List of Johnson solids

In geometry, polyhedra are three-dimensional objects where points are connected by lines to form polygons. The points, lines, and polygons of a polyhedron are referred to as its vertices, edges, and faces, respectively.^[1] A polyhedron is considered to be convex if:^[2]

The shortest path between any two of its vertices lies either within its interior or on its boundary.
None of its faces are coplanar—they do not share the same plane and do not "lie flat".
None of its edges are colinear—they are not segments of the same line.

A convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid. Some authors exclude uniform polyhedra from the definition. A uniform polyhedron is a polyhedron in which the faces are regular and they are isogonal; examples include Platonic and Archimedean solids as well as prisms and antiprisms.^[3] The Johnson solids are named after American mathematician Norman Johnson (1930–2017), who published a list of 92 such 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.^[4] Some of the 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 Johnson solids satisfying this criteria are the first six—equilateral square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda. The criteria 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.^[5] 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.^[6] The following table contains the 92 Johnson solids, with edge length $a$ . The table includes the solid's enumeration (denoted as $J_{n}$ ).^[7] It also includes the number of vertices, edges, and faces of each solid, as well as its symmetry group, surface area $A$ , and volume $V$ . 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 $\frac{36 0^{\circ}}{n}$ is denoted by $C_{n}$ , a cyclic group of order $n$ ; combining this with the reflection symmetry results in the symmetry of dihedral group $D_{n}$ of order $2 n$ .^[8] 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 $C_{n v}$ of order $2 n$ . The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry $D_{n h}$ of order $4 n$ . The antiprismatic symmetry $D_{n d}$ of order $4 n$ preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.^[9] The symmetry group $C_{n h}$ of order $2 n$ 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 $C_{1 h}$ of order 2, often denoted as $C_{s}$ .^[10] 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.^[11] A volume is a measurement of a region in three-dimensional space.^[12] 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.^[13]

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

References

↑ Meyer (2006), p. 418.
↑
- Litchenberg (1988)
- Boissonnat & Yvinec (1989)
↑
- Diudea (2018), p. 39
- Todesco (2020), p. 282
- Williams & Monteleone (2021), p. 23
↑
- Johnson (1966)
- Zalgaller (1969)
↑
- Cromwell (1997), p. 86–87, See the figure on p.89
- Johnson (1966)
↑
- Rajwade (2001), p. 84–88
- Slobodan, Obradović & Ðukanović (2015)
- Berman (1971), p. 350
↑ Uehara (2020), p. 62.
↑
- Powell (2010), p. 27
- Solomon (2003), p. 40
↑ Flusser, Suk & Zitofa (2017), p. 126.
↑
- Flusser, Suk & Zitofa (2017), p. 126
- Hergert & Geilhufe (2018), p. 56
↑ Walsh (2014), p. 284.
↑ Parker (1997), p. 264.
↑
↑ Johnson (1966).
↑ Berman (1971).

Bibliography

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.
Boissonnat, J. D.; Yvinec, M. (June 1989). Probing a scene of non convex polyhedra. Proceedings of the fifth annual symposium on Computational geometry. pp. 237–246. doi:10.1145/73833.73860.
Cromwell, P. R. (1997). Polyhedra. Cambridge University Press. ISBN 978-0-521-66405-9.
Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. 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. 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. 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

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 français) (6): 83–95.

[FOOTNOTEMeyer2006[httpsbooksgooglecombooksidez6H5Ho6E3cCpgPA418_418]-1] Meyer (2006), p. 418.

[2] 
Litchenberg (1988)
Boissonnat & Yvinec (1989)

[3] Litchenberg (1988)

[4] Boissonnat & Yvinec (1989)

[3] 
Diudea (2018), p. 39
Todesco (2020), p. 282
Williams & Monteleone (2021), p. 23

[6] Diudea (2018), p. 39

[7] Todesco (2020), p. 282

[8] Williams & Monteleone (2021), p. 23

[4] 
Johnson (1966)
Zalgaller (1969)

[10] Johnson (1966)

[11] Zalgaller (1969)

[5] 
Cromwell (1997), p. 86–87, See the figure on p.89
Johnson (1966)

[13] Cromwell (1997), p. 86–87, See the figure on p.89

[14] Johnson (1966)

[6] 
Rajwade (2001), p. 84–88
Slobodan, Obradović & Ðukanović (2015)
Berman (1971), p. 350

[16] Rajwade (2001), p. 84–88

[17] Slobodan, Obradović & Ðukanović (2015)

[18] Berman (1971), p. 350

[FOOTNOTEUehara2020[httpsbooksgooglecombooksid51juDwAAQBAJpgPA62_62]-7] Uehara (2020), p. 62.

[8] 
Powell (2010), p. 27
Solomon (2003), p. 40

[21] Powell (2010), p. 27

[22] Solomon (2003), p. 40

[FOOTNOTEFlusserSukZitofa2017[httpsbooksgooglecombooksidjwKLDQAAQBAJpgPA126_126]-9] Flusser, Suk & Zitofa (2017), p. 126.

[10] 
Flusser, Suk & Zitofa (2017), p. 126
Hergert & Geilhufe (2018), p. 56

[25] Flusser, Suk & Zitofa (2017), p. 126

[26] Hergert & Geilhufe (2018), p. 56

[FOOTNOTEWalsh2014[httpsbooksgooglecombooksidZhDdAwAAQBAJpgPA284_284]-11] Walsh (2014), p. 284.

[FOOTNOTEParker1997[httpsarchiveorgdetailsmcgrawhilldictio00park_0page264_264]-12] Parker (1997), p. 264.

[13] 
Cromwell (1997), p. 36
Berman (1971)
Timofeenko (2009)

[30] Cromwell (1997), p. 36

[31] Berman (1971)

[32] Timofeenko (2009)

[FOOTNOTEJohnson1966-14] Johnson (1966).

[FOOTNOTEBerman1971-15] Berman (1971).

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

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 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)

List of Johnson solids

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