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 (in which all vertices are symmetric to each other) 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 (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 mathematician Victor Zalgaller (1920–2020) in 1969.

This article lists the 92 non-uniform Johnson solids, accompanied by images. They are listed alongside their basic elements (vertices, edges, and faces), and their most important general characteristics, including symmetry groups ($C_{n}$, $D_{n}$, $C_{n\mathrm {v} }$, $D_{n\mathrm {h} }$, $D_{n\mathrm {d} }$, $C_{s}$), 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 (like for pyramids and prisms), 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 ${\textstyle {\frac {360^{\circ }}{n}}}$ is denoted by $C_{n}$, a cyclic group of order $n$; combining this with the reflection symmetry results in the symmetry of dihedral group $D_{n}$ of order $2n$. 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 $C_{n\mathrm {v} }$ of order $2n$. The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry $D_{n\mathrm {h} }$ of order $4n$. The antiprismatic symmetry $D_{n\mathrm {d} }$ of order $4n$ preserves the symmetry by rotating its half bottom and reflection across the horizontal plane. The symmetry group $C_{n\mathrm {h} }$ of order $2n$ 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 $C_{1\mathrm {h} }$ of order 2, often denoted as $C_{s}$.

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 (non-uniform) Johnson solids, with edge length 1. The table includes each solid's enumeration (denoted as $J_{n}$). It also includes the number of vertices, edges, and faces of each solid, as well as its symmetry group, surface area $A$, and volume $V$.

Table of the 92 Johnson solids

$J_{n}$	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	$C_{4v}$ of order 8	$1+{\sqrt {3}}$	2.7321	${\frac {\sqrt {2}}{6}}$	0.2357
2	Pentagonal pyramid		6	10	6	$C_{5v}$ of order 10	${\frac {1}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}$	3.8855	${\frac {5+{\sqrt {5}}}{24}}$	0.3015
3	Triangular cupola		9	15	8	$C_{3v}$ of order 6	$3+{\frac {5{\sqrt {3}}}{2}}$	7.3301	${\frac {5}{3{\sqrt {2}}}}$	1.1785
4	Square cupola		12	20	10	$C_{4v}$ of order 8	$7+2{\sqrt {2}}+{\sqrt {3}}$	11.5605	$1+{\frac {2{\sqrt {2}}}{3}}$	1.9428
5	Pentagonal cupola		15	25	12	$C_{5v}$ of order 10	${\frac {20+5{\sqrt {3}}+{\sqrt {725+310{\sqrt {5}}}}}{4}}$	16.5798	${\frac {5+4{\sqrt {5}}}{6}}$	2.3241
6	Pentagonal rotunda		20	35	17	$C_{5v}$ of order 10	${\frac {5{\sqrt {3}}+{\sqrt {650+290{\sqrt {5}}}}}{2}}$	22.3472	${\frac {45+17{\sqrt {5}}}{12}}$	6.9178
7	Elongated triangular pyramid		7	12	7	$C_{3v}$ of order 6	$3+{\sqrt {3}}$	4.7321	${\frac {{\sqrt {2}}+3{\sqrt {3}}}{12}}$	0.5509
8	Elongated square pyramid		9	16	9	$C_{4v}$ of order 8	$5+{\sqrt {3}}$	6.7321	$1+{\frac {\sqrt {2}}{6}}$	1.2357
9	Elongated pentagonal pyramid		11	20	11	$C_{5v}$ of order 10	${\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}$	8.8855	${\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}$	2.022
10	Gyroelongated square pyramid		9	20	13	$C_{4v}$ of order 8	$1+3{\sqrt {3}}$	6.1962	${\frac {{\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}}{6}}$	1.1927
11	Gyroelongated pentagonal pyramid		11	25	16	$C_{5v}$ of order 10	${\frac {15{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}$	8.2157	${\frac {25+9{\sqrt {5}}}{24}}$	1.8802
12	Triangular bipyramid		5	9	6	$D_{3h}$ of order 12	${\frac {3{\sqrt {3}}}{2}}$	2.5981	${\frac {\sqrt {2}}{6}}$	0.2357
13	Pentagonal bipyramid		7	15	10	$D_{5h}$ of order 20	${\frac {5{\sqrt {3}}}{2}}$	4.3301	${\frac {5+{\sqrt {5}}}{12}}$	0.6030
14	Elongated triangular bipyramid		8	15	9	$D_{3h}$ of order 12	$3+{\frac {3{\sqrt {3}}}{2}}$	5.5981	${\frac {\sqrt {2}}{6}}+{\frac {\sqrt {3}}{4}}$	0.6687
15	Elongated square bipyramid		10	20	12	$D_{4h}$ of order 16	$4+2{\sqrt {3}}$	7.4641	$1+{\frac {\sqrt {2}}{3}}$	1.4714
16	Elongated pentagonal bipyramid		12	25	15	$D_{5h}$ of order 20	$5+{\frac {5}{2}}{\sqrt {3}}$	9.3301	${\frac {5+{\sqrt {5}}+3{\sqrt {25+10{\sqrt {5}}}}}{12}}$	2.3235
17	Gyroelongated square bipyramid		10	24	16	$D_{4d}$ of order 16	$4{\sqrt {3}}$	6.9282	${\frac {{\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}}{3}}$	1.4284
18	Elongated triangular cupola		15	27	14	$C_{3v}$ of order 6	$9+{\frac {5}{2}}{\sqrt {3}}$	13.3301	${\frac {5{\sqrt {2}}+9{\sqrt {3}}}{6}}$	3.7766
19	Elongated square cupola		20	36	18	$C_{4v}$ of order 8	$15+2{\sqrt {2}}+{\sqrt {3}}$	19.5605	$3+{\frac {8{\sqrt {2}}}{3}}$	6.7712
20	Elongated pentagonal cupola		25	45	22	$C_{5v}$ of order 10	${\frac {60+5{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}$	26.5798	${\frac {5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}}{6}}$	10.0183
21	Elongated pentagonal rotunda		30	55	27	$C_{5v}$ of order 10	${\frac {20+5{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {25+10{\sqrt {5}}}}}{2}}$	32.3472	${\frac {45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}}{12}}$	14.612
22	Gyroelongated triangular cupola		15	33	20	$C_{3v}$ of order 6	$3+{\frac {11}{2}}{\sqrt {3}}$	12.5263	${\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}$	3.5161
23	Gyroelongated square cupola		20	44	26	$C_{4v}$ of order 8	$7+2{\sqrt {2}}+5{\sqrt {3}}$	18.4887	$1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}$	6.2108
24	Gyroelongated pentagonal cupola		25	55	32	$C_{5v}$ of order 10	$5+{\frac {25{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}$	25.2400	${\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}$	9.0733
25	Gyroelongated pentagonal rotunda		30	65	37	$C_{5v}$ of order 10	${\frac {15{\sqrt {3}}+\left(5+3{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}}{2}}$	31.0075	${\frac {45+17{\sqrt {5}}+10{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}}{12}}$	13.6671
26	Gyrobifastigium		8	14	8	$D_{2d}$ of order 8	$4+{\sqrt {3}}$	5.7321	${\frac {\sqrt {3}}{2}}$	0.8660
27	Triangular orthobicupola		12	24	14	$D_{3h}$ of order 12	$6+2{\sqrt {3}}$	9.4641	${\frac {5{\sqrt {2}}}{3}}$	2.3570
28	Square orthobicupola		16	32	18	$D_{4h}$ of order 16	$10+2{\sqrt {3}}$	13.4641	$2+{\frac {4{\sqrt {2}}}{3}}$	3.8856
29	Square gyrobicupola		16	32	18	$D_{4d}$ of order 16
30	Pentagonal orthobicupola		20	40	22	$D_{5h}$ of order 20	$10+5{\sqrt {1+{\frac {{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}}{10}}}}$	17.7711	${\frac {5+4{\sqrt {5}}}{3}}$	4.6481
31	Pentagonal gyrobicupola		20	40	22	$D_{5d}$ of order 20
32	Pentagonal orthocupolarotunda		25	50	27	$C_{5v}$ of order 10	$5+{\frac {1}{4}}{\sqrt {1900+490{\sqrt {5}}+210{\sqrt {75+30{\sqrt {5}}}}}}$	23.5385	${\frac {55+25{\sqrt {5}}}{12}}$	9.2418
33	Pentagonal gyrocupolarotunda		25	50	27	$C_{5v}$ of order 10	$5+{\frac {15}{4}}{\sqrt {3}}+{\frac {7}{4}}{\sqrt {25+10{\sqrt {5}}}}$	23.5385
34	Pentagonal orthobirotunda		30	60	32	$D_{5h}$ of order 20	$5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}$	29.306	${\frac {45+17{\sqrt {5}}}{6}}$	13.8355
35	Elongated triangular orthobicupola		18	36	20	$D_{3h}$ of order 12	$12+2{\sqrt {3}}$	15.4641	${\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}$	4.9551
36	Elongated triangular gyrobicupola		18	36	20	$D_{3d}$ of order 12
37	Elongated square gyrobicupola		24	48	26	$D_{4d}$ of order 16	$18+2{\sqrt {3}}$	21.4641	$4+{\frac {10{\sqrt {2}}}{3}}$	8.714
38	Elongated pentagonal orthobicupola		30	60	32	$D_{5h}$ of order 20	$20+5{\sqrt {1+{\frac {{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}}{10}}}}$	27.7711	${\frac {10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}}{6}}$	12.3423
39	Elongated pentagonal gyrobicupola		30	60	32	$D_{5d}$ of order 20
40	Elongated pentagonal orthocupolarotunda		35	70	37	$C_{5v}$ of order 10	$15+{\frac {1}{4}}{\sqrt {1900+490{\sqrt {5}}+210{\sqrt {75+30{\sqrt {5}}}}}}$	33.5385	${\frac {55+25{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}}{12}}$	16.936
41	Elongated pentagonal gyrocupolarotunda		35	70	37	$C_{5v}$ of order 10
42	Elongated pentagonal orthobirotunda		40	80	42	$D_{5h}$ of order 20	$10+{\sqrt {300+90{\sqrt {5}}+30{\sqrt {75+30{\sqrt {5}}}}}}$	39.306	${\frac {45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}}{6}}$	21.5297
43	Elongated pentagonal gyrobirotunda		40	80	42	$D_{5d}$ of order 20
44	Gyroelongated triangular bicupola		18	42	26	$D_{3}$ of order 6	$6+5{\sqrt {3}}$	14.6603	${\frac {5}{3}}{\sqrt {2}}+{\sqrt {2+2{\sqrt {3}}}}$	4.6946
45	Gyroelongated square bicupola		24	56	34	$D_{4}$ of order 8	$10+6{\sqrt {3}}$	20.3923	$2+{\frac {4}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}$	8.1536
46	Gyroelongated pentagonal bicupola		30	70	42	$D_{5}$ of order 10	$10+{\frac {15}{2}}{\sqrt {3}}+{\frac {1}{2}}{\sqrt {25+10{\sqrt {5}}}}$	26.4313	${\frac {5+4{\sqrt {5}}}{3}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}$	11.3974
47	Gyroelongated pentagonal cupolarotunda		35	80	47	$C_{5}$ of order 5	$5+{\frac {7}{4}}\left(5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)$	32.1988	${\frac {55+25{\sqrt {5}}}{12}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}$	15.9911
48	Gyroelongated pentagonal birotunda		40	90	52	$D_{5}$ of order 10	$10{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}$	37.9662	${\frac {45+17{\sqrt {5}}+5{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}}{6}}$	20.5848
49	Augmented triangular prism		7	13	8	$C_{2v}$ of order 4	$2+{\frac {3}{2}}{\sqrt {3}}$	4.5981	${\frac {\sqrt {2}}{6}}+{\frac {\sqrt {3}}{4}}$	0.6687
50	Biaugmented triangular prism		8	17	11	$C_{2v}$ of order 4	$1+{\frac {5}{2}}{\sqrt {3}}$	5.3301	${\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}$	0.9044
51	Triaugmented triangular prism		9	21	14	$D_{3h}$ of order 12	${\frac {7{\sqrt {3}}}{2}}$	6.0622	${\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}$	1.1401
52	Augmented pentagonal prism		11	19	10	$C_{2v}$ of order 4	$4+{\sqrt {3}}+{\frac {1}{2}}{\sqrt {25+10{\sqrt {5}}}}$	9.173	${\frac {1}{12}}{\sqrt {233+90{\sqrt {5}}+12{\sqrt {50+20{\sqrt {5}}}}}}$	1.9562
53	Biaugmented pentagonal prism		12	23	13	$C_{2v}$ of order 4	$3+2{\sqrt {3}}+{\frac {1}{2}}{\sqrt {25+10{\sqrt {5}}}}$	9.9051	${\frac {1}{12}}{\sqrt {257+90{\sqrt {5}}+24{\sqrt {50+20{\sqrt {5}}}}}}$	2.1919
54	Augmented hexagonal prism		13	22	11	$C_{2v}$ of order 4	$5+4{\sqrt {3}}$	11.9282	${\frac {\sqrt {2}}{6}}+{\frac {3{\sqrt {3}}}{2}}$	2.8338
55	Parabiaugmented hexagonal prism		14	26	14	$D_{2h}$ of order 8	$4+5{\sqrt {3}}$	12.6603	${\frac {\sqrt {2}}{3}}+{\frac {3{\sqrt {3}}}{2}}$	3.0695
56	Metabiaugmented hexagonal prism		14	26	14	$C_{2v}$ of order 4
57	Triaugmented hexagonal prism		15	30	17	$D_{3h}$ of order 12	$3+6{\sqrt {3}}$	13.3923	${\frac {1}{\sqrt {2}}}+{\frac {3{\sqrt {3}}}{2}}$	3.3052
58	Augmented dodecahedron		21	35	16	$C_{5v}$ of order 10	${\frac {5{\sqrt {3}}+11{\sqrt {25+10{\sqrt {5}}}}}{4}}$	21.0903	${\frac {95+43{\sqrt {5}}}{24}}$	7.9646
59	Parabiaugmented dodecahedron		22	40	20	$D_{5d}$ of order 20	${\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)$	21.5349	${\frac {25+11{\sqrt {5}}}{6}}$	8.2661
60	Metabiaugmented dodecahedron		22	40	20	$C_{2v}$ of order 4
61	Triaugmented dodecahedron		23	45	24	$C_{3v}$ of order 6	${\frac {3}{4}}\left(5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)$	21.9795	${\frac {5}{8}}\left(7+3{\sqrt {5}}\right)$	8.5676
62	Metabidiminished icosahedron		10	20	12	$C_{2v}$ of order 4	${\frac {5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{2}}$	7.7711	${\frac {5+2{\sqrt {5}}}{6}}$	1.5787
63	Tridiminished icosahedron		9	15	8	$C_{3v}$ of order 6	${\frac {5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}}{4}}$	7.3265	${\frac {5}{8}}+{\frac {7{\sqrt {5}}}{24}}$	1.2772
64	Augmented tridiminished icosahedron		10	18	10	$C_{3v}$ of order 6	${\frac {7{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}}{4}}$	8.1925	${\frac {15+2{\sqrt {2}}+7{\sqrt {5}}}{24}}$	1.3950
65	Augmented truncated tetrahedron		15	27	14	$C_{3v}$ of order 6	$3+{\frac {13}{2}}{\sqrt {3}}$	14.2583	${\frac {11}{2{\sqrt {2}}}}$	3.8891
66	Augmented truncated cube		28	48	22	$C_{4v}$ of order 8	$15+10{\sqrt {2}}+3{\sqrt {3}}$	34.3383	$8+{\frac {16{\sqrt {2}}}{3}}$	15.5425
67	Biaugmented truncated cube		32	60	30	$D_{4h}$ of order 16	$18+8{\sqrt {2}}+4{\sqrt {3}}$	36.2419	$9+6{\sqrt {2}}$	17.4853
68	Augmented truncated dodecahedron		65	105	42	$C_{5v}$ of order 10	$5+{\frac {25{\sqrt {3}}+110{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}$	102.1821	${\frac {505}{12}}+{\frac {81{\sqrt {5}}}{4}}$	87.3637
69	Parabiaugmented truncated dodecahedron		70	120	52	$D_{5d}$ of order 20	$10+25{\sqrt {5+2{\sqrt {5}}}}+{\frac {15{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{2}}$	103.3734	${\frac {515+251{\sqrt {5}}}{12}}$	89.6878
70	Metabiaugmented truncated dodecahedron		70	120	52	$C_{2v}$ of order 4
71	Triaugmented truncated dodecahedron		75	135	62	$C_{3v}$ of order 6	$15+{\frac {35{\sqrt {3}}+90{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {25+10{\sqrt {5}}}}}{4}}$	104.5648	${\frac {7}{12}}\left(75+37{\sqrt {5}}\right)$	92.0118
72	Gyrate rhombicosidodecahedron		60	120	62	$C_{5v}$ of order 10	$30+5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}$	59.306	$20+{\frac {29{\sqrt {5}}}{3}}$	41.6153
73	Parabigyrate rhombicosidodecahedron		60	120	62	$D_{5d}$ of order 20
74	Metabigyrate rhombicosidodecahedron		60	120	62	$C_{2v}$ of order 4
75	Trigyrate rhombicosidodecahedron		60	120	62	$C_{3v}$ of order 6
76	Diminished rhombicosidodecahedron		55	105	52	$C_{5v}$ of order 10	$25+{\frac {15{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+11{\sqrt {25+10{\sqrt {5}}}}}{4}}$	58.1147	${\frac {115}{6}}+9{\sqrt {5}}$	39.2913
77	Paragyrate diminished rhombicosidodecahedron		55	105	52	$C_{5v}$ of order 10
78	Metagyrate diminished rhombicosidodecahedron		55	105	52	$C_{s}$ of order 2
79	Bigyrate diminished rhombicosidodecahedron		55	105	52	$C_{s}$ of order 2
80	Parabidiminished rhombicosidodecahedron		50	90	42	$D_{5d}$ of order 20	$20+{\frac {5}{2}}\left({\sqrt {3}}+2{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {25+10{\sqrt {5}}}}\right)$	56.9233	${\frac {55+25{\sqrt {5}}}{3}}$	36.9672
81	Metabidiminished rhombicosidodecahedron		50	90	42	$C_{2v}$ of order 4
82	Gyrate bidiminished rhombicosidodecahedron		50	90	42	$C_{s}$ of order 2
83	Tridiminished rhombicosidodecahedron		45	75	32	$C_{3v}$ of order 6	$15+{\frac {5{\sqrt {3}}+30{\sqrt {5+2{\sqrt {5}}}}+9{\sqrt {25+10{\sqrt {5}}}}}{4}}$	55.732	${\frac {35}{2}}+{\frac {23{\sqrt {5}}}{3}}$	34.6432
84	Snub disphenoid		8	18	12	$D_{2d}$ of order 8	$3{\sqrt {3}}$	5.1962		0.8595
85	Snub square antiprism		16	40	26	$D_{4d}$ of order 16	$2+6{\sqrt {3}}$	12.3923		3.6012
86	Sphenocorona		10	22	14	$C_{2v}$ of order 4	$2+3{\sqrt {3}}$	7.1962	${\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}$	1.5154
87	Augmented sphenocorona		11	26	17	$C_{s}$ of order 2	$1+4{\sqrt {3}}$	7.9282	${\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}+{\frac {1}{3{\sqrt {2}}}}$	1.7511
88	Sphenomegacorona		12	28	18	$C_{2v}$ of order 4	$2+4{\sqrt {3}}$	8.9282		1.9481
89	Hebesphenomegacorona		14	33	21	$C_{2v}$ of order 4	$3+{\frac {9}{2}}{\sqrt {3}}$	10.7942		2.9129
90	Disphenocingulum		16	38	24	$D_{2d}$ of order 8	$4+5{\sqrt {3}}$	12.6603		3.7776
91	Bilunabirotunda		14	26	14	$D_{2h}$ of order 8	$2+2{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}$	12.346	${\frac {17+9{\sqrt {5}}}{12}}$	3.0937
92	Triangular hebesphenorotunda		18	36	20	$C_{3v}$ of order 6	$3+{\frac {19{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}}{4}}$	16.3887	${\frac {5}{2}}+{\frac {7{\sqrt {5}}}{6}}$	5.1087

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• Solomon, R. (2003). Abstract Algebra. American Mathematical Society. ISBN 978-0-821-84795-4.
• Slobodan, M.; Obradović, M.; Ðukanović, G. (2015). "Composite Concave Cupolae as Geometric and Architectural Forms" (PDF). Journal for Geometry and Graphics. 19 (1): 79–91.
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External links

• 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.