List of Johnson solids

Short description: none

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 $\frac{360^{\circ }}{n}$ 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 $C_s$. Template:Sticky-headerTemplate:Sort-under

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. The table includes each solid's enumeration (denoted as $J_n$). 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.

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	100px	5	8	5	$C_{4v}$ of order 8	$1+\sqrt{3}$	2.7321	$\frac{\sqrt{2}}{6}$	0.2357
2	Pentagonal pyramid	100px	6	10	6	$C_{5v}$ of order 10	$\frac{1}{2}\sqrt{\frac{5}{2}(10+\sqrt{5}+\sqrt{75+30\sqrt{5}})}$	3.8855	$\frac{5+\sqrt{5}}{24}$	0.3015
3	Triangular cupola	100px	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	100px	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	100px	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	100px	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	100px	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	100px	9	16	9	$C_{4v}$ of order 8	$5+\sqrt{3}$	6.7321	$1+\frac{\sqrt{2}}{6}$	1.2357
9	Elongated pentagonal pyramid	100px	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	100px	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	100px	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	100px	5	9	6	$D_{3h}$ of order 12	$\frac{3\sqrt{3}}{2}$	2.5981	$\frac{\sqrt{2}}{6}$	0.2357
13	Pentagonal bipyramid	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	30	65	37	$C_{5v}$ of order 10	$\frac{15\sqrt{3}+(5+3\sqrt{5})\sqrt{5+2\sqrt{5}}}{2}$	31.0075	$\frac{45+17\sqrt{5}+10\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}}{12}$	13.6671
26	Gyrobifastigium	100px	8	14	8	$D_{2d}$ of order 8	$4+\sqrt{3}$	5.7321	$\frac{\sqrt{3}}{2}$	0.8660
27	Triangular orthobicupola	100px	12	24	14	$D_{3h}$ of order 12	$6+2\sqrt{3}$	9.4641	$\frac{5\sqrt{2}}{3}$	2.3570
28	Square orthobicupola	100px	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	100px	16	32	18	$D_{4d}$ of order 16
30	Pentagonal orthobicupola	100px	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	100px	20	40	22	$D_{5d}$ of order 20
32	Pentagonal orthocupolarotunda	100px	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	100px	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	100px	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	100px	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	100px	18	36	20	$D_{3d}$ of order 12
37	Elongated square gyrobicupola	100px	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	100px	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	100px	30	60	32	$D_{5d}$ of order 20
40	Elongated pentagonal orthocupolarotunda	100px	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	100px	35	70	37	$C_{5v}$ of order 10
42	Elongated pentagonal orthobirotunda	100px	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	100px	40	80	42	$D_{5d}$ of order 20
44	Gyroelongated triangular bicupola	100px	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	100px	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	100px	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	100px	35	80	47	$C_5$ of order 5	$5+\frac{7}{4}(5\sqrt{3}+\sqrt{25+10\sqrt{5}})$	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	14	26	14	$C_{2v}$ of order 4
57	Triaugmented hexagonal prism	100px	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	100px	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	100px	22	40	20	$D_{5d}$ of order 20	$\frac{5}{2}(\sqrt{3}+\sqrt{25+10\sqrt{5}})$	21.5349	$\frac{25+11\sqrt{5}}{6}$	8.2661
60	Metabiaugmented dodecahedron	100px	22	40	20	$C_{2v}$ of order 4
61	Triaugmented dodecahedron	100px	23	45	24	$C_{3v}$ of order 6	$\frac{3}{4}(5\sqrt{3}+3\sqrt{25+10\sqrt{5}})$	21.9795	$\frac{5}{8}(7+3\sqrt{5})$	8.5676
62	Metabidiminished icosahedron	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	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	100px	70	120	52	$C_{2v}$ of order 4
71	Triaugmented truncated dodecahedron	100px	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	$\frac{7}{12}(75+37\sqrt{5})$	92.0118
72	Gyrate rhombicosidodecahedron	100px	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	100px	60	120	62	$D_{5d}$ of order 20
74	Metabigyrate rhombicosidodecahedron	100px	60	120	62	$C_{2v}$ of order 4
75	Trigyrate rhombicosidodecahedron	100px	60	120	62	$C_{3v}$ of order 6
76	Diminished rhombicosidodecahedron	100px	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	100px	55	105	52	$C_{5v}$ of order 10
78	Metagyrate diminished rhombicosidodecahedron	100px	55	105	52	$C_s$ of order 2
79	Bigyrate diminished rhombicosidodecahedron	100px	55	105	52	$C_s$ of order 2
80	Parabidiminished rhombicosidodecahedron	100px	50	90	42	$D_{5d}$ of order 20	$20+\frac{5}{2}(\sqrt{3}+2\sqrt{5+2\sqrt{5}}+\sqrt{25+10\sqrt{5}})$	56.9233	$\frac{55+25\sqrt{5}}{3}$	36.9672
81	Metabidiminished rhombicosidodecahedron	100px	50	90	42	$C_{2v}$ of order 4
82	Gyrate bidiminished rhombicosidodecahedron	100px	50	90	42	$C_s$ of order 2
83	Tridiminished rhombicosidodecahedron	100px	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	100px	8	18	12	$D_{2d}$ of order 8	$3\sqrt{3}$	5.1962		0.8595
85	Snub square antiprism	100px	16	40	26	$D_{4d}$ of order 16	$2+6\sqrt{3}$	12.3923		3.6012
86	Sphenocorona	100px	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	100px	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	100px	12	28	18	$C_{2v}$ of order 4	$2+4\sqrt{3}$	8.9282		1.9481
89	Hebesphenomegacorona	100px	14	33	21	$C_{2v}$ of order 4	$3+\frac{9}{2}\sqrt{3}$	10.7942		2.9129
90	Disphenocingulum	100px	16	38	24	$D_{2d}$ of order 8	$4+5\sqrt{3}$	12.6603		3.7776
91	Bilunabirotunda	100px	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	100px	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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• Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. doi:10.1007/978-93-86279-06-4. ISBN 978-9-386-27906-4. https://books.google.com/books?id=AGonAQAAIAAJ. 
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• Slobodan, M.; Obradović, M.; Ðukanović, G. (2015). "Composite Concave Cupolae as Geometric and Architectural Forms". Journal for Geometry and Graphics 19 (1): 79–91. https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf. 
• Timofeenko, A. V. (2009). "The Non-Platonic and Non-Archimedean Noncomposite Polyhedra". Journal of Mathematical Sciences 162 (5): 710–729. doi:10.1007/s10958-009-9655-0. 
• Todesco, G. M. (2020). "Hyperbolic Honeycomb". Imagine Math 7: Between Culture and Mathematics. Springer. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8. https://books.google.com/books?id=wtIBEAAAQBAJ. 
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External links

• "Johnson Solids". http://www.georgehart.com/virtual-polyhedra/johnson-info.html. 
• Bulatov, Vladimir. "Johnson solids". http://bulatov.org/polyhedra/johnson/.  – VRML models of Johnson solids
• Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières" (in fr). Topologie Structurale (6): 83–95. https://upcommons.upc.edu/bitstream/handle/2099/890/st6-11-a7.pdf.