List of Johnson solids

In geometry, a convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid. Some authors exclude uniform polyhedra from the definition; uniform polyhedra include Platonic and Archimedean solids as well as prisms and antiprisms.
The Johnson solids are named after American mathematician Norman Johnson, who published a list of 92 non-uniform Johnson polyhedra in 1966. His conjecture that the list was complete and no other examples existed was proven by Russian-Israeli mathematician Victor Zalgaller in 1969.
This article lists the 92 non-uniform Johnson solids, accompanied by images. They are listed alongside their basic elements, and their most important general characteristics, including symmetry groups, order, surface area, and volume; an overview of these follows first, before presenting the complete list of non-uniform Johnson solids.

Characteristics

Every polyhedron has its own characteristics, including symmetry and measurement. An object is said to have symmetry if there is a transformation that maps it to itself. All of those transformations may be composed in a group, alongside the group's number of elements, known as the order. In two-dimensional space, these transformations include rotating around the center of a polygon and reflecting an object around the perpendicular bisector of a polygon. The mensuration of polyhedra includes the surface area and volume. An area is a two-dimensional measurement calculated by the product of length and width; for a polyhedron, the surface area is the sum of the areas of all of its faces. A volume is a measurement of a region in three-dimensional space. The volume of a polyhedron may be ascertained in different ways: either through its base and height, by slicing it off into pieces and summing their individual volumes, or by finding the root of a polynomial representing the polyhedron.
A polygon that is rotated symmetrically by is denoted by, a cyclic group of order ; combining this with the reflection symmetry results in the symmetry of dihedral group of order. In three-dimensional symmetry point groups, the transformations preserving a polyhedron's symmetry include the rotation around the line passing through the base center, known as the axis of symmetry, and the reflection relative to perpendicular planes passing through the bisector of a base, which is known as the pyramidal symmetry of order. The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry of order. The antiprismatic symmetry of order preserves the symmetry by rotating its half bottom and reflection across the horizontal plane. The symmetry group of order preserves the symmetry by rotation around the axis of symmetry and reflection on the horizontal plane; the specific case preserving the symmetry by one full rotation is of order 2, often denoted as.

The solids

Seventeen Johnson solids may be categorized as elementary polyhedra, meaning they cannot be separated by a plane to create two small convex polyhedra with regular faces. The first six Johnson solids satisfy this criterion: the equilateral square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda. The criterion is also satisfied by eleven other Johnson solids, specifically the tridiminished icosahedron, parabidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron, snub disphenoid, snub square antiprism, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum, bilunabirotunda, and triangular hebesphenorotunda. The rest of the Johnson solids are not elementary, and they are constructed using the first six Johnson solids together with Platonic and Archimedean solids in various processes. Augmentation involves attaching the Johnson solids onto one or more faces of polyhedra, while elongation or gyroelongation involve joining them onto the bases of a prism or antiprism, respectively. Some others are constructed by diminishment, the removal of one of the first six solids from one or more of a polyhedron's faces.
The table below lists the 92 Johnson solids. The table includes each solid's enumeration. It also includes each solid's symmetry group and number of vertices, edges, and faces, as well as its surface area and volume when constructed with edge length 1.

	Solid name	Image	Vertices	Edges	Faces	Symmetry group and its order	Surface area, exact, with edge length 1	Surface area, approximate, with edge length 1	Volume, exact, with edge length 1	Volume, approximate, with edge length 1	-
1	Square pyramid		5	8	5	of order 8		2.7321		0.2357	-
2	Pentagonal pyramid		6	10	6	of order 10		3.8855		0.3015	-
3	Triangular cupola		9	15	8	of order 6		7.3301		1.1785	-
4	Square cupola		12	20	10	of order 8		11.5605		1.9428	-
5	Pentagonal cupola		15	25	12	of order 10		16.5798		2.3241	-
6	Pentagonal rotunda		20	35	17	of order 10		22.3472		6.9178	-
7	Elongated triangular pyramid		7	12	7	of order 6		4.7321		0.5509	-
8	Elongated square pyramid		9	16	9	of order 8		6.7321		1.2357	-
9	Elongated pentagonal pyramid		11	20	11	of order 10		8.8855		2.022	-
10	Gyroelongated square pyramid		9	20	13	of order 8		6.1962		1.1927	-
11	Gyroelongated pentagonal pyramid		11	25	16	of order 10		8.2157		1.8802	-
12	Triangular bipyramid		5	9	6	of order 12		2.5981		0.2357	-
13	Pentagonal bipyramid		7	15	10	of order 20		4.3301		0.6030	-
14	Elongated triangular bipyramid		8	15	9	of order 12		5.5981		0.6687	-
15	Elongated square bipyramid		10	20	12	of order 16		7.4641		1.4714	-
16	Elongated pentagonal bipyramid		12	25	15	of order 20		9.3301		2.3235	-
17	Gyroelongated square bipyramid		10	24	16	of order 16		6.9282		1.4284	-
18	Elongated triangular cupola		15	27	14	of order 6		13.3301		3.7766	-
19	Elongated square cupola		20	36	18	of order 8		19.5605		6.7712	-
20	Elongated pentagonal cupola		25	45	22	of order 10		26.5798		10.0183	-
21	Elongated pentagonal rotunda		30	55	27	of order 10		32.3472		14.612	-
22	Gyroelongated triangular cupola		15	33	20	of order 6		12.5263		3.5161	-
23	Gyroelongated square cupola		20	44	26	of order 8		18.4887		6.2108	-
24	Gyroelongated pentagonal cupola		25	55	32	of order 10		25.2400		9.0733	-
25	Gyroelongated pentagonal rotunda		30	65	37	of order 10		31.0075		13.6671	-
26	Gyrobifastigium		8	14	8	of order 8		5.7321		0.8660	-
27	Triangular orthobicupola		12	24	14	of order 12		9.4641		2.3570	-
28	Square orthobicupola		16	32	18	of order 16		13.4641		3.8856	-
29	Square gyrobicupola		16	32	18	of order 16		13.4641		3.8856	-
30	Pentagonal orthobicupola		20	40	22	of order 20		17.7711		4.6481	-
31	Pentagonal gyrobicupola		20	40	22	of order 20		17.7711		4.6481	-
32	Pentagonal orthocupolarotunda		25	50	27	of order 10		23.5385		9.2418	-
33	Pentagonal gyrocupolarotunda		25	50	27	of order 10		23.5385		9.2418	-
34	Pentagonal orthobirotunda		30	60	32	of order 20		29.306		13.8355	-
35	Elongated triangular orthobicupola		18	36	20	of order 12		15.4641		4.9551	-
36	Elongated triangular gyrobicupola		18	36	20	of order 12		15.4641		4.9551	-
37	Elongated square gyrobicupola		24	48	26	of order 16		21.4641		8.714	-
38	Elongated pentagonal orthobicupola		30	60	32	of order 20		27.7711		12.3423	-
39	Elongated pentagonal gyrobicupola		30	60	32	of order 20		27.7711		12.3423	-
40	Elongated pentagonal orthocupolarotunda		35	70	37	of order 10		33.5385		16.936	-
41	Elongated pentagonal gyrocupolarotunda		35	70	37	of order 10		33.5385		16.936
42	Elongated pentagonal orthobirotunda		40	80	42	of order 20		39.306		21.5297	-
43	Elongated pentagonal gyrobirotunda		40	80	42	of order 20		39.306		21.5297	-
44	Gyroelongated triangular bicupola		18	42	26	of order 6		14.6603		4.6946	-
45	Gyroelongated square bicupola		24	56	34	of order 8		20.3923		8.1536	-
46	Gyroelongated pentagonal bicupola		30	70	42	of order 10		26.4313		11.3974	-
47	Gyroelongated pentagonal cupolarotunda		35	80	47	of order 5		32.1988		15.9911	-
48	Gyroelongated pentagonal birotunda		40	90	52	of order 10		37.9662		20.5848	-
49	Augmented triangular prism		7	13	8	of order 4		4.5981		0.6687	-
50	Biaugmented triangular prism		8	17	11	of order 4		5.3301		0.9044	-
51	Triaugmented triangular prism		9	21	14	of order 12		6.0622		1.1401	-
52	Augmented pentagonal prism		11	19	10	of order 4		9.173		1.9562	-
53	Biaugmented pentagonal prism		12	23	13	of order 4		9.9051		2.1919	-
54	Augmented hexagonal prism		13	22	11	of order 4		11.9282		2.8338	-
55	Parabiaugmented hexagonal prism		14	26	14	of order 8		12.6603		3.0695	-
56	Metabiaugmented hexagonal prism		14	26	14	of order 4		12.6603		3.0695	-
57	Triaugmented hexagonal prism		15	30	17	of order 12		13.3923		3.3052	-
58	Augmented dodecahedron		21	35	16	of order 10		21.0903		7.9646	-
59	Parabiaugmented dodecahedron		22	40	20	of order 20		21.5349		8.2661	-
60	Metabiaugmented dodecahedron		22	40	20	of order 4		21.5349		8.2661	-
61	Triaugmented dodecahedron		23	45	24	of order 6		21.9795		8.5676	-
62	Metabidiminished icosahedron		10	20	12	of order 4		7.7711		1.5787	-
63	Tridiminished icosahedron		9	15	8	of order 6		7.3265		1.2772	-
64	Augmented tridiminished icosahedron		10	18	10	of order 6		8.1925		1.3950	-
65	Augmented truncated tetrahedron		15	27	14	of order 6		14.2583		3.8891	-
66	Augmented truncated cube		28	48	22	of order 8		34.3383		15.5425	-
67	Biaugmented truncated cube		32	60	30	of order 16		36.2419		17.4853	-
68	Augmented truncated dodecahedron		65	105	42	of order 10		102.1821		87.3637	-
69	Parabiaugmented truncated dodecahedron		70	120	52	of order 20		103.3734		89.6878	-
70	Metabiaugmented truncated dodecahedron		70	120	52	of order 4		103.3734		89.6878	-
71	Triaugmented truncated dodecahedron		75	135	62	of order 6		104.5648		92.0118	-
72	Gyrate rhombicosidodecahedron		60	120	62	of order 10		59.306		41.6153	-
73	Parabigyrate rhombicosidodecahedron		60	120	62	of order 20		59.306		41.6153	-
74	Metabigyrate rhombicosidodecahedron		60	120	62	of order 4		59.306		41.6153	-
75	Trigyrate rhombicosidodecahedron		60	120	62	of order 6		59.306		41.6153	-
76	Diminished rhombicosidodecahedron		55	105	52	of order 10		58.1147		39.2913	-
77	Paragyrate diminished rhombicosidodecahedron		55	105	52	of order 10		58.1147		39.2913	-
78	Metagyrate diminished rhombicosidodecahedron		55	105	52	of order 2		58.1147		39.2913	-
79	Bigyrate diminished rhombicosidodecahedron		55	105	52	of order 2		58.1147		39.2913	-
80	Parabidiminished rhombicosidodecahedron		50	90	42	of order 20		56.9233		36.9672	-
81	Metabidiminished rhombicosidodecahedron		50	90	42	of order 4		56.9233		36.9672	-
82	Gyrate bidiminished rhombicosidodecahedron		50	90	42	of order 2		56.9233		36.9672	-
83	Tridiminished rhombicosidodecahedron		45	75	32	of order 6		55.732		34.6432	-
84	Snub disphenoid		8	18	12	of order 8		5.1962		0.8595	-
85	Snub square antiprism		16	40	26	of order 16		12.3923		3.6012	-
86	Sphenocorona		10	22	14	of order 4		7.1962		1.5154	-
87	Augmented sphenocorona		11	26	17	of order 2		7.9282		1.7511	-
88	Sphenomegacorona		12	28	18	of order 4		8.9282		1.9481	-
89	Hebesphenomegacorona		14	33	21	of order 4		10.7942		2.9129	-
90	Disphenocingulum		16	38	24	of order 8		12.6603		3.7776	-
91	Bilunabirotunda		14	26	14	of order 8		12.346		3.0937	-
92	Triangular hebesphenorotunda		18	36	20	of order 6		16.3887		5.1087	-