List of Johnson solids

Short description: All 92 Johnson Solids listed here

In geometry, polyhedra is a three-dimensional object with lines meeting at a point that forms polygons. The points, lines, and polygons of polyhedra are respectively known as the vertices, edges, and faces. A polyhedron is said to be convex if, for every two points inside the polyhedron, there is a line connecting them that lies within the polyhedra as well; its faces are not coplanar (meaning every face are not in the same plane) and its edges are not colinear (meaning the edges are not in the same line). A polyhedron is said to be regular if every polygonal faces are equilateral and equiangular, and those with the polyhedron has vertex-transitive property are called a uniform polyhedron. A Johnson solid (or Johnson–Zalgaller solid) is a convex polyhedron with its faces are regular polygons. Some authors do not require that the Johnson solid not be uniform, meaning that the Johnson solids may not be Platonic solid, Archimedean solid, prism, or antiprism.

The 92 convex polyhedrons were published by Norman Johnson, conjecturing that there are no other solids. His conjecture was proved by Victor Zalgaller proved in 1969 that Johnson's list was complete. Pyramids, cupolae, and rotunda are the first six Johnson solids that have regular faces and convexity. These solids may be applied to construct another polyhedron that has the same properties, a process known as augmentation; attaching prism or antiprism to those is known as elongation or gyroelongation, respectively. Some others may be constructed by diminishment, the removal of those from the component of polyhedra, or by snubification, a construction by cutting loose the edges, lifting the faces and rotate in certain angle, after which adding the equilateral triangles between them.

Every polyhedra has own characteristics, including symmetry and measurement. An object is said to be symmetrical if there is such transformation preserving the immunity to change. All of those transformations may be composed in a concept of group, alongside the number of elements, known as 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. A polygon that is rotated symmetrically in [math]\displaystyle{ \frac{360^\circ}{n} }[/math] is denoted by C_n, a cyclic group of order n; combining with the reflection symmetry results in the symmetry of dihedral group D_n of order 2n. In three-dimensional symmetry point groups, the transformation of polyhedra's symmetry includes the rotation around the line passing through the base center, known as axis of symmetry, and reflection relative to perpendicular planes passing through the bisector of a base; this is known as the pyramidal symmetry C_nv of order 2n. Relatedly, polyhedra that preserve their symmetry by rotating it horizontally in [math]\displaystyle{ 180^\circ }[/math] are known as prismatic symmetry D_nv of order 2n. The antiprismatic symmetry D_nd of order 4n preserving the symmetry by rotating its half bottom and reflection across the horizontal plane. The symmetry group C_nh of order 2n preserve the symmetry by rotation around the axis of symmetry and reflection on horizontal plane; one case that preserves the symmetry by one full rotation and one reflection horizontal plane is C_1h of order 2, or simply denoted as C_s. 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, and the surface area is the overall area of all faces of polyhedra that is measured by summing all of them. A volume is a measurement of the region in three-dimensional space.

The following table contains the 92 Johnson solids of the edge length a. Each of the columns includes the enumeration of Johnson solid (J_n), the number of vertices, edges, and faces, symmetry, surface area A and volume V.

Table of all 92 Johnson solids

Jn	Solid name	Image	Vertices	Edges	Faces	Symmetry group and its order	Surface area and volume
1	Equilateral square pyramid		5	8	5	C4v of order 8	[math]\displaystyle{ \begin{align} A &= \left(1 + \sqrt{3}\right)a^2 \\ &\approx 2.7321a^2 \\ V &= \frac{\sqrt{2}}{6}a^3 \\ &\approx 0.2357a^3 \end{align} }[/math]
2	Pentagonal pyramid		6	10	6	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{a^2}{2}\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)} \\ &\approx 3.8855a^2 \\ V &= \left(\frac{5 + \sqrt{5}}{24}\right)a^3 \\ &\approx 0.3015a^3 \end{align} }[/math]
3	Triangular cupola		9	15	8	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \left(3+\frac{5\sqrt{3}}{2} \right) a^2 \\ &\approx 7.3301a^2 \\ V &= \left(\frac{5}{3\sqrt{2}}\right) a^3 \\ &\approx 1.1785a^3 \end{align} }[/math]
4	Square cupola		12	20	10	C4v of order 8	[math]\displaystyle{ \begin{align} A &= \left(7+2\sqrt{2}+\sqrt{3}\right)a^2 \\ &\approx 11.5605a^2 \\ V &= \left(1+\frac{2\sqrt{2}}{3}\right)a^3 \\ &\approx 1.9428a^3 \end{align} }[/math]
5	Pentagonal cupola		15	25	12	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \left(\frac{1}{4}\left(20+5\sqrt{3}+\sqrt{5\left(145+62\sqrt{5}\right)}\right)\right)a^2 \\ &\approx 16.5798a^2 \\ V &= \left(\frac{1}{6}\left(5+4\sqrt{5}\right)\right)a^3 \\ &\approx 2.3241a^3 \end{align} }[/math]
6	Pentagonal rotunda		20	35	17	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \left(\frac{1}{2}\left(5\sqrt{3}+\sqrt{10\left(65+29\sqrt{5}\right)}\right)\right)a^2 \\ &\approx 22.3472a^2 \\ V &= \left(\frac{1}{12}\left(45+17\sqrt{5}\right)\right)a^3 \\ &\approx 6.9178a^3 \end{align} }[/math]
7	Elongated triangular pyramid		7	12	7	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \left(3+\sqrt{3}\right)a^2 \\ &\approx 4.7321a^2 \\ V &= \left(\frac{1}{12}\left(\sqrt{2}+3\sqrt{3}\right)\right)a^3 \\ &\approx 0.5509a^3 \end{align} }[/math]
8	Elongated square pyramid		9	16	9	C4v of order 8	[math]\displaystyle{ \begin{align} A &= \left( 5 + \sqrt{3} \right)a^2 \\ &\approx 6.7321a^2 \\ V &= \left( 1 + \frac{\sqrt{2}}{6}\right)a^3 \\ &\approx 1.2357a^3 \end{align} }[/math]
9	Elongated pentagonal pyramid		11	20	11	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{20 + 5\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4}a^2 \\ &\approx 8.8855a^2 \\ V &= \left(\frac{5 + \sqrt{5} + 6\sqrt{25 + 10\sqrt{5}}}{24} \right)a^3 \\ &\approx 2.022a^3 \end{align} }[/math]
10	Gyroelongated square pyramid		9	20	13	C4v of order 8	[math]\displaystyle{ \begin{align} A &= (1 + 3\sqrt{3})a^2 \\ &\approx 6.1962a^2 \\ V &= \frac{1}{6} \left(\sqrt{2}+2 \sqrt{4+3 \sqrt{2}}\right)a^3 \\ &\approx 1.1927a^3 \end{align} }[/math]
11	Gyroelongated pentagonal pyramid		11	25	16	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(15 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 8.2157a^2 \\ V &= \frac{1}{24} \left(25+9 \sqrt{5}\right)a^3 \\ &\approx 1.8802a^3 \end{align} }[/math]
12	Triangular bipyramid		5	9	6	D3h of order 12	[math]\displaystyle{ \begin{align} A &= \frac{3\sqrt{3}}{2}a^2 \\ &\approx 2.5981a^2 \\ V &= \frac{\sqrt{2}}{6}a^3 \\ &\approx 0.2358a^3 \end{align} }[/math]
13	Pentagonal bipyramid		7	15	10	D5h of order 20	[math]\displaystyle{ \begin{align} A &= \frac{5 \sqrt{3}}{2}a^2 \\ &\approx 4.3301a^2 \\ V &= \frac{1}{12} \left(5+\sqrt{5}\right)a^3 \\ &\approx 0.603a^3 \end{align} }[/math]
14	Elongated triangular bipyramid		8	15	9	D3h of order 12	[math]\displaystyle{ \begin{align} A &= \frac{3}{2} \left(2+\sqrt{3}\right)a^2 \\ &\approx 5.5981a^2 \\ V &= \frac{1}{12} \left(2 \sqrt{2}+3 \sqrt{3}\right)a^3 \\ &\approx 0.6687a^3 \end{align} }[/math]
15	Elongated square bipyramid		10	20	12	D4h of order 16	[math]\displaystyle{ \begin{align} A &= 2 \left(2+\sqrt{3}\right)a^2 \\ &\approx 7.4641a^2 \\ V &= \frac{1}{3} \left(3+\sqrt{2}\right)a^3 \\ &\approx 1.4714a^3 \end{align} }[/math]
16	Elongated pentagonal bipyramid		12	25	15	D5h of order 20	[math]\displaystyle{ \begin{align} A &= \frac{5}{2} \left(2+\sqrt{3}\right)a^2 \\ &\approx 9.3301a^2 \\ V &= \frac{1}{12} \left(5+\sqrt{5}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \\ &\approx 2.3235a^3 \end{align} }[/math]
17	Elongated square bipyramid		10	24	16	D4d of order 16	[math]\displaystyle{ \begin{align} A &= 4 \sqrt{3}a^2 \\ &\approx 6.9282a^2 \\ V &= \frac{1}{12} \left(5+\sqrt{5}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \\ &\approx 2.3235a^3 \end{align} }[/math]
18	Elongated triangular cupola		15	27	14	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{2} \left(18+5 \sqrt{3}\right)a^2 \\ &\approx 13.3301a^2 \\ V &= \frac{1}{3} \left(\sqrt{2}+\sqrt{4+3 \sqrt{2}}\right)a^3 \\ &\approx 1.4284a^3 \end{align} }[/math]
19	Elongated square cupola		20	36	18	C4v of order 8	[math]\displaystyle{ \begin{align} A &= (15+2 \sqrt{2}+\sqrt{3})a^2\\ &\approx 19.5605a^2 \\ V &= \left(3+\frac{8 \sqrt{2}}{3}\right)a^3 \\ &\approx 6.7712a^3 \end{align} }[/math]
20	Elongated pentagonal cupola		25	45	22	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(60+5 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 26.5798a^2 \\ V &= \frac{1}{6} \left(5+4 \sqrt{5}+15 \sqrt{5+2 \sqrt{5}}\right)a^3 \\ &\approx 10.0183a^3 \end{align} }[/math]
21	Elongated pentagonal rotunda		30	55	27	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{2}a^2 \left(20+5 \sqrt{3}+5 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\ &\approx 32.3472a^2 \\ V &= \frac{1}{12}a^3 \left(45+17 \sqrt{5}+30 \sqrt{5+2 \sqrt{5}}\right) \\ &\approx 14.612a^3 \end{align} }[/math]
22	Gyroelongated triangular cupola		15	33	20	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{2} \left(6+11 \sqrt{3}\right)a^2 \\ &\approx 12.5263a^2 \\ V &= \frac{1}{3} \sqrt{\frac{61}{2}+18 \sqrt{3}+30 \sqrt{1+\sqrt{3}}}a^3 \\ &\approx 3.5161a^3 \end{align} }[/math]
23	Gyroelongated square cupola		20	44	26	C4v of order 8	[math]\displaystyle{ \begin{align} A &= (7+2 \sqrt{2}+5 \sqrt{3})a^2 \\ &\approx 18.4887a^2 \\ V &= \left(1+\frac{2}{3}\sqrt{2} + \frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right)a^3 \\ &\approx 6.2108a^3 \end{align} }[/math]
24	Gyroelongated pentagonal cupola		25	55	32	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(20+25 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 25.2400a^2 \\ V &= \left(\frac{5}{6}+\frac{2}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ &\approx 9.0733a^3 \end{align} }[/math]
25	Gyroelongated pentagonal rotunda		30	65	37	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{2}\left( 15\sqrt{3}+\left(5+3\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2 \\ &\approx 31.0075a^2 \\ V &= \left(\frac{45}{12}+\frac{17}{12}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ &\approx 13.6671a^3 \end{align} }[/math]
26	Gyrobifastigium		8	14	8	D2d of order 8	[math]\displaystyle{ \begin{align} A &= \left(4+\sqrt{3}\right)a^2 \\ &\approx 5.7321a^2 \\ V &= \left(\frac{\sqrt{3}}{2}\right)a^3 \\ &\approx 0.866a^3 \end{align} }[/math]
27	Triangular orthobicupola		12	24	14	D3h of order 12	[math]\displaystyle{ \begin{align} A &= 2\left(3+\sqrt{3}\right)a^2 \\ &\approx 9.4641a^2 \\ V &= \frac{5\sqrt{2}}{3}a^3 \\ &\approx 2.357a^3 \end{align} }[/math]
28	Square orthobicupola		16	32	18	D4h of order 16	[math]\displaystyle{ \begin{align} A &= 2(\sqrt{5} + \sqrt{3})a^2 \\ &\approx 13.4641a^2 \\ V &= \left(2 + \frac{4\sqrt{2}}{3}\right)a^3 \\ &\approx 3.8856a^3 \end{align} }[/math]
29	Square gyrobicupola		16	32	18	D4d of order 16	[math]\displaystyle{ \begin{align} A &= 2(\sqrt{5} + \sqrt{3})a^2 \\ &\approx 13.4641a^2 \\ V &= \left(2 + \frac{4\sqrt{2}}{3}\right)a^3 \\ &\approx 3.8856a^3 \end{align} }[/math]
30	Pentagonal orthobicupola		20	40	22	D5h of order 20	[math]\displaystyle{ \begin{align} A &= \left(10+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 17.7711a^2 \\ V &= \frac{1}{3}\left(5+4\sqrt{5}\right)a^3 \\ &\approx 4.6481a^3 \end{align} }[/math]
31	Pentagonal gyrobicupola		20	40	22	D5d of order 20	[math]\displaystyle{ \begin{align} A &= \left(10+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 17.7711a^2 \\ V &= \frac{1}{3}\left(5+4\sqrt{5}\right)a^3 \\ &\approx 4.6481a^3 \end{align} }[/math]
32	Pentagonal orthocupolarotunda		25	50	27	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \left(5+\frac{1}{4}\sqrt{1900+490\sqrt{5}+210\sqrt{75+30\sqrt{5}}}\right)a^2 \\ &\approx 23.5385a^2 \\ V &= \frac{5}{12}\left(11+5\sqrt{5}\right)a^3 \\ &\approx 9.2418a^3 \end{align} }[/math]
33	Pentagonal gyrocupolarotunda		25	50	27	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \left(5+\frac{15}{4}\sqrt{3}+\frac{7}{4}\sqrt{25+10\sqrt{5}}\right)a^2 \\ &\approx 23.5385a^2 \\ V &= \frac{5}{12}\left(11+5\sqrt{5}\right)a^3 \\ &\approx 9.2418a^3 \end{align} }[/math]
34	Pentagonal orthobirotunda		30	60	32	D5h of order 20	[math]\displaystyle{ \begin{align} A &= \left((5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 29.306a^2 \\ V &= \frac{1}{6}(45 + 17\sqrt{5})a^3 \\ &\approx 13.8355a^3 \end{align} }[/math]
35	Elongated triangular orthobicupola		18	36	20	D3h of order 12	[math]\displaystyle{ \begin{align} A &= 2(6 + \sqrt{3})a^2 \\ &\approx 15.4641a^2 \\ V &= \left(\frac{5 \sqrt{2}}{3} + \frac{3 \sqrt{3}}{2}\right)a^3 \\ &\approx 4.9551a^3 \end{align} }[/math]
36	Elongated triangular gyrobicupola		18	36	20	D3d of order 12	[math]\displaystyle{ \begin{align} A &= 2(6 + \sqrt{3})a^2 \\ &\approx 15.4641a^2 \\ V &= \left(\frac{5 \sqrt{2}}{3} + \frac{3 \sqrt{3}}{2}\right)a^3 \\ &\approx 4.9551a^3 \end{align} }[/math]
37	Elongated square gyrobicupola		24	48	26	D4d of order 16	[math]\displaystyle{ \begin{align} A &= 2(9 + \sqrt{3})a^2 \\ &\approx 21.4641a^2 \\ V &= \left(4 + \frac{10\sqrt{2}}{3}\right)a^3 \\ &\approx 8.714a^3 \end{align} }[/math]
38	Elongated pentagonal orthobicupola		30	60	32	D5h of order 20	[math]\displaystyle{ \begin{align} A &= \left(20+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 27.7711a^2 \\ V &= \frac{1}{6}\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ &\approx 12.3423a^3 \end{align} }[/math]
39	Elongated pentagonal gyrobicupola		30	60	32	D5d of order 20	[math]\displaystyle{ \begin{align} A &= \left(20+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 27.7711a^2 \\ V &= \frac{1}{6}\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ &\approx 12.3423a^3 \end{align} }[/math]
40	Elongated pentagonal orthocupolarotunda		35	70	37	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4}\left(60+\sqrt{10\left(190+49\sqrt{5}+21\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 33.5385a^2 \\ V &= \frac{5}{12}\left(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}}\right)a^3 \\ &\approx 16.936a^3 \end{align} }[/math]
41	Elongated pentagonal gyrocupolarotunda		35	70	37	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4}\left(60+\sqrt{10\left(190+49\sqrt{5}+21\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 33.5385a^2 \\ V &= \frac{5}{12}\left(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}}\right)a^3 \\ &\approx 16.936a^3 \end{align} }[/math]
42	Elongated pentagonal orthobirotunda		40	80	42	D5h of order 20	[math]\displaystyle{ \begin{align} A &= \left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 39.306a^2 \\ V &= \frac{1}{6}\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ &\approx 21.5297a^3 \end{align} }[/math]
43	Elongated pentaognal gyrobirotunda		40	80	42	D5d of order 20	[math]\displaystyle{ \begin{align} A &= \left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \\ &\approx 39.306a^2 \\ V &= \frac{1}{6}\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \\ &\approx 21.5297a^3 \end{align} }[/math]
44	Gyroelongated triangular bicupola		18	42	26	D3 of order 6	[math]\displaystyle{ \begin{align} A &= \left(6+5\sqrt{3}\right)a^2 \\ &\approx 14.6603a^2 \\ V &= \sqrt{2} \left(\frac{5}{3}+\sqrt{1+\sqrt{3}}\right) a^3 \\ &\approx 4.6946a^3 \end{align} }[/math]
45	Gyroelongated square bicupola		24	56	34	D4 of order 8	[math]\displaystyle{ \begin{align} A &= \left(10+6\sqrt{3}\right) a^2 \\ &\approx 20.3923a^2 \\ V &= \left(2+\frac{4}{3}\sqrt{2} + \frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right) a^3 \\ &\approx 8.1536a^3 \end{align} }[/math]
46	Gyroelongated pentagonal bicupola		30	70	42	D5 of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{2}\left(20+15\sqrt{3}+\sqrt{25+10\sqrt{5}}\right)a^2 \\ &\approx 26.4313a^2 \\ V &= \left(\frac{5}{3}+\frac{4}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ &\approx 11.3974a^3 \end{align} }[/math]
47	Gyroelongated pentagonal cupolarotunda		35	80	47	C5 of order 5	[math]\displaystyle{ \begin{align} A &= \frac{1}{4}\left(20+35\sqrt{3}+7\sqrt{25+10\sqrt{5}}\right)a^2 \\ &\approx 32.1988a^2 \\ V &= \left(\frac{55}{12}+\frac{25}{12}\sqrt{5}+ \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ &\approx 15.9911a^3 \end{align} }[/math]
48	Gyroelongated pentagonal birotunda		40	90	52	D5 of order 10	[math]\displaystyle{ \begin{align} A &= \left(10\sqrt{3} + 3\sqrt{25+10\sqrt{5}}\right) a^2 \\ &\approx 37.9662a^2 \\ V &= \left(\frac{45}{6}+\frac{17}{6}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3 \\ &\approx 20.5848a^3 \end{align} }[/math]
49	Augmented triangular prism		7	13	8	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{1}{2}(4 + 3\sqrt{3})a^2 \\ &\approx 4.5981a^2 \\ V &= \frac{1}{12}(2\sqrt{2} + 3\sqrt{3})a^3 \\ &\approx 0.6687a^3 \end{align} }[/math]
50	Biaugmented triangular prism		8	17	11	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{1}{2}(2 + 5\sqrt{3})a^2 \\ &\approx 5.3301a^2 \\ V &= \left(\frac{59}{144} + \frac{1}{\sqrt{6}}\right)a^3 \\ &\approx 0.9044a^3 \end{align} }[/math]
51	Triaugmented triangular prism		9	21	14	D3h of order 12	[math]\displaystyle{ \begin{align} A &= \frac{7\sqrt{3}}{2}a^2 \\ &\approx 6.0622a^2 \\ V &= \frac{2\sqrt{2}+\sqrt{3}}{4}a^3 \\ &\approx 1.1401a^3 \end{align} }[/math]
52	Augmented pentagonal prism		11	19	10	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{1}{2} \left(8+2 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 9.173a^2 \\ V &= \frac{1}{12} \sqrt{233+90 \sqrt{5}+12 \sqrt{50+20 \sqrt{5}}}a^3 \\ &\approx 1.9562a^3 \end{align} }[/math]
53	Biaugmented pentagonal prism		12	23	13	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{1}{2}a^2 \left(6+4 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right) \\ &\approx 9.9051a^2 \\ V &= \frac{1}{12}a^3 \sqrt{257+90 \sqrt{5}+24 \sqrt{50+20 \sqrt{5}}} \\ &\approx 2.1919a^3 \end{align} }[/math]
54	Augmented hexagonal prism		13	22	11	C2v of order 4	[math]\displaystyle{ \begin{align} A &= (5+4 \sqrt{3})a^2 \\ &\approx 11.9282a^2 \\ V &= \frac{1}{6} \left(\sqrt{2}+9 \sqrt{3}\right)a^3 \\ &\approx 2.8338a^3 \end{align} }[/math]
55	Parabiaugmented hexagonal prism		14	26	14	D2h of order 8	[math]\displaystyle{ \begin{align} A &= (4+5 \sqrt{3})a^2 \\ &\approx 12.6603a^2 \\ V &= \frac{1}{6} \left(2 \sqrt{2}+9 \sqrt{3}\right)a^3 \\ &\approx 3.0695a^3 \end{align} }[/math]
56	Metabiaugmented hexagonal prism		14	26	14	C2v of order 4	[math]\displaystyle{ \begin{align} A &= (4+5 \sqrt{3})a^2 \\ &\approx 12.6603a^2 \\ V &= \frac{1}{6} \left(2 \sqrt{2}+9 \sqrt{3}\right)a^3 \\ &\approx 3.0695a^3 \end{align} }[/math]
57	Triaugmented hexagonal prism		15	30	17	D3h of order 12	[math]\displaystyle{ \begin{align} A &= 3 \left(1+2 \sqrt{3}\right)a^2 \\ &\approx 13.3923a^2 \\ V &= \left(\frac{1}{\sqrt{2}}+\frac{3 \sqrt{3}}{2}\right)a^3 \\ &\approx 3.3052a^3 \end{align} }[/math]
58	Augmented dodecahedron		21	35	16	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(5 \sqrt{3}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 21.0903a^2 \\ V &= \frac{1}{24} \left(95+43 \sqrt{5}\right)a^3 \\ &\approx 7.9646a^3 \end{align} }[/math]
59	Parabiaugmented dodecahedron		22	40	20	D5d of order 20	[math]\displaystyle{ \begin{align} A &= \frac{5}{2} \left(\sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 21.5349a^2 \\ V &= \frac{1}{6} \left(25+11 \sqrt{5}\right)a^3 \\ &\approx 8.2661a^3 \end{align} }[/math]
60	Metabiaugmented dodecahedron		22	40	20	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{5}{2} \left(\sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 21.5349a^2 \\ V &= \frac{1}{6} \left(25+11 \sqrt{5}\right)a^3 \\ &\approx 8.2661a^3 \end{align} }[/math]
61	Triaugmented dodecahedron		23	45	24	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{3}{4} \left(5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 21.9795a^2 \\ V &= \frac{5}{8} \left(7+3 \sqrt{5}\right)a^3 \\ &\approx 8.5676a^3 \end{align} }[/math]
62	Metabidiminished icosahedron		10	20	12	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{1}{2} \left(5 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 7.7711a^2 \\ V &= \frac{1}{6} \left(5+2 \sqrt{5}\right)a^3 \\ &\approx 1.5787a^3 \end{align} }[/math]
63	Tridiminished icosahedron		9	15	8	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^3 \\ &\approx 7.3265a^3 \\ V &= \left(\frac{5}{8}+\frac{7 \sqrt{5}}{24}\right)a^3 \\ &\approx 1.2772a^3 \end{align} }[/math]
64	Augmented tridiminished icosahedron		10	18	10	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(7 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 8.1925a^2 \\ V &= \frac{1}{24} \left(15+2 \sqrt{2}+7 \sqrt{5}\right)a^3 \\ &\approx 1.395a^3 \end{align} }[/math]
65	Augmented truncated tetrahedron		15	27	14	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{2} \left(6+13 \sqrt{3}\right)a^2 \\ &\approx 14.2583a^2 \\ V &= \frac{11}{2 \sqrt{2}}a^3 \\ &\approx 3.8891a^3 \end{align} }[/math]
66	Augmented truncated cube		28	48	22	C4v of order 8	[math]\displaystyle{ \begin{align} A &= (15+10 \sqrt{2}+3 \sqrt{3})a^2 \\ &\approx 34.3383a^2 \\ V &= \left(8+\frac{16 \sqrt{2}}{3}\right)a^3 \\ &\approx 15.5425a^3 \end{align} }[/math]
67	Biaugmented truncated cube		32	60	30	D4h of order 16	[math]\displaystyle{ \begin{align} A &= 2 \left(9+4 \sqrt{2}+2 \sqrt{3}\right)a^2 \\ &\approx 36.2419a^2 \\ V &= (9+6 \sqrt{2})a^3 \\ &\approx 17.4853a^3 \end{align} }[/math]
68	Augmented truncated dodecahedron		65	105	42	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(20+25 \sqrt{3}+110 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 102.1821a^2 \\ V &= \left(\frac{505}{12}+\frac{81 \sqrt{5}}{4}\right)a^3 \\ &\approx 87.3637a^3 \end{align} }[/math]
69	Parabiaugmented truncated dodecahedron		70	120	52	D5d of order 20	[math]\displaystyle{ \begin{align} A &= \frac{1}{2} \left(20+15 \sqrt{3}+50 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 103.3734a^2 \\ V &= \frac{1}{12} \left(515+251 \sqrt{5}\right)a^3 \\ &\approx 89.6878a^3 \end{align} }[/math]
70	Metabiaugmented truncated dodecahedron		70	120	52	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{1}{2} \left(20+15 \sqrt{3}+50 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 103.3734a^2 \\ V &= \frac{1}{12} \left(515+251 \sqrt{5}\right)a^3 \\ &\approx 89.6878a^3 \end{align} }[/math]
71	Triaugmented truncated dodecahedron		75	135	62	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(60+35 \sqrt{3}+90 \sqrt{5+2 \sqrt{5}}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 104.5648a^2 \\ V &= \frac{7}{12} \left(75+37 \sqrt{5}\right)a^3 \\ &\approx 92.0118a^3 \end{align} }[/math]
72	Gyrate rhombicosidodecahedron		60	120	62	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 59.306a^2 \\ V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ &\approx 41.6153a^3 \end{align} }[/math]
73	Parabigyrate rhombicosidodecahedron		60	120	62	D5d of order 20	[math]\displaystyle{ \begin{align} A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 59.306a^2 \\ V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ &\approx 41.6153a^3 \end{align} }[/math]
74	Metabigyrate rhombicosidodecahedron		60	120	62	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 59.306a^2 \\ V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ &\approx 41.6153a^3 \end{align} }[/math]
75	Trigyrate rhombicosidodecahedron		60	120	62	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \left(30+5 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 59.306a^2 \\ V &= \left(20+\frac{29 \sqrt{5}}{3}\right)a^3 \\ &\approx 41.6153a^3 \end{align} }[/math]
76	Diminished rhombicosidodecahedron		55	105	52	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 58.1147a^2 \\ V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ &\approx 39.2913a^3 \end{align} }[/math]
77	Paragyrate diminished rhombicosidodecahedron		55	105	52	C5v of order 10	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 58.1147a^2 \\ V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ &\approx 39.2913a^3 \end{align} }[/math]
78	Metagyrate diminished rhombicosidodecahedron		55	105	52	Cs of order 2	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 58.1147a^2 \\ V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ &\approx 39.2913a^3 \end{align} }[/math]
79	Bigyrate diminished rhombicosidodecahedron		55	105	52	Cs of order 2	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(100+15 \sqrt{3}+10 \sqrt{5+2 \sqrt{5}}+11 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 58.1147a^2 \\ V &= \left(\frac{115}{6}+9 \sqrt{5}\right)a^3 \\ &\approx 39.2913a^3 \end{align} }[/math]
80	Parabidiminished rhombicosidodecahedron		50	90	42	D5d of order 20	[math]\displaystyle{ \begin{align} A &= \frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 56.9233a^2 \\ V &= \frac{5}{3} \left(11+5 \sqrt{5}\right)a^3 \\ &\approx 36.9672a^3 \end{align} }[/math]
81	Metabidiminished rhombicosidodecahedron		50	90	42	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 56.9233a^2 \\ V &= \frac{5}{3} \left(11+5 \sqrt{5}\right)a^3 \\ &\approx 36.9672a^3 \end{align} }[/math]
82	Gyrate bidiminished rhombicosidodecahedron		50	90	42	Cs of order 2	[math]\displaystyle{ \begin{align} A &= \frac{5}{2} \left(8+\sqrt{3}+2 \sqrt{5+2 \sqrt{5}}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 56.9233a^2 \\ V &= \frac{5}{3} \left(11+5 \sqrt{5}\right)a^3 \\ &\approx 36.9672a^3 \end{align} }[/math]
83	Tridiminished rhombicosidodecahedron		45	75	32	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(60+5 \sqrt{3}+30 \sqrt{5+2 \sqrt{5}}+9 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 55.732a^2 \\ V &= \left(\frac{35}{2}+\frac{23 \sqrt{5}}{3}\right)a^3 \\ &\approx 34.6432a^3 \end{align} }[/math]
84	Snub disphenoid		8	18	12	D2d of order 8	[math]\displaystyle{ \begin{align} A &= 2 \left(1+3 \sqrt{3}\right)a^2 \\ &\approx 12.3923a^2 \\ V &\approx 0.8595a^3 \end{align} }[/math]
85	Snub square antiprism		16	40	26	D4d of order 16	[math]\displaystyle{ \begin{align} A &= 2 \left(1+3 \sqrt{3}\right)a^2 \\ &\approx 12.3923a^2 \\ V &\approx 3.6012a^3 \end{align} }[/math]
86	Sphenocorona		10	22	14	C2v of order 4	[math]\displaystyle{ \begin{align} A &= (2+3 \sqrt{3})a^2 \\ &\approx 7.1962a^2 \\ V &= \frac{1}{2}a^3 \sqrt{1+3 \sqrt{\frac{3}{2}}+\sqrt{13+3 \sqrt{6}}} \\ &\approx 1.5154a^3 \end{align} }[/math]
87	Augmented sphenocorona		11	26	17	Cs of order 2	[math]\displaystyle{ \begin{align} A &= (1+4 \sqrt{3})a^2 \\ &\approx 7.9282a^2 \\ V &= \frac{1}{2}a^3\sqrt{1 + 3 \sqrt{\frac{3}{2}} + \sqrt{13 + 3 \sqrt{6}}}+\frac{1}{3\sqrt{2}} \\ &\approx 1.7511a^3 \end{align} }[/math]
88	Sphenomegacorona		12	28	18	C2v of order 4	[math]\displaystyle{ \begin{align} A &= 2 \left(1+2 \sqrt{3}\right)a^2 \\ &\approx 8.9282a^2 \\ V &\approx 1.9481a^3 \end{align} }[/math]
89	Hebesphenomegacorona		14	33	21	C2v of order 4	[math]\displaystyle{ \begin{align} A &= \frac{3}{2} \left(2+3 \sqrt{3}\right)a^2 \\ &\approx 10.7942a^2 \\ V &\approx 2.9129a^3 \end{align} }[/math]
90	Disphenocingulum		16	38	24	D2d of order 8	[math]\displaystyle{ \begin{align} A &= (4+5 \sqrt{3})a^2 \\ &\approx 12.6603a^2 \\ V &\approx 3.7776a^3 \end{align} }[/math]
91	Bilunabirotunda		14	26	14	D2h of order 8	[math]\displaystyle{ \begin{align} A &= \left(2+2 \sqrt{3}+\sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 12.346a^2 \\ V &= \frac{1}{12} \left(17+9 \sqrt{5}\right)a^3 \\ &\approx 3.0937a^3 \end{align} }[/math]
92	Triangular hebespenorotunda		18	36	20	C3v of order 6	[math]\displaystyle{ \begin{align} A &= \frac{1}{4} \left(12+19 \sqrt{3}+3 \sqrt{5 \left(5+2 \sqrt{5}\right)}\right)a^2 \\ &\approx 16.3887a^2 \\ V &= \left(\frac{5}{2}+\frac{7 \sqrt{5}}{6}\right)a^3 \\ &\approx 5.1087a^3 \end{align} }[/math]

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References

• Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. 
• Boissonnat, J. D.; Yvinec, M. (June 1989). "Probing a scene of non convex polyhedra". 237–246. doi:10.1145/73833.73860. 
• Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. https://archive.org/details/polyhedra0000crom. 
• Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Springer. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2. 
• Flusser, Jan; Suk, Tomas; Zitofa, Barbara (2017). 2D and 3D Image Analysis by Moments. John & Sons Wiley. 
• Hergert, Wolfram; Geilhufe, Matthias (2018). Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica. John & Sons Wiley. ISBN 978-3-527-41300-3. 
• Holme, Audun (2010). Geometry: Our Cultural Heritage. Springer. doi:10.1007/978-3-642-14441-7. ISBN 978-3-642-14441-7. https://books.google.com/books?id=zXwQGo8jyHUC. 
• "Convex Solids with Regular Faces". Canadian Journal of Mathematics 18: 169–200. 1966. doi:10.4153/CJM-1966-021-8. 
• Litchenberg, Dorovan R. (1988). "Pyramids, Prisms, Antiprisms, and Deltahedra". The Mathematics Teacher 81 (4): 261-265. 
• Meyer, W. (2006). Geometry and Its Applications. Academic Press. ISBN 978-0-12-369427-0. https://books.google.com/books?id=ez6H5Ho6E3cC. 
• Parker, Sybil P. (1997). Dictionary of Mathematics. McGraw-Hill. 
• Powell, Richard C. (2010). Symmetry, Group Theory, and the Physical Properties of Crystals. Springer. doi:10.1007/978-1-4419-7598-0. ISBN 978-1-4419-7598-0. 
• 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 978-93-86279-06-4. 
• Solomon, Ronald (2003). Abstract Algebra. American Mathematical Society. ISBN 978-0-8218-4795-4. https://books.google.com/books?id=ouvZKQiykf4C. 
• Slobodan, Mišić; Obradović, Marija; Ðukanović, Gordana (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. 
• Todesco, Gian Marco (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. 
• Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5. https://books.google.com/books?id=51juDwAAQBAJ. 
• Walsh, Edward T. (2014). A First Course in Geometry. Dover Publications. ISBN 978-0-486-78020-7. https://books.google.com/books?id=ZhDdAwAAQBAJ. 
• Williams, Kim; Monteleone, Cosino (2021). Daniele Barbaro’s Perspective of 1568. Springer. doi:10.1007/978-3-030-76687-0. ISBN 978-3-030-76687-0. 
• Convex Polyhedra with Regular Faces. Consultants Bureau. 1969. 

External links

• Gagnon, Sylvain (1982). "Convex polyhedra with regular faces" (in fr). Topologie Structurale [Structural Topology] (6): 83-95. https://upcommons.upc.edu/bitstream/handle/2099/890/st6-11-a7.pdf. 
• Hart, George W.. "Johnson Solid". http://www.georgehart.com/virtual-polyhedra/johnson-info.html. 
• "Johnson Polyhedra: Polyhedra with Regular Polygon Faces". https://web.archive.org/web/20130601082835/http://www.uwgb.edu/dutchs/symmetry/johnsonp.htm.  See all of the categorized 92 Johnson solids images on one page.
• "Johnson Solids". http://www.orchidpalms.com/polyhedra/johnson/johnson.html. 
• Vladimir, Bulatov. "VRML models of Johnson Solids". http://bulatov.org/polyhedra/johnson/.