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.[1] 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.[2] 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.[3]

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.[4] 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.[5]

The following table contains the 92 Johnson solids, with edge length  a {\displaystyle a} . The table includes the solid's enumeration (denoted as  J n {\displaystyle J_{n}} ).[6] It also includes the number of vertices, edges, and faces of each solid, as well as its symmetry group, surface area  A {\displaystyle A} , and volume  V {\displaystyle V} . 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. A polygon that is rotated symmetrically by  360 ∘ n {\textstyle {\frac {360^{\circ }}{n}}} is denoted by  C n {\displaystyle C_{n}} , a cyclic group of order  n {\displaystyle n} ; combining this with the reflection symmetry results in the symmetry of dihedral group D n {\displaystyle D_{n}} of order  2 n {\displaystyle 2n} .[7] 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 v {\displaystyle C_{n\mathrm {v} }} of order  2 n {\displaystyle 2n} . The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry D n h {\displaystyle D_{n\mathrm {h} }} of order  4 n {\displaystyle 4n} . The antiprismatic symmetry D n d {\displaystyle D_{n\mathrm {d} }} of order  4 n {\displaystyle 4n} preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.[8] The symmetry group  C n h {\displaystyle C_{n\mathrm {h} }} of order  2 n {\displaystyle 2n} preserves the symmetry by rotation around the axis of symmetry and reflection on the horizontal plane; the specific case preserving the symmetry by one full rotation is  C 1 h {\displaystyle C_{1\mathrm {h} }} of order 2, often denoted as  C s {\displaystyle C_{s}} .[9] 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.[10] A volume is a measurement of a region in three-dimensional space.[11] 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.[12]

The 92 Johnson solids  J n {\displaystyle J_{n}} | Solid name  | Image  | Vertices  | Edges  | Faces  | Symmetry group and its order[13] | Surface area and volume[14]

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1  | Equilateral square pyramid |  | 5  | 8  | 5  |  C 4 v {\displaystyle C_{4v}} of order 8  |  A = ( 1 + 3 ) a 2 ≈ 2.7321 a 2 V = 2 6 a 3 ≈ 0.2357 a 3 {\displaystyle {\begin{aligned}A&=\left(1+{\sqrt {3}}\right)a^{2}\\\&\approx 2.7321a^{2}\\\V&={\frac {\sqrt {2}}{6}}a^{3}\\\&\approx 0.2357a^{3}\end{aligned}}}
2  | Pentagonal pyramid |  | 6  | 10  | 6  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = a 2 2 5 2 ( 10 + 5 + 75 + 30 5 ) ≈ 3.8855 a 2 V = ( 5 + 5 24 ) a 3 ≈ 0.3015 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
3  | Triangular cupola |  | 9  | 15  | 8  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = ( 3 + 5 3 2 ) a 2 ≈ 7.3301 a 2 V = ( 5 3 2 ) a 3 ≈ 1.1785 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
4  | Square cupola |  | 12  | 20  | 10  |  C 4 v {\displaystyle C_{4v}} of order 8  |  A = ( 7 + 2 2 + 3 ) a 2 ≈ 11.5605 a 2 V = ( 1 + 2 2 3 ) a 3 ≈ 1.9428 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
5  | Pentagonal cupola |  | 15  | 25  | 12  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = ( 1 4 ( 20 + 5 3 + 5 ( 145 + 62 5 ) ) ) a 2 ≈ 16.5798 a 2 V = ( 1 6 ( 5 + 4 5 ) ) a 3 ≈ 2.3241 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
6  | Pentagonal rotunda |  | 20  | 35  | 17  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = ( 1 2 ( 5 3 + 10 ( 65 + 29 5 ) ) ) a 2 ≈ 22.3472 a 2 V = ( 1 12 ( 45 + 17 5 ) ) a 3 ≈ 6.9178 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
7  | Elongated triangular pyramid |  | 7  | 12  | 7  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = ( 3 + 3 ) a 2 ≈ 4.7321 a 2 V = ( 1 12 ( 2 + 3 3 ) ) a 3 ≈ 0.5509 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
8  | Elongated square pyramid |  | 9  | 16  | 9  |  C 4 v {\displaystyle C_{4v}} of order 8  |  A = ( 5 + 3 ) a 2 ≈ 6.7321 a 2 V = ( 1 + 2 6 ) a 3 ≈ 1.2357 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
9  | Elongated pentagonal pyramid |  | 11  | 20  | 11  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 20 + 5 3 + 25 + 10 5 4 a 2 ≈ 8.8855 a 2 V = ( 5 + 5 + 6 25 + 10 5 24 ) a 3 ≈ 2.022 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
10  | Gyroelongated square pyramid |  | 9  | 20  | 13  |  C 4 v {\displaystyle C_{4v}} of order 8  |  A = ( 1 + 3 3 ) a 2 ≈ 6.1962 a 2 V = 1 6 ( 2 + 2 4 + 3 2 ) a 3 ≈ 1.1927 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
11  | Gyroelongated pentagonal pyramid |  | 11  | 25  | 16  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 15 3 + 5 ( 5 + 2 5 ) ) a 2 ≈ 8.2157 a 2 V = 1 24 ( 25 + 9 5 ) a 3 ≈ 1.8802 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
12  | Triangular bipyramid |  | 5  | 9  | 6  |  D 3 h {\displaystyle D_{3h}} of order 12  |  A = 3 3 2 a 2 ≈ 2.5981 a 2 V = 2 6 a 3 ≈ 0.2358 a 3 {\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}a^{2}\\\&\approx 2.5981a^{2}\\\V&={\frac {\sqrt {2}}{6}}a^{3}\\\&\approx 0.2358a^{3}\end{aligned}}}
13  | Pentagonal bipyramid |  | 7  | 15  | 10  |  D 5 h {\displaystyle D_{5h}} of order 20  |  A = 5 3 2 a 2 ≈ 4.3301 a 2 V = 1 12 ( 5 + 5 ) a 3 ≈ 0.603 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
14  | Elongated triangular bipyramid |  | 8  | 15  | 9  |  D 3 h {\displaystyle D_{3h}} of order 12  |  A = 3 2 ( 2 + 3 ) a 2 ≈ 5.5981 a 2 V = 1 12 ( 2 2 + 3 3 ) a 3 ≈ 0.6687 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
15  | Elongated square bipyramid |  | 10  | 20  | 12  |  D 4 h {\displaystyle D_{4h}} of order 16  |  A = 2 ( 2 + 3 ) a 2 ≈ 7.4641 a 2 V = 1 3 ( 3 + 2 ) a 3 ≈ 1.4714 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
16  | Elongated pentagonal bipyramid |  | 12  | 25  | 15  |  D 5 h {\displaystyle D_{5h}} of order 20  |  A = 5 2 ( 2 + 3 ) a 2 ≈ 9.3301 a 2 V = 1 12 ( 5 + 5 + 3 5 ( 5 + 2 5 ) ) a 3 ≈ 2.3235 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
17  | Gyroelongated square bipyramid |  | 10  | 24  | 16  |  D 4 d {\displaystyle D_{4d}} of order 16  |  A = 4 3 a 2 ≈ 6.9282 a 2 V = 1 3 ( 2 + 4 + 3 2 ) a 3 ≈ 1.4284 a 3 {\displaystyle {\begin{aligned}A&=4{\sqrt {3}}a^{2}\\\&\approx 6.9282a^{2}\\\V&={\frac {1}{3}}\left({\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\\&\approx 1.4284a^{3}\end{aligned}}}
18  | Elongated triangular cupola |  | 15  | 27  | 14  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 2 ( 18 + 5 3 ) a 2 ≈ 13.3301 a 2 V = 1 6 ( 5 2 + 9 3 ) a 3 ≈ 3.7766 a 3 {\displaystyle {\begin{aligned}A&={\frac {1}{2}}\left(18+5{\sqrt {3}}\right)a^{2}\\\&\approx 13.3301a^{2}\\\V&={\frac {1}{6}}\left(5{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\\&\approx 3.7766a^{3}\end{aligned}}}
19  | Elongated square cupola |  | 20  | 36  | 18  |  C 4 v {\displaystyle C_{4v}} of order 8  |  A = ( 15 + 2 2 + 3 ) a 2 ≈ 19.5605 a 2 V = ( 3 + 8 2 3 ) a 3 ≈ 6.7712 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
20  | Elongated pentagonal cupola |  | 25  | 45  | 22  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 60 + 5 3 + 10 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 26.5798 a 2 V = 1 6 ( 5 + 4 5 + 15 5 + 2 5 ) a 3 ≈ 10.0183 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
21  | Elongated pentagonal rotunda |  | 30  | 55  | 27  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 2 a 2 ( 20 + 5 3 + 5 5 + 2 5 + 3 5 ( 5 + 2 5 ) ) ≈ 32.3472 a 2 V = 1 12 a 3 ( 45 + 17 5 + 30 5 + 2 5 ) ≈ 14.612 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
22  | Gyroelongated triangular cupola |  | 15  | 33  | 20  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 2 ( 6 + 11 3 ) a 2 ≈ 12.5263 a 2 V = 1 3 61 2 + 18 3 + 30 1 + 3 a 3 ≈ 3.5161 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
23  | Gyroelongated square cupola |  | 20  | 44  | 26  |  C 4 v {\displaystyle C_{4v}} of order 8  |  A = ( 7 + 2 2 + 5 3 ) a 2 ≈ 18.4887 a 2 V = ( 1 + 2 3 2 + 2 3 4 + 2 2 + 2 146 + 103 2 ) a 3 ≈ 6.2108 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
24  | Gyroelongated pentagonal cupola |  | 25  | 55  | 32  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 20 + 25 3 + 10 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 25.2400 a 2 V = ( 5 6 + 2 3 5 + 5 6 2 650 + 290 5 − 2 5 − 2 ) a 3 ≈ 9.0733 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
25  | Gyroelongated pentagonal rotunda |  | 30  | 65  | 37  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 2 ( 15 3 + ( 5 + 3 5 ) 5 + 2 5 ) a 2 ≈ 31.0075 a 2 V = ( 45 12 + 17 12 5 + 5 6 2 650 + 290 5 − 2 5 − 2 ) a 3 ≈ 13.6671 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
26  | Gyrobifastigium |  | 8  | 14  | 8  |  D 2 d {\displaystyle D_{2d}} of order 8  |  A = ( 4 + 3 ) a 2 ≈ 5.7321 a 2 V = ( 3 2 ) a 3 ≈ 0.866 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
27  | Triangular orthobicupola |  | 12  | 24  | 14  |  D 3 h {\displaystyle D_{3h}} of order 12  |  A = 2 ( 3 + 3 ) a 2 ≈ 9.4641 a 2 V = 5 2 3 a 3 ≈ 2.357 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
28  | Square orthobicupola |  | 16  | 32  | 18  |  D 4 h {\displaystyle D_{4h}} of order 16  |  A = 2 ( 5 + 3 ) a 2 ≈ 13.4641 a 2 V = ( 2 + 4 2 3 ) a 3 ≈ 3.8856 a 3 {\displaystyle {\begin{aligned}A&=2(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{aligned}}}
29  | Square gyrobicupola |  | 16  | 32  | 18  |  D 4 d {\displaystyle D_{4d}} of order 16  |  A = 2 ( 5 + 3 ) a 2 ≈ 13.4641 a 2 V = ( 2 + 4 2 3 ) a 3 ≈ 3.8856 a 3 {\displaystyle {\begin{aligned}A&=2(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{aligned}}}
30  | Pentagonal orthobicupola |  | 20  | 40  | 22  |  D 5 h {\displaystyle D_{5h}} of order 20  |  A = ( 10 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 ≈ 17.7711 a 2 V = 1 3 ( 5 + 4 5 ) a 3 ≈ 4.6481 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
31  | Pentagonal gyrobicupola |  | 20  | 40  | 22  |  D 5 d {\displaystyle D_{5d}} of order 20  |  A = ( 10 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 ≈ 17.7711 a 2 V = 1 3 ( 5 + 4 5 ) a 3 ≈ 4.6481 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
32  | Pentagonal orthocupolarotunda |  | 25  | 50  | 27  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = ( 5 + 1 4 1900 + 490 5 + 210 75 + 30 5 ) a 2 ≈ 23.5385 a 2 V = 5 12 ( 11 + 5 5 ) a 3 ≈ 9.2418 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
33  | Pentagonal gyrocupolarotunda |  | 25  | 50  | 27  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = ( 5 + 15 4 3 + 7 4 25 + 10 5 ) a 2 ≈ 23.5385 a 2 V = 5 12 ( 11 + 5 5 ) a 3 ≈ 9.2418 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
34  | Pentagonal orthobirotunda |  | 30  | 60  | 32  |  D 5 h {\displaystyle D_{5h}} of order 20  |  A = ( ( 5 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 29.306 a 2 V = 1 6 ( 45 + 17 5 ) a 3 ≈ 13.8355 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
35  | Elongated triangular orthobicupola |  | 18  | 36  | 20  |  D 3 h {\displaystyle D_{3h}} of order 12  |  A = 2 ( 6 + 3 ) a 2 ≈ 15.4641 a 2 V = ( 5 2 3 + 3 3 2 ) a 3 ≈ 4.9551 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
36  | Elongated triangular gyrobicupola |  | 18  | 36  | 20  |  D 3 d {\displaystyle D_{3d}} of order 12  |  A = 2 ( 6 + 3 ) a 2 ≈ 15.4641 a 2 V = ( 5 2 3 + 3 3 2 ) a 3 ≈ 4.9551 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
37  | Elongated square gyrobicupola |  | 24  | 48  | 26  |  D 4 d {\displaystyle D_{4d}} of order 16  |  A = 2 ( 9 + 3 ) a 2 ≈ 21.4641 a 2 V = ( 4 + 10 2 3 ) a 3 ≈ 8.714 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
38  | Elongated pentagonal orthobicupola |  | 30  | 60  | 32  |  D 5 h {\displaystyle D_{5h}} of order 20  |  A = ( 20 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 ≈ 27.7711 a 2 V = 1 6 ( 10 + 8 5 + 15 5 + 2 5 ) a 3 ≈ 12.3423 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
39  | Elongated pentagonal gyrobicupola |  | 30  | 60  | 32  |  D 5 d {\displaystyle D_{5d}} of order 20  |  A = ( 20 + 5 2 ( 10 + 5 + 75 + 30 5 ) ) a 2 ≈ 27.7711 a 2 V = 1 6 ( 10 + 8 5 + 15 5 + 2 5 ) a 3 ≈ 12.3423 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
40  | Elongated pentagonal orthocupolarotunda |  | 35  | 70  | 37  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 60 + 10 ( 190 + 49 5 + 21 75 + 30 5 ) ) a 2 ≈ 33.5385 a 2 V = 5 12 ( 11 + 5 5 + 6 5 + 2 5 ) a 3 ≈ 16.936 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
41  | Elongated pentagonal gyrocupolarotunda |  | 35  | 70  | 37  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 60 + 10 ( 190 + 49 5 + 21 75 + 30 5 ) ) a 2 ≈ 33.5385 a 2 V = 5 12 ( 11 + 5 5 + 6 5 + 2 5 ) a 3 ≈ 16.936 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
42  | Elongated pentagonal orthobirotunda |  | 40  | 80  | 42  |  D 5 h {\displaystyle D_{5h}} of order 20  |  A = ( 10 + 30 ( 10 + 3 5 + 75 + 30 5 ) ) a 2 ≈ 39.306 a 2 V = 1 6 ( 45 + 17 5 + 15 5 + 2 5 ) a 3 ≈ 21.5297 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
43  | Elongated pentagonal gyrobirotunda |  | 40  | 80  | 42  |  D 5 d {\displaystyle D_{5d}} of order 20  |  A = ( 10 + 30 ( 10 + 3 5 + 75 + 30 5 ) ) a 2 ≈ 39.306 a 2 V = 1 6 ( 45 + 17 5 + 15 5 + 2 5 ) a 3 ≈ 21.5297 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
44  | Gyroelongated triangular bicupola |  | 18  | 42  | 26  |  D 3 {\displaystyle D_{3}} of order 6  |  A = ( 6 + 5 3 ) a 2 ≈ 14.6603 a 2 V = 2 ( 5 3 + 1 + 3 ) a 3 ≈ 4.6946 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
45  | Gyroelongated square bicupola |  | 24  | 56  | 34  |  D 4 {\displaystyle D_{4}} of order 8  |  A = ( 10 + 6 3 ) a 2 ≈ 20.3923 a 2 V = ( 2 + 4 3 2 + 2 3 4 + 2 2 + 2 146 + 103 2 ) a 3 ≈ 8.1536 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
46  | Gyroelongated pentagonal bicupola |  | 30  | 70  | 42  |  D 5 {\displaystyle D_{5}} of order 10  |  A = 1 2 ( 20 + 15 3 + 25 + 10 5 ) a 2 ≈ 26.4313 a 2 V = ( 5 3 + 4 3 5 + 5 6 2 650 + 290 5 − 2 5 − 2 ) a 3 ≈ 11.3974 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
47  | Gyroelongated pentagonal cupolarotunda |  | 35  | 80  | 47  |  C 5 {\displaystyle C_{5}} of order 5  |  A = 1 4 ( 20 + 35 3 + 7 25 + 10 5 ) a 2 ≈ 32.1988 a 2 V = ( 55 12 + 25 12 5 + 5 6 2 650 + 290 5 − 2 5 − 2 ) a 3 ≈ 15.9911 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
48  | Gyroelongated pentagonal birotunda |  | 40  | 90  | 52  |  D 5 {\displaystyle D_{5}} of order 10  |  A = ( 10 3 + 3 25 + 10 5 ) a 2 ≈ 37.9662 a 2 V = ( 45 6 + 17 6 5 + 5 6 2 650 + 290 5 − 2 5 − 2 ) a 3 ≈ 20.5848 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
49  | Augmented triangular prism |  | 7  | 13  | 8  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 1 2 ( 4 + 3 3 ) a 2 ≈ 4.5981 a 2 V = 1 12 ( 2 2 + 3 3 ) a 3 ≈ 0.6687 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
50  | Biaugmented triangular prism |  | 8  | 17  | 11  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 1 2 ( 2 + 5 3 ) a 2 ≈ 5.3301 a 2 V = 59 144 + 1 6 a 3 ≈ 0.9044 a 3 {\displaystyle {\begin{aligned}A&={\frac {1}{2}}(2+5{\sqrt {3}})a^{2}\\\&\approx 5.3301a^{2}\\\V&={\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}a^{3}\\\&\approx 0.9044a^{3}\end{aligned}}}
51  | Triaugmented triangular prism |  | 9  | 21  | 14  |  D 3 h {\displaystyle D_{3h}} of order 12  |  A = 7 3 2 a 2 ≈ 6.0622 a 2 V = 2 2 + 3 4 a 3 ≈ 1.1401 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
52  | Augmented pentagonal prism |  | 11  | 19  | 10  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 1 2 ( 8 + 2 3 + 5 ( 5 + 2 5 ) ) a 2 ≈ 9.173 a 2 V = 1 12 233 + 90 5 + 12 50 + 20 5 a 3 ≈ 1.9562 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
53  | Biaugmented pentagonal prism |  | 12  | 23  | 13  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 1 2 a 2 ( 6 + 4 3 + 5 ( 5 + 2 5 ) ) ≈ 9.9051 a 2 V = 1 12 a 3 257 + 90 5 + 24 50 + 20 5 ≈ 2.1919 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
54  | Augmented hexagonal prism |  | 13  | 22  | 11  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = ( 5 + 4 3 ) a 2 ≈ 11.9282 a 2 V = 1 6 ( 2 + 9 3 ) a 3 ≈ 2.8338 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
55  | Parabiaugmented hexagonal prism |  | 14  | 26  | 14  |  D 2 h {\displaystyle D_{2h}} of order 8  |  A = ( 4 + 5 3 ) a 2 ≈ 12.6603 a 2 V = 1 6 ( 2 2 + 9 3 ) a 3 ≈ 3.0695 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
56  | Metabiaugmented hexagonal prism |  | 14  | 26  | 14  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = ( 4 + 5 3 ) a 2 ≈ 12.6603 a 2 V = 1 6 ( 2 2 + 9 3 ) a 3 ≈ 3.0695 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
57  | Triaugmented hexagonal prism |  | 15  | 30  | 17  |  D 3 h {\displaystyle D_{3h}} of order 12  |  A = 3 ( 1 + 2 3 ) a 2 ≈ 13.3923 a 2 V = ( 1 2 + 3 3 2 ) a 3 ≈ 3.3052 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
58  | Augmented dodecahedron |  | 21  | 35  | 16  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 5 3 + 11 5 ( 5 + 2 5 ) ) a 2 ≈ 21.0903 a 2 V = 1 24 ( 95 + 43 5 ) a 3 ≈ 7.9646 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
59  | Parabiaugmented dodecahedron |  | 22  | 40  | 20  |  D 5 d {\displaystyle D_{5d}} of order 20  |  A = 5 2 ( 3 + 5 ( 5 + 2 5 ) ) a 2 ≈ 21.5349 a 2 V = 1 6 ( 25 + 11 5 ) a 3 ≈ 8.2661 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
60  | Metabiaugmented dodecahedron |  | 22  | 40  | 20  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 5 2 ( 3 + 5 ( 5 + 2 5 ) ) a 2 ≈ 21.5349 a 2 V = 1 6 ( 25 + 11 5 ) a 3 ≈ 8.2661 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
61  | Triaugmented dodecahedron |  | 23  | 45  | 24  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 3 4 ( 5 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 21.9795 a 2 V = 5 8 ( 7 + 3 5 ) a 3 ≈ 8.5676 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
62  | Metabidiminished icosahedron |  | 10  | 20  | 12  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 1 2 ( 5 3 + 5 ( 5 + 2 5 ) ) a 2 ≈ 7.7711 a 2 V = 1 6 ( 5 + 2 5 ) a 3 ≈ 1.5787 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
63  | Tridiminished icosahedron |  | 9  | 15  | 8  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 4 ( 5 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 7.3265 a 2 V = ( 5 8 + 7 5 24 ) a 3 ≈ 1.2772 a 3 {\displaystyle {\begin{aligned}A&={\frac {1}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\\&\approx 7.3265a^{2}\\\V&=\left({\frac {5}{8}}+{\frac {7{\sqrt {5}}}{24}}\right)a^{3}\\\&\approx 1.2772a^{3}\end{aligned}}}
64  | Augmented tridiminished icosahedron |  | 10  | 18  | 10  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 4 ( 7 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 8.1925 a 2 V = 1 24 ( 15 + 2 2 + 7 5 ) a 3 ≈ 1.395 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
65  | Augmented truncated tetrahedron |  | 15  | 27  | 14  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 2 ( 6 + 13 3 ) a 2 ≈ 14.2583 a 2 V = 11 2 2 a 3 ≈ 3.8891 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
66  | Augmented truncated cube |  | 28  | 48  | 22  |  C 4 v {\displaystyle C_{4v}} of order 8  |  A = ( 15 + 10 2 + 3 3 ) a 2 ≈ 34.3383 a 2 V = ( 8 + 16 2 3 ) a 3 ≈ 15.5425 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
67  | Biaugmented truncated cube |  | 32  | 60  | 30  |  D 4 h {\displaystyle D_{4h}} of order 16  |  A = 2 ( 9 + 4 2 + 2 3 ) a 2 ≈ 36.2419 a 2 V = ( 9 + 6 2 ) a 3 ≈ 17.4853 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
68  | Augmented truncated dodecahedron |  | 65  | 105  | 42  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 20 + 25 3 + 110 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 102.1821 a 2 V = ( 505 12 + 81 5 4 ) a 3 ≈ 87.3637 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
69  | Parabiaugmented truncated dodecahedron |  | 70  | 120  | 52  |  D 5 d {\displaystyle D_{5d}} of order 20  |  A = 1 2 ( 20 + 15 3 + 50 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 103.3734 a 2 V = 1 12 ( 515 + 251 5 ) a 3 ≈ 89.6878 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
70  | Metabiaugmented truncated dodecahedron |  | 70  | 120  | 52  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 1 2 ( 20 + 15 3 + 50 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 103.3734 a 2 V = 1 12 ( 515 + 251 5 ) a 3 ≈ 89.6878 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
71  | Triaugmented truncated dodecahedron |  | 75  | 135  | 62  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 4 ( 60 + 35 3 + 90 5 + 2 5 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 104.5648 a 2 V = 7 12 ( 75 + 37 5 ) a 3 ≈ 92.0118 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
72  | Gyrate rhombicosidodecahedron |  | 60  | 120  | 62  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = ( 30 + 5 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 59.306 a 2 V = ( 20 + 29 5 3 ) a 3 ≈ 41.6153 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
73  | Parabigyrate rhombicosidodecahedron |  | 60  | 120  | 62  |  D 5 d {\displaystyle D_{5d}} of order 20  |  A = ( 30 + 5 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 59.306 a 2 V = ( 20 + 29 5 3 ) a 3 ≈ 41.6153 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
74  | Metabigyrate rhombicosidodecahedron |  | 60  | 120  | 62  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = ( 30 + 5 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 59.306 a 2 V = ( 20 + 29 5 3 ) a 3 ≈ 41.6153 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
75  | Trigyrate rhombicosidodecahedron |  | 60  | 120  | 62  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = ( 30 + 5 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 59.306 a 2 V = ( 20 + 29 5 3 ) a 3 ≈ 41.6153 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
76  | Diminished rhombicosidodecahedron |  | 55  | 105  | 52  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 100 + 15 3 + 10 5 + 2 5 + 11 5 ( 5 + 2 5 ) ) a 2 ≈ 58.1147 a 2 V = ( 115 6 + 9 5 ) a 3 ≈ 39.2913 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
77  | Paragyrate diminished rhombicosidodecahedron |  | 55  | 105  | 52  |  C 5 v {\displaystyle C_{5v}} of order 10  |  A = 1 4 ( 100 + 15 3 + 10 5 + 2 5 + 11 5 ( 5 + 2 5 ) ) a 2 ≈ 58.1147 a 2 V = ( 115 6 + 9 5 ) a 3 ≈ 39.2913 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
78  | Metagyrate diminished rhombicosidodecahedron |  | 55  | 105  | 52  |  C s {\displaystyle C_{s}} of order 2  |  A = 1 4 ( 100 + 15 3 + 10 5 + 2 5 + 11 5 ( 5 + 2 5 ) ) a 2 ≈ 58.1147 a 2 V = ( 115 6 + 9 5 ) a 3 ≈ 39.2913 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
79  | Bigyrate diminished rhombicosidodecahedron |  | 55  | 105  | 52  |  C s {\displaystyle C_{s}} of order 2  |  A = 1 4 ( 100 + 15 3 + 10 5 + 2 5 + 11 5 ( 5 + 2 5 ) ) a 2 ≈ 58.1147 a 2 V = ( 115 6 + 9 5 ) a 3 ≈ 39.2913 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
80  | Parabidiminished rhombicosidodecahedron |  | 50  | 90  | 42  |  D 5 d {\displaystyle D_{5d}} of order 20  |  A = 5 2 ( 8 + 3 + 2 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 56.9233 a 2 V = 5 3 ( 11 + 5 5 ) a 3 ≈ 36.9672 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
81  | Metabidiminished rhombicosidodecahedron |  | 50  | 90  | 42  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 5 2 ( 8 + 3 + 2 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 56.9233 a 2 V = 5 3 ( 11 + 5 5 ) a 3 ≈ 36.9672 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
82  | Gyrate bidiminished rhombicosidodecahedron |  | 50  | 90  | 42  |  C s {\displaystyle C_{s}} of order 2  |  A = 5 2 ( 8 + 3 + 2 5 + 2 5 + 5 ( 5 + 2 5 ) ) a 2 ≈ 56.9233 a 2 V = 5 3 ( 11 + 5 5 ) a 3 ≈ 36.9672 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
83  | Tridiminished rhombicosidodecahedron |  | 45  | 75  | 32  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 4 ( 60 + 5 3 + 30 5 + 2 5 + 9 5 ( 5 + 2 5 ) ) a 2 ≈ 55.732 a 2 V = ( 35 2 + 23 5 3 ) a 3 ≈ 34.6432 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
84  | Snub disphenoid |  | 8  | 18  | 12  |  D 2 d {\displaystyle D_{2d}} of order 8  |  A = 3 3 a 2 ≈ 5.1962 a 2 V ≈ 0.8595 a 3 {\displaystyle {\begin{aligned}A&=3{\sqrt {3}}a^{2}\\\&\approx 5.1962a^{2}\\\V&\approx 0.8595a^{3}\end{aligned}}}
85  | Snub square antiprism |  | 16  | 40  | 26  |  D 4 d {\displaystyle D_{4d}} of order 16  |  A = 2 ( 1 + 3 3 ) a 2 ≈ 12.3923 a 2 V ≈ 3.6012 a 3 {\displaystyle {\begin{aligned}A&=2\left(1+3{\sqrt {3}}\right)a^{2}\\\&\approx 12.3923a^{2}\\\V&\approx 3.6012a^{3}\end{aligned}}}
86  | Sphenocorona |  | 10  | 22  | 14  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = ( 2 + 3 3 ) a 2 ≈ 7.1962 a 2 V = 1 2 a 3 1 + 3 3 2 + 13 + 3 6 ≈ 1.5154 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
87  | Augmented sphenocorona |  | 11  | 26  | 17  |  C s {\displaystyle C_{s}} of order 2  |  A = ( 1 + 4 3 ) a 2 ≈ 7.9282 a 2 V = 1 2 a 3 1 + 3 3 2 + 13 + 3 6 + 1 3 2 ≈ 1.7511 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
88  | Sphenomegacorona |  | 12  | 28  | 18  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 2 ( 1 + 2 3 ) a 2 ≈ 8.9282 a 2 V ≈ 1.9481 a 3 {\displaystyle {\begin{aligned}A&=2\left(1+2{\sqrt {3}}\right)a^{2}\\\&\approx 8.9282a^{2}\\\V&\approx 1.9481a^{3}\end{aligned}}}
89  | Hebesphenomegacorona |  | 14  | 33  | 21  |  C 2 v {\displaystyle C_{2v}} of order 4  |  A = 3 2 ( 2 + 3 3 ) a 2 ≈ 10.7942 a 2 V ≈ 2.9129 a 3 {\displaystyle {\begin{aligned}A&={\frac {3}{2}}\left(2+3{\sqrt {3}}\right)a^{2}\\\&\approx 10.7942a^{2}\\\V&\approx 2.9129a^{3}\end{aligned}}}
90  | Disphenocingulum |  | 16  | 38  | 24  |  D 2 d {\displaystyle D_{2d}} of order 8  |  A = ( 4 + 5 3 ) a 2 ≈ 12.6603 a 2 V ≈ 3.7776 a 3 {\displaystyle {\begin{aligned}A&=(4+5{\sqrt {3}})a^{2}\\\&\approx 12.6603a^{2}\\\V&\approx 3.7776a^{3}\end{aligned}}}
91  | Bilunabirotunda |  | 14  | 26  | 14  |  D 2 h {\displaystyle D_{2h}} of order 8  |  A = ( 2 + 2 3 + 5 ( 5 + 2 5 ) ) a 2 ≈ 12.346 a 2 V = 1 12 ( 17 + 9 5 ) a 3 ≈ 3.0937 a 3 {\displaystyle {\begin{aligned}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{aligned}}}
92  | Triangular hebesphenorotunda |  | 18  | 36  | 20  |  C 3 v {\displaystyle C_{3v}} of order 6  |  A = 1 4 ( 12 + 19 3 + 3 5 ( 5 + 2 5 ) ) a 2 ≈ 16.3887 a 2 V = ( 5 2 + 7 5 6 ) a 3 ≈ 5.1087 a 3 {\displaystyle {\begin{aligned}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{aligned}}}

## References

[edit]

1. ^ Araki, Horiyama & Uehara (2015).
2. ^

* Diudea (2018), p. 39
* Todesco (2020), p. 282
* Williams & Monteleone (2021), p. 23

3. ^

* Johnson (1966)
* Zalgaller (1969)

4. ^

* Cromwell (1997), p. 86–87, See the figure on p.89
* Johnson (1966)

5. ^

* Rajwade (2001), p. 84–88
* Slobodan, Obradović & Ðukanović (2015)
* Berman (1971), p. 350

6. ^ Uehara (2020), p. 62.
7. ^

* Powell (2010), p. 27
* Solomon (2003), p. 40

8. ^ Flusser, Suk & Zitofa (2017), p. 126.
9. ^

* Flusser, Suk & Zitofa (2017), p. 126
* Hergert & Geilhufe (2018), p. 56

10. ^ Walsh (2014), p. 284.
11. ^ Parker (1997), p. 264.
12. ^

* Cromwell (1997), p. 36
* Berman (1971)
* Timofeenko (2009)

13. ^ Johnson (1966).
14. ^ Berman (1971).

## Bibliography

[edit]

* Araki, Yoshiaki; Horiyama, Takashi; Uehara, Ryuhei (2015). "Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid". In Rahman, M. Sohel; Tomita, Etsuji (eds.). WALCOM: Algorithms and Computation. Lecture Notes in Computer Science. Vol. 8973\. Cham: Springer International Publishing. pp. 294–305\. doi:10.1007/978-3-319-15612-5_26. ISBN 978-3-319-15612-5.
* Berman, M. (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352\. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
* Cromwell, P. R. (1997). Polyhedra. Cambridge University Press. ISBN 978-0-521-66405-9.
* Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10\. Springer. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
* Flusser, J.; Suk, T.; Zitofa, B. (2017). 2D and 3D Image Analysis by Moments. John Wiley & Sons. ISBN 978-1-119-03935-8.
* Hergert, W.; Geilhufe, M. (2018). Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica. John Wiley & Sons. ISBN 978-3-527-41300-3.
* Holme, A. (2010). Geometry: Our Cultural Heritage. Springer. doi:10.1007/978-3-642-14441-7. ISBN 978-3-642-14441-7.
* Johnson, N. (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200\. doi:10.4153/CJM-1966-021-8.
* Litchenberg, D. R. (1988). "Pyramids, Prisms, Antiprisms, and Deltahedra". The Mathematics Teacher. 81 (4): 261–265\. doi:10.5951/MT.81.4.0261. JSTOR 27965792.
* Meyer, W. (2006). Geometry and Its Applications. Academic Press. ISBN 978-0-123-69427-0.
* Parker, S. P. (1997). Dictionary of Mathematics. McGraw Hill. ISBN 978-0-070-52433-0.
* Powell, R. C. (2010). Symmetry, Group Theory, and the Physical Properties of Crystals. Lecture Notes in Physics. Vol. 824\. Springer. doi:10.1007/978-1-4419-7598-0. ISBN 978-1-441-97598-0.
* 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 978-9-386-27906-4.
* 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.
* 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". In Emmer, M.; Abate, M. (eds.). Imagine Math 7: Between Culture and Mathematics. Springer. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8.
* Uehara, R. (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. doi:10.1007/978-981-15-4470-5. ISBN 978-9-811-54470-5.
* Walsh, E. T. (2014). A First Course in Geometry. Dover. ISBN 978-0-486-78020-7.
* Williams, K.; Monteleone, C. (2021). Daniele Barbaro's Perspective of 1568. Springer. doi:10.1007/978-3-030-76687-0. ISBN 978-3-030-76687-0.
* Zalgaller, V. A. (1969). Convex Polyhedra with Regular Faces. Springer. ISBN 978-1-489-95671-2.

## External links

[edit]

* 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.

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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
* 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

Other 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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