List of Johnson solids

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. 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.

The complete list is here with sorting by column. Some polyhedra can be constructed that are only approximately regular planar polygon faces, and informally called near-miss Johnson solid although there can be no definitive count of them.

J_n	Solid name	V	E	F	F₃	F₄	F₅	F₆	F₈	F₁₀	Symmetry group	Order
1	Square pyramid	5	8	5	4	1					C_4v, [4], (*44)	8
2	Pentagonal pyramid	6	10	6	5		1				C_5v, [5], (*55)	10
3	Triangular cupola	9	15	8	4	3		1			C_3v, [3], (*33)	6
4	Square cupola	12	20	10	4	5			1		C_4v, [4], (*44)	8
5	Pentagonal cupola	15	25	12	5	5	1			1	C_5v, [5], (*55)	10
6	Pentagonal rotunda	20	35	17	10		6			1	C_5v, [5], (*55)	10
7	Elongated triangular pyramid	7	12	7	4	3					C_3v, [3], (*33)	6
8	Elongated square pyramid	9	16	9	4	5					C_4v, [4], (*44)	8
9	Elongated pentagonal pyramid	11	20	11	5	5	1				C_5v, [5], (*55)	10
10	Gyroelongated square pyramid	9	20	13	12	1					C_4v, [4], (*44)	8
11	Gyroelongated pentagonal pyramid	11	25	16	15		1				C_5v, [5], (*55)	10
12	Triangular bipyramid	5	9	6	6						D_3h, [3,2], (*223)	12
13	Pentagonal bipyramid	7	15	10	10						D_5h, [5,2], (*225)	20
14	Elongated triangular bipyramid	8	15	9	6	3					D_3h, [3,2], (*223)	12
15	Elongated square bipyramid	10	20	12	8	4					D_4h, [4,2], (*224)	16
16	Elongated pentagonal bipyramid	12	25	15	10	5					D_5h, [5,2], (*225)	20
17	Gyroelongated square bipyramid	10	24	16	16						D_4d, [2⁺,8], (2*4)	16
18	Elongated triangular cupola	15	27	14	4	9		1			C_3v, [3], (*33)	6
19	Elongated square cupola	20	36	18	4	13			1		C_4v, [4], (*44)	8
20	Elongated pentagonal cupola	25	45	22	5	15	1			1	C_5v, [5], (*55)	10
21	Elongated pentagonal rotunda	30	55	27	10	10	6			1	C_5v, [5], (*55)	10
22	Gyroelongated triangular cupola	15	33	20	16	3		1			C_3v, [3], (*33)	6
23	Gyroelongated square cupola	20	44	26	20	5			1		C_4v, [4], (*44)	8
24	Gyroelongated pentagonal cupola	25	55	32	25	5	1			1	C_5v, [5], (*55)	10
25	Gyroelongated pentagonal rotunda	30	65	37	30		6			1	C_5v, [5], (*55)	10
26	Gyrobifastigium	8	14	8	4	4					D_2d, [2⁺,4], (2*2)	8
27	Triangular orthobicupola	12	24	14	8	6					D_3h, [3,2], (*223)	12
28	Square orthobicupola	16	32	18	8	10					D_4h, [4,2], (*224)	16
29	Square gyrobicupola	16	32	18	8	10					D_4d, [2⁺,8], (2*4)	16
30	Pentagonal orthobicupola	20	40	22	10	10	2				D_5h, [5,2], (*225)	20
31	Pentagonal gyrobicupola	20	40	22	10	10	2				D_5d, [2⁺,10], (2*5)	20
32	Pentagonal orthocupolarotunda	25	50	27	15	5	7				C_5v, [5], (*55)	10
33	Pentagonal gyrocupolarotunda	25	50	27	15	5	7				C_5v, [5], (*55)	10
34	Pentagonal orthobirotunda	30	60	32	20		12				D_5h, [5,2], (*225)	20
35	Elongated triangular orthobicupola	18	36	20	8	12					D_3h, [3,2], (*223)	12
36	Elongated triangular gyrobicupola	18	36	20	8	12					D_3d, [2⁺,6], (2*3)	12
37	Elongated square gyrobicupola	24	48	26	8	18					D_4d, [2⁺,8], (2*4)	16
38	Elongated pentagonal orthobicupola	30	60	32	10	20	2				D_5h, [5,2], (*225)	20
39	Elongated pentagonal gyrobicupola	30	60	32	10	20	2				D_5d, [2⁺,10], (2*5)	20
40	Elongated pentagonal orthocupolarotunda	35	70	37	15	15	7				C_5v, [5], (*55)	10
41	Elongated pentagonal gyrocupolarotunda	35	70	37	15	15	7				C_5v, [5], (*55)	10
42	Elongated pentagonal orthobirotunda	40	80	42	20	10	12				D_5h, [5,2], (*225)	20
43	Elongated pentagonal gyrobirotunda	40	80	42	20	10	12				D_5d, [2⁺,10], (2*5)	20
44	Gyroelongated triangular bicupola	18	42	26	20	6					D₃, [3,2]⁺,(223)	6
45	Gyroelongated square bicupola	24	56	34	24	10					D₄, [4,2]⁺, (224)	8
46	Gyroelongated pentagonal bicupola	30	70	42	30	10	2				D₅, [5,2]⁺, (225)	10
47	Gyroelongated pentagonal cupolarotunda	35	80	47	35	5	7				C₅, [5]⁺, (55)	5
48	Gyroelongated pentagonal birotunda	40	90	52	40		12				D₅, [5,2]⁺, (225)	10
49	Augmented triangular prism	7	13	8	6	2					C_2v, [2], (*22)	4
50	Biaugmented triangular prism	8	17	11	10	1					C_2v, [2], (*22)	4
51	Triaugmented triangular prism	9	21	14	14						D_3h, [3,2], (*223)	12
52	Augmented pentagonal prism	11	19	10	4	4	2				C_2v, [2], (*22)	4
53	Biaugmented pentagonal prism	12	23	13	8	3	2				C_2v, [2], (*22)	4
54	Augmented hexagonal prism	13	22	11	4	5		2			C_2v, [2], (*22)	4
55	Parabiaugmented hexagonal prism	14	26	14	8	4		2			D_2h, [2,2], (*222)	8
56	Metabiaugmented hexagonal prism	14	26	14	8	4		2			C_2v, [2], (*22)	4
57	Triaugmented hexagonal prism	15	30	17	12	3		2			D_3h, [3,2], (*223)	12
58	Augmented dodecahedron	21	35	16	5		11				C_5v, [5], (*55)	10
59	Parabiaugmented dodecahedron	22	40	20	10		10				D_5d, [2⁺,10], (2*5)	20
60	Metabiaugmented dodecahedron	22	40	20	10		10				C_2v, [2], (*22)	4
61	Triaugmented dodecahedron	23	45	24	15		9				C_3v, [3], (*33)	6
62	Metabidiminished icosahedron	10	20	12	10		2				C_2v, [2], (*22)	4
63	Tridiminished icosahedron	9	15	8	5		3				C_3v, [3], (*33)	6
64	Augmented tridiminished icosahedron	10	18	10	7		3				C_3v, [3], (*33)	6
65	Augmented truncated tetrahedron	15	27	14	8	3		3			C_3v, [3], (*33)	6
66	Augmented truncated cube	28	48	22	12	5			5		C_4v, [4], (*44)	8
67	Biaugmented truncated cube	32	60	30	16	10			4		D_4h, [4,2], (*224)	16
68	Augmented truncated dodecahedron	65	105	42	25	5	1			11	C_5v, [5], (*55)	10
69	Parabiaugmented truncated dodecahedron	70	120	52	30	10	2			10	D_5d, [2⁺,10], (2*5)	20
70	Metabiaugmented truncated dodecahedron	70	120	52	30	10	2			10	C_2v, [2], (*22)	4
71	Triaugmented truncated dodecahedron	75	135	62	35	15	3			9	C_3v, [3], (*33)	6
72	Gyrate rhombicosidodecahedron	60	120	62	20	30	12				C_5v, [5], (*55)	10
73	Parabigyrate rhombicosidodecahedron	60	120	62	20	30	12				D_5d, [2⁺,10], (2*5)	20
74	Metabigyrate rhombicosidodecahedron	60	120	62	20	30	12				C_2v, [2], (*22)	4
75	Trigyrate rhombicosidodecahedron	60	120	62	20	30	12				C_3v, [3], (*33)	6
76	Diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_5v, [5], (*55)	10
77	Paragyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_5v, [5], (*55)	10
78	Metagyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_s, [ ], (*11)	2
79	Bigyrate diminished rhombicosidodecahedron	55	105	52	15	25	11			1	C_s, [ ], (*11)	2
80	Parabidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	D_5d, [2⁺,10], (2*5)	20
81	Metabidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	C_2v, [2], (*22)	4
82	Gyrate bidiminished rhombicosidodecahedron	50	90	42	10	20	10			2	C_s, [ ], (*11)	2
83	Tridiminished rhombicosidodecahedron	45	75	32	5	15	9			3	C_3v, [3], (*33)	6
84	Snub disphenoid	8	18	12	12						D_2d, [2⁺,4], (2*2)	8
85	Snub square antiprism	16	40	26	24	2					D_4d, [2⁺,8], (2*4)	16
86	Sphenocorona	10	22	14	12	2					C_2v, [2], (*22)	4
87	Augmented sphenocorona	11	26	17	16	1					C_s, [ ], (*11)	2
88	Sphenomegacorona	12	28	18	16	2					C_2v, [2], (*22)	4
89	Hebesphenomegacorona	14	33	21	18	3					C_2v, [2], (*22)	4
90	Disphenocingulum	16	38	24	20	4					D_2d, [2⁺,4], (2*2)	8
91	Bilunabirotunda	14	26	14	8	2	4				D_2h, [2,2], (*222)	8
92	Triangular hebesphenorotunda	18	36	20	13	3	3	1			C_3v, [3], (*33)	6

Legend:

J_n – Johnson Solid Number
Net – Flattened (unfolded) image
V – Number of Vertices
E – Number of Edges
F – Number of Faces (total)
F₃-F₁₀ – Number of faces by side counts

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.

External links

Sylvain Gagnon, "Convex polyhedra with regular faces", Structural Topology, No. 6, 1982, 83-95.
Johnson Solids by George W. Hart.
Images of all 92 solids, categorized, on one page
Weisstein, Eric W. "Johnson Solid". MathWorld.

VRML models of Johnson Solids by Jim McNeill
VRML models of Johnson Solids by Vladimir Bulatov

Johnson solids navigator

Prismatoids and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda

Modified pyramids and dipyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular dipyramid square dipyramid pentagonal dipyramid elongated triangular dipyramid elongated square dipyramid elongated pentagonal dipyramid gyroelongated square dipyramid

Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda

Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism

Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron

Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron

Miscellaneous	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)

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