is not specific enough. We must distinguish between a solid that has had two parallel faces altered and one that has had two oblique faces altered. When the faces altered are parallel, we add para- to the name of the operation. (A parabiaugmented solid has had two parallel faces augmented.) When they are not, we add meta- to the name of the operation. (A metabiaugmented solid has had two oblique faces augmented.)

Enumeration

Prismatoids and rotundae

J_n	Solid name	V	E	F	F₃	F₄	F₅	F₆	F₈	F₁₀	Symmetry

1	Square pyramid	5	8	5	4	1					C_4v
2	Pentagonal pyramid	6	10	6	5		1				C_5v
3	Triangular cupola	9	15	8	4	3		1			C_3v
4	Square cupola	12	20	10	4	5			1		C_4v
5	Pentagonal cupola	15	25	12	5	5	1			1	C_5v
6	Pentagonal rotunda	20	35	17	10		6			1	C_5v

Modified pyramids and dipyramids

J_n	Solid name	V	E	F	F₃	F₄	F₅	Symmetry

7	Elongated triangular pyramid	7	12	7	4	3		C_3v
8	Elongated square pyramid or (augmented cube) or (augmented square prism)	9	16	9	4	5		C_4v
9	Elongated pentagonal pyramid	11	20	11	5	5	1	C_5v
10	Gyroelongated square pyramid	9	20	13	12	1		C_4v
11	Gyroelongated pentagonal pyramid or (diminished icosahedron)	11	25	16	15		1	C_5v
12	Triangular dipyramid	5	9	6	6			D_3h
13	Pentagonal dipyramid	7	15	10	10			D_5h
14	Elongated triangular dipyramid	8	15	9	6	3		D_3h
15	Elongated square dipyramid or (biaugmented cube) or (biaugmented square prism)	10	20	12	8	4		D_4h
16	Elongated pentagonal dipyramid	12	25	15	10	5		D_5h
17	Gyroelongated square dipyramid	10	24	16	16			D_4d

Modified cupolas and rotunda

elongated cupola
elongated rotunda
elongated birotunda
elongated cupolarotunda
elongated bicupola
gyroelongated cupola
gyroelongated rotunda
bicupola
cupolarotunda
gyroelongated bicupola
gyroelongated birotunda
gyroelongated cupolarotunda

J_n	Solid name	V	E	F	F₃	F₄	F₅	F₆	F₈	F₁₀	Symmetry

18	Elongated triangular cupola	15	27	14	4	9		1			C_3v
19	Elongated square cupola (diminished rhombicuboctahedron)	20	36	18	4	13			1		C_4v
20	Elongated pentagonal cupola	25	45	22	5	15	1			1	C_5v
21	Elongated pentagonal rotunda	30	55	27	10	10	6			1	C_5v
22	Gyroelongated triangular cupola	15	33	20	16	3		1			C_3v
23	Gyroelongated square cupola	20	44	26	20	5			1		C_4v
24	Gyroelongated pentagonal cupola	25	55	32	25	5	1			1	C_5v
25	Gyroelongated pentagonal rotunda	30	65	37	30		6			1	C_5v
26	Gyrobifastigium	8	14	8	4	4					D_2d
27	Triangular orthobicupola (gyrate cuboctahedron)	12	24	14	8	6					D_3h
28	Square orthobicupola	16	32	18	8	10					D_4h
29	Square gyrobicupola	16	32	18	8	10					D_4d
30	Pentagonal orthobicupola	20	40	22	10	10	2				D_5h
31	Pentagonal gyrobicupola	20	40	22	10	10	2				D_5d
32	Pentagonal orthocupolarotunda	25	50	27	15	5	7				C_5v
33	Pentagonal gyrocupolarotunda	25	50	27	15	5	7				C_5v
34	Pentagonal orthobirotunda (gyrate icosidodecahedron)	30	60	32	20		12				D_5h
35	Elongated triangular orthobicupola	18	36	20	8	12					D_3h
36	Elongated triangular gyrobicupola	18	36	20	8	12					D_3d
37	Elongated square gyrobicupola (gyrate rhombicuboctahedron)	24	48	26	8	18					D_4d
38	Elongated pentagonal orthobicupola	30	60	32	10	20	2				D_5h
39	Elongated pentagonal gyrobicupola	30	60	32	10	20	2				D_5d
40	Elongated pentagonal orthocupolarotunda	35	70	37	15	15	7				C_5v
41	Elongated pentagonal gyrocupolarotunda	35	70	37	15	15	7				C_5v
42	Elongated pentagonal orthobirotunda	40	80	42	20	10	12				D_5h
43	Elongated pentagonal gyrobirotunda	40	80	42	20	10	12				D_5d
44	Gyroelongated triangular bicupola (2 chiral forms)	18	42	26	20	6					D₃
45	Gyroelongated square bicupola (2 chiral forms)	24	56	34	24	10					D₄
46	Gyroelongated pentagonal bicupola (2 chiral forms)	30	70	42	30	10	2				D₅
47	Gyroelongated pentagonal cupolarotunda (2 chiral forms)	35	80	47	35	5	7				C₅
48	Gyroelongated pentagonal birotunda (2 chiral forms)	40	90	52	40		12				D₅

Augmented prisms

J_n	Solid name	V	E	F	F₃	F₄	F₅	F₆	Symmetry

49	Augmented triangular prism	7	13	8	6	2			C_2v
50	Biaugmented triangular prism	8	17	11	10	1			C_2v
51	Triaugmented triangular prism	9	21	14	14				D_3h
52	Augmented pentagonal prism	11	19	10	4	4	2		C_2v
53	Biaugmented pentagonal prism	12	23	13	8	3	2		C_2v
54	Augmented hexagonal prism	13	22	11	4	5		2	C_2v
55	Parabiaugmented hexagonal prism	14	26	14	8	4		2	D_2h
56	Metabiaugmented hexagonal prism	14	26	14	8	4		2	C_2v
57	Triaugmented hexagonal prism	15	30	17	12	3		2	D_3h

Modified Platonic solids

Augmented dodecahedrons
Diminished icosahedrons

J_n	Solid name	V	E	F	F₃	F₅	Symmetry

58	Augmented dodecahedron	21	35	16	5	11	C_5v
59	Parabiaugmented dodecahedron	22	40	20	10	10	D_5d
60	Metabiaugmented dodecahedron	22	40	20	10	10	C_2v
61	Triaugmented dodecahedron	23	45	24	15	9	C_3v
62	Metabidiminished icosahedron	10	20	12	10	2	C_2v
63	Tridiminished icosahedron	9	15	8	5	3	C_3v
64	Augmented tridiminished icosahedron	10	18	10	7	3	C_3v

Modified Archimedean solids

augmented truncated tetrahedron
augmented truncated cube
augmented truncated dodecahedron
gyrate rhombicosadodecahedron
diminished rhombicosadodecahedron
gyrate diminished rhombicosadodecahedron
diminished rhombicosadodecahedron
gyrate diminished rhombicosadodecahedron
diminished rhombicosadodecahedron

J_n	Solid name	V	E	F	F₃	F₄	F₅	F₆	F₈	F₁₀	Symmetry

65	Augmented truncated tetrahedron	15	27	14	8	3		3			C_3v
66	Augmented truncated cube	28	48	22	12	5			5		C_4v
67	Biaugmented truncated cube	32	60	30	16	10			4		D_4h
68	Augmented truncated dodecahedron	65	105	42	25	5	1			11	C_5v
69	Parabiaugmented truncated dodecahedron	70	120	52	30	10	2			10	D_5d
70	Metabiaugmented truncated dodecahedron	70	120	52	30	10	2			10	C_2v
71	Triaugmented truncated dodecahedron	75	135	62	35	15	3			9	C_3v
72	Gyrate rhombicosidodecahedron	60	120	62	20	30	12				C_5v
73	Parabigyrate rhombicosidodecahedron	60	120	62	20	30	12				D_5d
74	Metabigyrate rhombicosidodecahedron	60	120	62	20	30	12				C_2v
75	Trigyrate rhombicosidodecahedron	60	120	62	20	30	12				C_3v
76	Diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_5v
77	Paragyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_5v
78	Metagyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_s
79	Bigyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_s
80	Parabidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	D_5d
81	Metabidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	C_2v
82	Gyrate bidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	C_s
83	Tridiminished rhombicosidodecahedron	45	75	32	5	15	9			3	C_3v

Miscellaneous

J_n	Solid name	V	E	F	F₃	F₄	F₅	F₆	Symmetry

84	Snub disphenoid (Siamese dodecahedron)	8	18	12	12				D_2d
85	Snub square antiprism	16	40	26	24	2			D_4d
86	Sphenocorona	10	22	14	12	2			C_2v
87	Augmented sphenocorona	11	26	17	16	1			C_s
88	Sphenomegacorona	12	28	18	16	2			C_2v
89	Hebesphenomegacorona	14	33	21	18	3			C_2v
90	Disphenocingulum	16	38	24	20	4			D_2d
91	Bilunabirotunda	14	26	14	8	2	4		D_2h
92	Triangular hebesphenorotunda	18	36	20	13	3	3	1	C_3v

References

Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.