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.[1] 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;[2] 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).[3] A polyhedron is said to be regular if every polygonal faces are equilateral and equiangular,[4] and those with the polyhedron has vertex-transitive property are called a uniform polyhedron.[5] 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.[6]

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.[7] 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.[8]

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 Cn, a cyclic group of order n; combining with the reflection symmetry results in the symmetry of dihedral group Dn of order 2n.[9] 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 Cnv of order 2n. Relatedly, polyhedra that preserve their symmetry by rotating it horizontally in [math]\displaystyle{ 180^\circ }[/math] are known as prismatic symmetry Dnv of order 2n. The antiprismatic symmetry Dnd of order 4n preserving the symmetry by rotating its half bottom and reflection across the horizontal plane.[10] The symmetry group Cnh 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 C1h of order 2, or simply denoted as Cs.[11] 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.[12] A volume is a measurement of the region in three-dimensional space.[13]

The following table contains the 92 Johnson solids of the edge length a. Each of the columns includes the enumeration of Johnson solid (Jn),[14] 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[15] | Surface area and volume[16]

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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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## Notes

1. ↑ Meyer (2006), p. 418.
2. ↑ Litchenberg (1988).
3. ↑ Boissonnat & Yvinec (1989).
4. ↑ Cromwell (1997), p. 77.
5. ↑ Diudea (2018), p. 40.
6. ↑ Todesco (2020), p. 282; Williams & Monteleone (2021), p. 23.
7. ↑ Johnson (1966); Zalgaller (1969).
8. ↑ Rajwade (2001), p. 84-88; Slobodan, Obradović & Ðukanović (2015); Berman (1971), p. 350; Holme (2010), p. 99.
9. ↑ Powell (2010), p. 27; Solomon (2003), p. 40.
10. ↑ Flusser, Suk & Zitofa (2017), p. 126.
11. ↑ Flusser, Suk & Zitofa (2017), p. 126; Hergert & Geilhufe (2018), p. 56.
12. ↑ Walsh (2014), p. 284.
13. ↑ Parker (1997), p. 264.
14. ↑ Uehara (2020), p. 62.
15. ↑ Johnson (1966).
16. ↑ Berman (1971).

## 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/.

* v
* t
* e

Johnson solids

Pyramids, cupolae and rotundae|

* square pyramid
* pentagonal pyramid
* triangular cupola
* square cupola
* pentagonal cupola
* pentagonal rotunda

Modified pyramids|

* elongated triangular pyramid
* elongated square pyramid
* elongated pentagonal pyramid
* gyroelongated square pyramid
* gyroelongated pentagonal pyramid
* triangular bipyramid
* square bipyramid
* pentagonal bipyramid
* elongated triangular bipyramid
* elongated square bipyramid
* elongated pentagonal bipyramid
* gyroelongated square bipyramid

Modified cupolae and rotundae|

* elongated triangular cupola
* elongated square cupola
* elongated pentagonal cupola
* elongated pentagonal rotunda
* gyroelongated triangular cupola
* gyroelongated square cupola
* gyroelongated pentagonal cupola
* gyroelongated pentagonal rotunda
* gyrobifastigium
* triangular orthobicupola
* square orthobicupola
* square gyrobicupola
* pentagonal orthobicupola
* pentagonal gyrobicupola
* pentagonal orthocupolarotunda
* pentagonal gyrocupolarotunda
* pentagonal orthobirotunda
* elongated triangular orthobicupola
* elongated triangular gyrobicupola
* elongated square gyrobicupola
* elongated pentagonal orthobicupola
* elongated pentagonal gyrobicupola
* elongated pentagonal orthocupolarotunda
* elongated pentagonal gyrocupolarotunda
* elongated pentagonal orthobirotunda
* elongated pentagonal gyrobirotunda
* gyroelongated triangular bicupola
* gyroelongated square bicupola
* gyroelongated pentagonal bicupola
* gyroelongated pentagonal cupolarotunda
* gyroelongated pentagonal birotunda

Augmented prisms|

* augmented triangular prism
* biaugmented triangular prism
* triaugmented triangular prism
* augmented pentagonal prism
* biaugmented pentagonal prism
* augmented hexagonal prism
* parabiaugmented hexagonal prism
* metabiaugmented hexagonal prism
* triaugmented hexagonal prism

Modified Platonic solids|

* augmented dodecahedron
* parabiaugmented dodecahedron
* metabiaugmented dodecahedron
* triaugmented dodecahedron
* metabidiminished icosahedron
* tridiminished icosahedron
* augmented tridiminished icosahedron

Modified Archimedean solids|

* augmented truncated tetrahedron
* augmented truncated cube
* biaugmented truncated cube
* augmented truncated dodecahedron
* parabiaugmented truncated dodecahedron
* metabiaugmented truncated dodecahedron
* triaugmented truncated dodecahedron
* gyrate rhombicosidodecahedron
* parabigyrate rhombicosidodecahedron
* metabigyrate rhombicosidodecahedron
* trigyrate rhombicosidodecahedron
* diminished rhombicosidodecahedron
* paragyrate diminished rhombicosidodecahedron
* metagyrate diminished rhombicosidodecahedron
* bigyrate diminished rhombicosidodecahedron
* parabidiminished rhombicosidodecahedron
* metabidiminished rhombicosidodecahedron
* gyrate bidiminished rhombicosidodecahedron
* tridiminished rhombicosidodecahedron

Elementary solids|

* snub disphenoid
* snub square antiprism
* sphenocorona
* augmented sphenocorona
* sphenomegacorona
* hebesphenomegacorona
* disphenocingulum
* bilunabirotunda
* triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)

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