In geometry, aconvex polyhedron whose faces areregular polygons is known as aJohnson solid, or sometimes as a Johnson–Zalgaller solid.^[1] Some authors excludeuniform polyhedra (in whichall vertices are symmetric to each other) from the definition; uniform polyhedra includePlatonic andArchimedean solids as well asprisms andantiprisms.^[2]The Johnson solids are named after American mathematicianNorman Johnson (1930–2017), 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 mathematicianVictor Zalgaller (1920–2020) in 1969.^[3]

This article lists the 92 non-uniform Johnson solids, accompanied by images. They are listed alongside their basic elements (vertices,edges, andfaces), and their most importantgeneral characteristics, includingsymmetry groups (${\displaystyle C_n}$,${\displaystyle D_n}$,${\displaystyle C_{n\mathrm{v}}}$,${\displaystyle D_{n\mathrm{h}}}$,${\displaystyle D_{n\mathrm{d}}}$,${\displaystyle C_s}$),order,surface area, andvolume; an overview of these follows first, before presenting the complete list of non-uniform Johnson solids.

Every polyhedron has its owncharacteristics, includingsymmetry and measurement. An object is said to have symmetry if there is atransformation that maps it to itself. All of those transformations may be composed in agroup, alongside the group's number ofelements, known as theorder. In two-dimensional space, these transformations includerotating around the center of a polygon andreflecting an object around theperpendicular bisector of a polygon. The mensuration of polyhedra includes thesurface area andvolume. Anarea 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.^[4] A volume is a measurement of a region in three-dimensional space.^[5] The volume of a polyhedron may be ascertained in different ways: either through its base and height (like forpyramids andprisms), by slicing it off into pieces and summing their individual volumes, or by finding theroot of apolynomial representing the polyhedron.^[6]

A polygon that is rotated symmetrically by$\frac{{360}^{\circ }}{n}$ is denoted by${\displaystyle C_n}$, acyclic group of order${\displaystyle n}$; combining this with the reflection symmetry results in the symmetry ofdihedral group${\displaystyle D_n}$ of order${\displaystyle 2n}$.^[7] Inthree-dimensional symmetry point groups, the transformations preserving a polyhedron's symmetry include the rotation around the line passing through the base center, known as theaxis of symmetry, and the reflection relative to perpendicular planes passing through the bisector of a base, which is known as thepyramidal symmetry${\displaystyle C_{n\mathrm{v}}}$ of order${\displaystyle 2n}$. The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as theprismatic symmetry${\displaystyle D_{n\mathrm{h}}}$ of order${\displaystyle 4n}$. Theantiprismatic symmetry${\displaystyle D_{n\mathrm{d}}}$ of order${\displaystyle 4n}$ preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.^[8] The symmetry group${\displaystyle C_{n\mathrm{h}}}$ of order${\displaystyle 2n}$ 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${\displaystyle C_{1\mathrm{h}}}$ of order 2, often denoted as${\displaystyle C_s}$.^[9]

Seventeen Johnson solids may be categorized aselementary 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: theequilateral square pyramid,pentagonal pyramid,triangular cupola,square cupola,pentagonal cupola, andpentagonal rotunda. The criterion is also satisfied by eleven other Johnson solids, specifically thetridiminished icosahedron,parabidiminished rhombicosidodecahedron,tridiminished rhombicosidodecahedron,snub disphenoid,snub square antiprism,sphenocorona,sphenomegacorona,hebesphenomegacorona,disphenocingulum,bilunabirotunda, andtriangular hebesphenorotunda.^[10] 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, whileelongation orgyroelongation involve joining them onto the bases of a prism or antiprism, respectively. Some others are constructed bydiminishment, the removal of one of the first six solids from one or more of a polyhedron's faces.^[11]

The table below lists the 92 (non-uniform) Johnson solids. The table includes each solid's enumeration (denoted as${\displaystyle J_n}$).^[12] It also includes each solid'ssymmetry group and number of vertices, edges, and faces, as well as its surface area and volume when constructed with edge length 1. For simplicity, the table uses the quantity${\displaystyle \alpha =5+2\sqrt{5}}$.

Table of the 92 Johnson solids

${\displaystyle J_n}$	Solid name	Image	Vertices	Edges	Faces	Symmetry group andorder[13]	Surface area, exact[14]	Surface area, approx.[14]	Volume, exact[14]	Volume, approx.[14]
1	Square pyramid		5	8	5	${\displaystyle C_{4v}}$ of order 8	${\displaystyle 1+\sqrt{3}}$	2.7321	${\displaystyle \frac{\sqrt{2}}{6}}$	0.2357
2	Pentagonal pyramid		6	10	6	${\displaystyle C_{5v}}$ of order 10	${\displaystyle \frac{1}{2}\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{15\alpha }\right)}}$	3.8855	${\displaystyle \frac{5+\sqrt{5}}{24}}$	0.3015
3	Triangular cupola		9	15	8	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 3+\frac{5\sqrt{3}}{2}}$	7.3301	${\displaystyle \frac{5}{3\sqrt{2}}}$	1.1785
4	Square cupola		12	20	10	${\displaystyle C_{4v}}$ of order 8	${\displaystyle 7+2\sqrt{2}+\sqrt{3}}$	11.5605	${\displaystyle 1+\frac{2\sqrt{2}}{3}}$	1.9428
5	Pentagonal cupola		15	25	12	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 5+\frac{5\sqrt{3}+\sqrt{155\alpha -50}}{4}}$	16.5798	${\displaystyle \frac{5+4\sqrt{5}}{6}}$	2.3241
6	Pentagonal rotunda		20	35	17	${\displaystyle C_{5v}}$ of order 10	${\displaystyle \frac{5\sqrt{3}+\sqrt{145\alpha -75}}{2}}$	22.3472	${\displaystyle \frac{45+17\sqrt{5}}{12}}$	6.9178
7	Elongated triangular pyramid		7	12	7	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 3+\sqrt{3}}$	4.7321	${\displaystyle \frac{\sqrt{2}+3\sqrt{3}}{12}}$	0.5509
8	Elongated square pyramid		9	16	9	${\displaystyle C_{4v}}$ of order 8	${\displaystyle 5+\sqrt{3}}$	6.7321	${\displaystyle 1+\frac{\sqrt{2}}{6}}$	1.2357
9	Elongated pentagonal pyramid		11	20	11	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 5+\frac{5\sqrt{3}+\sqrt{5\alpha }}{4}}$	8.8855	${\displaystyle \frac{5+\sqrt{5}+6\sqrt{5\alpha }}{24}}$	2.022
10	Gyroelongated square pyramid		9	20	13	${\displaystyle C_{4v}}$ of order 8	${\displaystyle 1+3\sqrt{3}}$	6.1962	${\displaystyle \frac{\sqrt{2}+2\sqrt{4+3\sqrt{2}}}{6}}$	1.1927
11	Gyroelongated pentagonal pyramid (diminished icosahedron)		11	25	16	${\displaystyle C_{5v}}$ of order 10	${\displaystyle \frac{15\sqrt{3}+\sqrt{5\alpha }}{4}}$	8.2157	${\displaystyle \frac{25+9\sqrt{5}}{24}}$	1.8802
12	Triangular bipyramid		5	9	6	${\displaystyle D_{3h}}$ of order 12	${\displaystyle \frac{3\sqrt{3}}{2}}$	2.5981	${\displaystyle \frac{\sqrt{2}}{6}}$	0.2357
13	Pentagonal bipyramid		7	15	10	${\displaystyle D_{5h}}$ of order 20	${\displaystyle \frac{5\sqrt{3}}{2}}$	4.3301	${\displaystyle \frac{5+\sqrt{5}}{12}}$	0.6030
14	Elongated triangular bipyramid		8	15	9	${\displaystyle D_{3h}}$ of order 12	${\displaystyle 3+\frac{3\sqrt{3}}{2}}$	5.5981	${\displaystyle \frac{\sqrt{2}}{6}+\frac{\sqrt{3}}{4}}$	0.6687
15	Elongated square bipyramid		10	20	12	${\displaystyle D_{4h}}$ of order 16	${\displaystyle 4+2\sqrt{3}}$	7.4641	${\displaystyle 1+\frac{\sqrt{2}}{3}}$	1.4714
16	Elongated pentagonal bipyramid		12	25	15	${\displaystyle D_{5h}}$ of order 20	${\displaystyle 5+\frac{5}{2}\sqrt{3}}$	9.3301	${\displaystyle \frac{5+\sqrt{5}+3\sqrt{5\alpha }}{12}}$	2.3235
17	Gyroelongated square bipyramid		10	24	16	${\displaystyle D_{4d}}$ of order 16	${\displaystyle 4\sqrt{3}}$	6.9282	${\displaystyle \frac{\sqrt{2}+\sqrt{4+3\sqrt{2}}}{3}}$	1.4284
18	Elongated triangular cupola		15	27	14	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 9+\frac{5}{2}\sqrt{3}}$	13.3301	${\displaystyle \frac{5\sqrt{2}+9\sqrt{3}}{6}}$	3.7766
19	Elongated square cupola		20	36	18	${\displaystyle C_{4v}}$ of order 8	${\displaystyle 15+2\sqrt{2}+\sqrt{3}}$	19.5605	${\displaystyle 3+\frac{8\sqrt{2}}{3}}$	6.7712
20	Elongated pentagonal cupola		25	45	22	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 15+\frac{5\sqrt{3}+10\sqrt{\alpha }+\sqrt{5\alpha }}{4}}$	26.5798	${\displaystyle \frac{5+4\sqrt{5}+15\sqrt{\alpha }}{6}}$	10.0183
21	Elongated pentagonal rotunda		30	55	27	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 10+\frac{5\sqrt{3}+5\sqrt{\alpha }+3\sqrt{5\alpha }}{2}}$	32.3472	${\displaystyle \frac{45+17\sqrt{5}+30\sqrt{\alpha }}{12}}$	14.612
22	Gyroelongated triangular cupola		15	33	20	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 3+\frac{11}{2}\sqrt{3}}$	12.5263	${\displaystyle \frac{1}{3}\sqrt{\frac{61}{2}+18\sqrt{3}+30\sqrt{1+\sqrt{3}}}}$	3.5161
23	Gyroelongated square cupola		20	44	26	${\displaystyle C_{4v}}$ of order 8	${\displaystyle 7+2\sqrt{2}+5\sqrt{3}}$	18.4887	${\displaystyle 1+\frac{2}{3}\sqrt{2}+\frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}}$	6.2108
24	Gyroelongated pentagonal cupola		25	55	32	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 5+\frac{25\sqrt{3}+10\sqrt{\alpha }+\sqrt{5\alpha }}{4}}$	25.2400	${\displaystyle \frac{5}{6}+\frac{2}{3}\sqrt{5}+\frac{5}{6}\sqrt{2\sqrt{145\alpha -75}-2\sqrt{5}-2}}$	9.0733
25	Gyroelongated pentagonal rotunda		30	65	37	${\displaystyle C_{5v}}$ of order 10	${\displaystyle \frac{15\sqrt{3}+\left(\alpha +\sqrt{5}\right)\sqrt{\alpha }}{2}}$	31.0075	${\displaystyle \frac{45+17\sqrt{5}+10\sqrt{2\sqrt{145\alpha -75}-2\sqrt{5}-2}}{12}}$	13.6671
26	Gyrobifastigium		8	14	8	${\displaystyle D_{2d}}$ of order 8	${\displaystyle 4+\sqrt{3}}$	5.7321	${\displaystyle \frac{\sqrt{3}}{2}}$	0.8660
27	Triangular orthobicupola		12	24	14	${\displaystyle D_{3h}}$ of order 12	${\displaystyle 6+2\sqrt{3}}$	9.4641	${\displaystyle \frac{5\sqrt{2}}{3}}$	2.3570
28	Square orthobicupola		16	32	18	${\displaystyle D_{4h}}$ of order 16	${\displaystyle 10+2\sqrt{3}}$	13.4641	${\displaystyle 2+\frac{4\sqrt{2}}{3}}$	3.8856
29	Square gyrobicupola		16	32	18	${\displaystyle D_{4d}}$ of order 16
30	Pentagonal orthobicupola		20	40	22	${\displaystyle D_{5h}}$ of order 20	${\displaystyle 10+\frac{\sqrt{75+5\alpha +10\sqrt{15\alpha }}}{2}}$	17.7711	${\displaystyle \frac{5+4\sqrt{5}}{3}}$	4.6481
31	Pentagonal gyrobicupola		20	40	22	${\displaystyle D_{5d}}$ of order 20
32	Pentagonal orthocupolarotunda		25	50	27	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 5+\frac{\sqrt{675+245\alpha +210\sqrt{15\alpha }}}{4}}$	23.5385	${\displaystyle \frac{55+25\sqrt{5}}{12}}$	9.2418
33	Pentagonal gyrocupolarotunda		25	50	27	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 5+\frac{15}{4}\sqrt{3}+\frac{7}{4}\sqrt{5\alpha }}$	23.5385
34	Pentagonal orthobirotunda		30	60	32	${\displaystyle D_{5h}}$ of order 20	${\displaystyle 5\sqrt{3}+3\sqrt{5\alpha }}$	29.306	${\displaystyle \frac{45+17\sqrt{5}}{6}}$	13.8355
35	Elongated triangular orthobicupola		18	36	20	${\displaystyle D_{3h}}$ of order 12	${\displaystyle 12+2\sqrt{3}}$	15.4641	${\displaystyle \frac{5\sqrt{2}}{3}+\frac{3\sqrt{3}}{2}}$	4.9551
36	Elongated triangular gyrobicupola		18	36	20	${\displaystyle D_{3d}}$ of order 12
37	Elongated square gyrobicupola		24	48	26	${\displaystyle D_{4d}}$ of order 16	${\displaystyle 18+2\sqrt{3}}$	21.4641	${\displaystyle 4+\frac{10\sqrt{2}}{3}}$	8.714
38	Elongated pentagonal orthobicupola		30	60	32	${\displaystyle D_{5h}}$ of order 20	${\displaystyle 20+\frac{\sqrt{75+5\alpha +10\sqrt{15\alpha }}}{2}}$	27.7711	${\displaystyle \frac{10+8\sqrt{5}+15\sqrt{\alpha }}{6}}$	12.3423
39	Elongated pentagonal gyrobicupola		30	60	32	${\displaystyle D_{5d}}$ of order 20
40	Elongated pentagonal orthocupolarotunda		35	70	37	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 15+\frac{\sqrt{675+245\alpha +210\sqrt{15\alpha }}}{4}}$	33.5385	${\displaystyle \frac{55+25\sqrt{5}+30\sqrt{\alpha }}{12}}$	16.936
41	Elongated pentagonal gyrocupolarotunda		35	70	37	${\displaystyle C_{5v}}$ of order 10
42	Elongated pentagonal orthobirotunda		40	80	42	${\displaystyle D_{5h}}$ of order 20	${\displaystyle 10+\sqrt{75+45\alpha +30\sqrt{15\alpha }}}$	39.306	${\displaystyle \frac{45+17\sqrt{5}+15\sqrt{\alpha }}{6}}$	21.5297
43	Elongated pentagonal gyrobirotunda		40	80	42	${\displaystyle D_{5d}}$ of order 20
44	Gyroelongated triangular bicupola		18	42	26	${\displaystyle D_3}$ of order 6	${\displaystyle 6+5\sqrt{3}}$	14.6603	${\displaystyle \frac{5}{3}\sqrt{2}+\sqrt{2+2\sqrt{3}}}$	4.6946
45	Gyroelongated square bicupola		24	56	34	${\displaystyle D_4}$ of order 8	${\displaystyle 10+6\sqrt{3}}$	20.3923	${\displaystyle 2+\frac{4}{3}\sqrt{2}+\frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}}$	8.1536
46	Gyroelongated pentagonal bicupola		30	70	42	${\displaystyle D_5}$ of order 10	${\displaystyle 10+\frac{15}{2}\sqrt{3}+\frac{1}{2}\sqrt{5\alpha }}$	26.4313	${\displaystyle \frac{5+4\sqrt{5}}{3}+\frac{5}{6}\sqrt{2\sqrt{145\alpha -75}-\alpha +3}}$	11.3974
47	Gyroelongated pentagonal cupolarotunda		35	80	47	${\displaystyle C_5}$ of order 5	${\displaystyle 5+\frac{7}{4}\left(5\sqrt{3}+\sqrt{5\alpha }\right)}$	32.1988	${\displaystyle \frac{55+25\sqrt{5}}{12}+\frac{5}{6}\sqrt{2\sqrt{145\alpha -75}-\alpha +3}}$	15.9911
48	Gyroelongated pentagonal birotunda		40	90	52	${\displaystyle D_5}$ of order 10	${\displaystyle 10\sqrt{3}+3\sqrt{5\alpha }}$	37.9662	${\displaystyle \frac{45+17\sqrt{5}+5\sqrt{2\sqrt{145\alpha -75}-\alpha +3}}{6}}$	20.5848
49	Augmented triangular prism		7	13	8	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 2+\frac{3}{2}\sqrt{3}}$	4.5981	${\displaystyle \frac{\sqrt{2}}{6}+\frac{\sqrt{3}}{4}}$	0.6687
50	Biaugmented triangular prism		8	17	11	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 1+\frac{5}{2}\sqrt{3}}$	5.3301	${\displaystyle \sqrt{\frac{59}{144}+\frac{1}{\sqrt{6}}}}$	0.9044
51	Triaugmented triangular prism		9	21	14	${\displaystyle D_{3h}}$ of order 12	${\displaystyle \frac{7\sqrt{3}}{2}}$	6.0622	${\displaystyle \frac{2\sqrt{2}+\sqrt{3}}{4}}$	1.1401
52	Augmented pentagonal prism		11	19	10	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 4+\sqrt{3}+\frac{1}{2}\sqrt{5\alpha }}$	9.173	${\displaystyle \frac{1}{12}\sqrt{8+45\alpha +12\sqrt{10\alpha }}}$	1.9562
53	Biaugmented pentagonal prism		12	23	13	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 3+2\sqrt{3}+\frac{1}{2}\sqrt{5\alpha }}$	9.9051	${\displaystyle \frac{1}{12}\sqrt{32+45\alpha +24\sqrt{10\alpha }}}$	2.1919
54	Augmented hexagonal prism		13	22	11	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 5+4\sqrt{3}}$	11.9282	${\displaystyle \frac{\sqrt{2}}{6}+\frac{3\sqrt{3}}{2}}$	2.8338
55	Parabiaugmented hexagonal prism		14	26	14	${\displaystyle D_{2h}}$ of order 8	${\displaystyle 4+5\sqrt{3}}$	12.6603	${\displaystyle \frac{\sqrt{2}}{3}+\frac{3\sqrt{3}}{2}}$	3.0695
56	Metabiaugmented hexagonal prism		14	26	14	${\displaystyle C_{2v}}$ of order 4
57	Triaugmented hexagonal prism		15	30	17	${\displaystyle D_{3h}}$ of order 12	${\displaystyle 3+6\sqrt{3}}$	13.3923	${\displaystyle \frac{1}{\sqrt{2}}+\frac{3\sqrt{3}}{2}}$	3.3052
58	Augmented dodecahedron		21	35	16	${\displaystyle C_{5v}}$ of order 10	${\displaystyle \frac{5\sqrt{3}+11\sqrt{5\alpha }}{4}}$	21.0903	${\displaystyle \frac{95+43\sqrt{5}}{24}}$	7.9646
59	Parabiaugmented dodecahedron		22	40	20	${\displaystyle D_{5d}}$ of order 20	${\displaystyle \frac{5}{2}\left(\sqrt{3}+\sqrt{5\alpha }\right)}$	21.5349	${\displaystyle \frac{25+11\sqrt{5}}{6}}$	8.2661
60	Metabiaugmented dodecahedron		22	40	20	${\displaystyle C_{2v}}$ of order 4
61	Triaugmented dodecahedron		23	45	24	${\displaystyle C_{3v}}$ of order 6	${\displaystyle \frac{3}{4}\left(5\sqrt{3}+3\sqrt{5\alpha }\right)}$	21.9795	${\displaystyle \frac{5}{8}\left(7+3\sqrt{5}\right)}$	8.5676
62	Metabidiminished icosahedron		10	20	12	${\displaystyle C_{2v}}$ of order 4	${\displaystyle \frac{5\sqrt{3}+\sqrt{5\alpha }}{2}}$	7.7711	${\displaystyle \frac{\alpha }{6}}$	1.5787
63	Tridiminished icosahedron		9	15	8	${\displaystyle C_{3v}}$ of order 6	${\displaystyle \frac{5\sqrt{3}+3\sqrt{5\alpha }}{4}}$	7.3265	${\displaystyle \frac{5}{8}+\frac{7\sqrt{5}}{24}}$	1.2772
64	Augmented tridiminished icosahedron		10	18	10	${\displaystyle C_{3v}}$ of order 6	${\displaystyle \frac{7\sqrt{3}+3\sqrt{5\alpha }}{4}}$	8.1925	${\displaystyle \frac{15+2\sqrt{2}+7\sqrt{5}}{24}}$	1.3950
65	Augmented truncated tetrahedron		15	27	14	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 3+\frac{13}{2}\sqrt{3}}$	14.2583	${\displaystyle \frac{11}{2\sqrt{2}}}$	3.8891
66	Augmented truncated cube		28	48	22	${\displaystyle C_{4v}}$ of order 8	${\displaystyle 15+10\sqrt{2}+3\sqrt{3}}$	34.3383	${\displaystyle 8+\frac{16\sqrt{2}}{3}}$	15.5425
67	Biaugmented truncated cube		32	60	30	${\displaystyle D_{4h}}$ of order 16	${\displaystyle 18+8\sqrt{2}+4\sqrt{3}}$	36.2419	${\displaystyle 9+6\sqrt{2}}$	17.4853
68	Augmented truncated dodecahedron		65	105	42	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 5+\frac{25\sqrt{3}+110\sqrt{\alpha }+\sqrt{5\alpha }}{4}}$	102.1821	${\displaystyle \frac{505}{12}+\frac{81\sqrt{5}}{4}}$	87.3637
69	Parabiaugmented truncated dodecahedron		70	120	52	${\displaystyle D_{5d}}$ of order 20	${\displaystyle 10+25\sqrt{\alpha }+\frac{15\sqrt{3}+\sqrt{5\alpha }}{2}}$	103.3734	${\displaystyle \frac{515+251\sqrt{5}}{12}}$	89.6878
70	Metabiaugmented truncated dodecahedron		70	120	52	${\displaystyle C_{2v}}$ of order 4
71	Triaugmented truncated dodecahedron		75	135	62	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 15+\frac{35\sqrt{3}+90\sqrt{\alpha }+3\sqrt{5\alpha }}{4}}$	104.5648	${\displaystyle \frac{7}{12}\left(75+37\sqrt{5}\right)}$	92.0118
72	Gyrate rhombicosidodecahedron		60	120	62	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 30+5\sqrt{3}+3\sqrt{5\alpha }}$	59.306	${\displaystyle 20+\frac{29\sqrt{5}}{3}}$	41.6153
73	Parabigyrate rhombicosidodecahedron		60	120	62	${\displaystyle D_{5d}}$ of order 20
74	Metabigyrate rhombicosidodecahedron		60	120	62	${\displaystyle C_{2v}}$ of order 4
75	Trigyrate rhombicosidodecahedron		60	120	62	${\displaystyle C_{3v}}$ of order 6
76	Diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_{5v}}$ of order 10	${\displaystyle 25+\frac{15\sqrt{3}+10\sqrt{\alpha }+11\sqrt{5\alpha }}{4}}$	58.1147	${\displaystyle \frac{115}{6}+9\sqrt{5}}$	39.2913
77	Paragyrate diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_{5v}}$ of order 10
78	Metagyrate diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_s}$ of order 2
79	Bigyrate diminished rhombicosidodecahedron		55	105	52	${\displaystyle C_s}$ of order 2
80	Parabidiminished rhombicosidodecahedron		50	90	42	${\displaystyle D_{5d}}$ of order 20	${\displaystyle 20+\frac{5}{2}\left(\sqrt{3}+2\sqrt{\alpha }+\sqrt{5\alpha }\right)}$	56.9233	${\displaystyle \frac{55+25\sqrt{5}}{3}}$	36.9672
81	Metabidiminished rhombicosidodecahedron		50	90	42	${\displaystyle C_{2v}}$ of order 4
82	Gyrate bidiminished rhombicosidodecahedron		50	90	42	${\displaystyle C_s}$ of order 2
83	Tridiminished rhombicosidodecahedron		45	75	32	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 15+\frac{5\sqrt{3}+30\sqrt{\alpha }+9\sqrt{5\alpha }}{4}}$	55.732	${\displaystyle \frac{35}{2}+\frac{23\sqrt{5}}{3}}$	34.6432
84	Snub disphenoid		8	18	12	${\displaystyle D_{2d}}$ of order 8	${\displaystyle 3\sqrt{3}}$	5.1962		0.8595
85	Snub square antiprism		16	40	26	${\displaystyle D_{4d}}$ of order 16	${\displaystyle 2+6\sqrt{3}}$	12.3923		3.6012
86	Sphenocorona		10	22	14	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 2+3\sqrt{3}}$	7.1962	${\displaystyle \frac{1}{2}\sqrt{1+3\sqrt{\frac{3}{2}}+\sqrt{13+3\sqrt{6}}}}$	1.5154
87	Augmented sphenocorona		11	26	17	${\displaystyle C_s}$ of order 2	${\displaystyle 1+4\sqrt{3}}$	7.9282	${\displaystyle \frac{1}{2}\sqrt{1+3\sqrt{\frac{3}{2}}+\sqrt{13+3\sqrt{6}}}+\frac{1}{3\sqrt{2}}}$	1.7511
88	Sphenomegacorona		12	28	18	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 2+4\sqrt{3}}$	8.9282		1.9481
89	Hebesphenomegacorona		14	33	21	${\displaystyle C_{2v}}$ of order 4	${\displaystyle 3+\frac{9}{2}\sqrt{3}}$	10.7942		2.9129
90	Disphenocingulum		16	38	24	${\displaystyle D_{2d}}$ of order 8	${\displaystyle 4+5\sqrt{3}}$	12.6603		3.7776
91	Bilunabirotunda		14	26	14	${\displaystyle D_{2h}}$ of order 8	${\displaystyle 2+2\sqrt{3}+\sqrt{5\alpha }}$	12.346	${\displaystyle \frac{17+9\sqrt{5}}{12}}$	3.0937
92	Triangular hebesphenorotunda		18	36	20	${\displaystyle C_{3v}}$ of order 6	${\displaystyle 3+\frac{19\sqrt{3}+3\sqrt{5\alpha }}{4}}$	16.3887	${\displaystyle \frac{5}{2}+\frac{7\sqrt{5}}{6}}$	5.1087

• Hart, George W."Johnson Solids".
• Bulatov, Vladimir."Johnson solids". – VRML models of Johnson solids
• Gagnon, Sylvain (1982)."Les polyèdres convexes aux faces régulières" [Convex polyhedra with regular faces](PDF).Topologie Structurale [Structural Topology] (in French) (6):83–95.

• Johnson solids
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