In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid,[1] is a convex polyhedron whose faces[2] are regular polygons and that is not a uniform polyhedron.[3][4] There are 92 such solids:
- 48 composed of the primitive pyramids, cupolas, and rotundas assembled in various ways together with prisms and antiprisms;
- 35 formed by modifying uniform polyhedra, by augmenting, diminishing, or gyrating with primitives; and
- 9 which are not derived from "cut-and-paste" manipulations of uniform solids.
Definition and background
A convex polyhedron is the convex hull of a finite set of points in 3-dimensional space, not all in a plane.[5] Its boundary is a finite union of polygons, no two in the same plane; those polygons are called the faces. A Johnson solid is a convex polyhedron[2] whose faces are all regular polygons,[6] but not a uniform polyhedron;[3][4] the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms.
The solids are named after Norman Johnson and Victor Zalgaller.[7] Johnson (1966) published a list of 92 such solids and assigned them their names and numbers. Zalgaller (1969) proved Johnson's conjecture[8] that there were none beyond these 92.
A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a near-miss Johnson solid.[9]
Naming and construction of solids
- invalid, - Platonic, - Archimedean, - Gyrated sections.
The naming of Johnson solids follows a flexible and precise descriptive formula that allows many solids to be named in multiple different ways without compromising the accuracy of each name as a description. The names of the Johnson solids are described in the following sections.
Pyramids, cupolas, rotundas
The first 48 Johnson solids are constructed from pyramids, cupolas, or rotundas, combined with prisms or antiprisms. The following prefixes are attached to the word to indicate specific combinations of shapes:[10]
- Bi- indicates that two copies of the solid are joined base-to-base.
- For cupolas and rotundas, ortho- indicates that like faces meet.
- For cupolas and rotundas, gyro- indicates that unlike faces meet.
- Elongated indicates a prism is joined to the base of the solid, or between the bases.
- Gyroelongated indicates an antiprism is joined to the base of the solid, or between the bases.
Using this nomenclature, a pentagonal bipyramid is a solid constructed by attaching two bases of pentagonal pyramids. Triangular orthobicupola is constructed by two triangular cupolas along their bases.
Excluded solids: - coplanar, - Platonic, - Archimedean.
| Fastigium | |
|---|---|
| gyrobi- | 26 Gyrobifastigium  |
Modified uniform polyhedra
The next 35 Johnson solids are constructed by modifying uniform polyhedra such as prisms, Platonic, or Archimedean solids by adding, subtracting, or rotating pyramids or cupolas. The following prefixes are attached to the word to indicate additions, subtractions, or rotations:[10]
- Augmented indicates a pyramid or cupola is added to one or more faces of the solid in question.
- Diminished indicates a pyramid or cupola is removed from one or more faces of the solid in question.
- Gyrate indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up.
The three operations—augmentation, diminution, and gyration—can be performed multiple times for certain large solids. Bi- & Tri- indicate a double and triple operation respectively. For example, a bigyrate solid has two rotated cupolas, and a tridiminished solid has three removed pyramids or cupolas. In certain large solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. Para- indicates the former, that the solid in question has altered parallel faces, and meta- the latter, altered oblique faces. For example, a parabiaugmented solid has had two parallel faces augmented, and a metabigyrate solid has had two oblique cupolas gyrated.[10]
Non cut-and-paste
The last 9 Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson with the following nomenclature:[10]
- A lune is a figure of two triangles attached to opposite sides of a square. Prefixes indicating a complex of lunes are:
- Spheno- is a wedgelike complex of two adjacent lunes. Dispheno- indicates two such complexes.
- Hebespheno- is a blunt complex of three adjacent lunes.
- Suffixes indicating a complex of triangles are:
- -corona is a crownlike complex of eight triangles.
- -megacorona is a larger crownlike complex of twelve triangles.
- -cingulum is a belt of twelve triangles.
- Suffix -rotunda indicates a complex of two or three pentagons with triangles between them, bearing a structural resemblance to the pentagonal rotunda.
| Snub polyhedra | |
|---|---|
| 84 Snub disphenoid  |
85 Snub square antiprism  |
| Formed from lunes and triangles | |
| 86 Sphenocorona  |
87 Augmented sphenocorona  |
| 88 Sphenomegacorona  |
89 Hebesphenomegacorona  |
| 90 Disphenocingulum  | |
| Rotundoids | |
| 91 Bilunabirotunda  |
92 Triangular hebesphenorotunda  |
Notable subsets
Deltahedra
Five Johnson solids are deltahedra, with only triangle faces.
Elementary solids
Seventeen Johnson solids may be categorized as elementary polyhedra, meaning they cannot be separated by a plane to create two small convex polyhedra with regular faces. The first six Johnson solids satisfy this criterion:
- J1 square pyramid
- J2 pentagonal pyramid
- J3 triangular cupola
- J4 square cupola
- J5 pentagonal cupola
- J6 pentagonal rotunda
The criterion is also satisfied by eleven other Johnson solids:[11]
Chiral solids
The five gyroelongated bicupolas or birotundas are chiral and have distinct left-handed and right-handed forms.
Circumscribable solids
Twenty five of the Johnson solids have vertices that exist on the surface of a sphere. All of them can be seen to be related to a Platonic or Archimedean solid by gyration, diminishment, or dissection.[12]
| Octahedron | Icosahedron | |||
|---|---|---|---|---|
J1 |
J2 |
J11 |
J62 |
J63 |
| Cuboctahedron | Rhombicuboctahedron | Icosidodecahedron | ||||
|---|---|---|---|---|---|---|
J3 |
J27 |
J4 |
J19 |
J37 |
J6 |
J34 |
| Rhombicosidodecahedron | ||||||
J5 |
J72 |
J73 |
J74 |
J75 |
J76 |
J77 |
J78 |
J79 |
J80 |
J81 |
J82 |
J83 | |
Characteristics of solids
Every polyhedron has its own characteristics, including symmetry and measurement. An object is said to have symmetry if there is a transformation that maps it to itself. All of those transformations may be composed in a group, alongside the group's number of elements, known as the order. In two-dimensional space, these transformations include rotating around the center of a polygon and reflecting an object around the perpendicular bisector of a polygon. The mensuration of polyhedra includes the surface area and volume. An area is a two-dimensional measurement calculated by the product of length and width; for a polyhedron, the surface area is the sum of the areas of all of its faces.[13] A volume is a measurement of a region in three-dimensional space.[14] 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.[15]
A polygon that is rotated symmetrically by
360
∘
n
{\textstyle {\frac {360^{\circ }}{n}}}
The table below lists the properties of the 92 (non-uniform) Johnson solids. The table includes each solid's enumeration (denoted as
J
n
{\displaystyle J_{n}}
|
J
n
{\displaystyle J_{n}}
|
Solid name | Image | Vertices | Edges | Faces | Symmetry group and order[20] | Surface area, exact[21] | Surface area, approx.[21] |
Volume, exact[21] | Volume, approx.[21] |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | Square pyramid | |
5 | 8 | 5 |
C
4
v
{\displaystyle C_{4v}}
 of order 8 |
1
+
3
{\displaystyle 1+{\sqrt {3}}}
 |
2.7321 |
2
6
{\displaystyle {\frac {\sqrt {2}}{6}}}
 |
0.2357 |
| 2 | Pentagonal pyramid | |
6 | 10 | 6 |
C
5
v
{\displaystyle C_{5v}}
 of order 10 |
1
2
5
2
(
10
+
5
+
15
α
)
{\displaystyle {\frac {1}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {15\alpha }}\right)}}}
 |
3.8855 |
5
+
5
24
{\displaystyle {\frac {5+{\sqrt {5}}}{24}}}
 |
0.3015 |
| 3 | Triangular cupola | |
9 | 15 | 8 |
C
3
v
{\displaystyle C_{3v}}
 of order 6 |
3
+
5
3
2
{\displaystyle 3+{\frac {5{\sqrt {3}}}{2}}}
 |
7.3301 |
5
3
2
{\displaystyle {\frac {5}{3{\sqrt {2}}}}}
 |
1.1785 |
| 4 | Square cupola | |
12 | 20 | 10 |
C
4
v
{\displaystyle C_{4v}}
of order 8 |
7
+
2
2
+
3
{\displaystyle 7+2{\sqrt {2}}+{\sqrt {3}}}
 |
11.5605 |
1
+
2
2
3
{\displaystyle 1+{\frac {2{\sqrt {2}}}{3}}}
 |
1.9428 |
| 5 | Pentagonal cupola | |
15 | 25 | 12 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
+
5
3
+
155
α
−
50
4
{\displaystyle 5+{\frac {5{\sqrt {3}}+{\sqrt {155\alpha -50}}}{4}}}
 |
16.5798 |
5
+
4
5
6
{\displaystyle {\frac {5+4{\sqrt {5}}}{6}}}
 |
2.3241 |
| 6 | Pentagonal rotunda | |
20 | 35 | 17 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
3
+
145
α
−
75
2
{\displaystyle {\frac {5{\sqrt {3}}+{\sqrt {145\alpha -75}}}{2}}}
 |
22.3472 |
45
+
17
5
12
{\displaystyle {\frac {45+17{\sqrt {5}}}{12}}}
 |
6.9178 |
| 7 | Elongated triangular pyramid |  |
7 | 12 | 7 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
3
+
3
{\displaystyle 3+{\sqrt {3}}}
 |
4.7321 |
2
+
3
3
12
{\displaystyle {\frac {{\sqrt {2}}+3{\sqrt {3}}}{12}}}
 |
0.5509 |
| 8 | Elongated square pyramid |  |
9 | 16 | 9 |
C
4
v
{\displaystyle C_{4v}}
of order 8 |
5
+
3
{\displaystyle 5+{\sqrt {3}}}
 |
6.7321 |
1
+
2
6
{\displaystyle 1+{\frac {\sqrt {2}}{6}}}
 |
1.2357 |
| 9 | Elongated pentagonal pyramid |  |
11 | 20 | 11 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
+
5
3
+
5
α
4
{\displaystyle 5+{\frac {5{\sqrt {3}}+{\sqrt {5\alpha }}}{4}}}
 |
8.8855 |
5
+
5
+
6
5
α
24
{\displaystyle {\frac {5+{\sqrt {5}}+6{\sqrt {5\alpha }}}{24}}}
 |
2.022 |
| 10 | Gyroelongated square pyramid |  |
9 | 20 | 13 |
C
4
v
{\displaystyle C_{4v}}
of order 8 |
1
+
3
3
{\displaystyle 1+3{\sqrt {3}}}
 |
6.1962 |
2
+
2
4
+
3
2
6
{\displaystyle {\frac {{\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}}{6}}}
 |
1.1927 |
| 11 | Gyroelongated pentagonal pyramid (diminished icosahedron) | |
11 | 25 | 16 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
15
3
+
5
α
4
{\displaystyle {\frac {15{\sqrt {3}}+{\sqrt {5\alpha }}}{4}}}
 |
8.2157 |
25
+
9
5
24
{\displaystyle {\frac {25+9{\sqrt {5}}}{24}}}
 |
1.8802 |
| 12 | Triangular bipyramid |  |
5 | 9 | 6 |
D
3
h
{\displaystyle D_{3h}}
 of order 12 |
3
3
2
{\displaystyle {\frac {3{\sqrt {3}}}{2}}}
 |
2.5981 |
2
6
{\displaystyle {\frac {\sqrt {2}}{6}}}
|
0.2357 |
| 13 | Pentagonal bipyramid |  |
7 | 15 | 10 |
D
5
h
{\displaystyle D_{5h}}
 of order 20 |
5
3
2
{\displaystyle {\frac {5{\sqrt {3}}}{2}}}
 |
4.3301 |
5
+
5
12
{\displaystyle {\frac {5+{\sqrt {5}}}{12}}}
 |
0.6030 |
| 14 | Elongated triangular bipyramid |  |
8 | 15 | 9 |
D
3
h
{\displaystyle D_{3h}}
of order 12 |
3
+
3
3
2
{\displaystyle 3+{\frac {3{\sqrt {3}}}{2}}}
 |
5.5981 |
2
6
+
3
4
{\displaystyle {\frac {\sqrt {2}}{6}}+{\frac {\sqrt {3}}{4}}}
 |
0.6687 |
| 15 | Elongated square bipyramid |  |
10 | 20 | 12 |
D
4
h
{\displaystyle D_{4h}}
 of order 16 |
4
+
2
3
{\displaystyle 4+2{\sqrt {3}}}
 |
7.4641 |
1
+
2
3
{\displaystyle 1+{\frac {\sqrt {2}}{3}}}
 |
1.4714 |
| 16 | Elongated pentagonal bipyramid |  |
12 | 25 | 15 |
D
5
h
{\displaystyle D_{5h}}
of order 20 |
5
+
5
2
3
{\displaystyle 5+{\frac {5}{2}}{\sqrt {3}}}
 |
9.3301 |
5
+
5
+
3
5
α
12
{\displaystyle {\frac {5+{\sqrt {5}}+3{\sqrt {5\alpha }}}{12}}}
 |
2.3235 |
| 17 | Gyroelongated square bipyramid |  |
10 | 24 | 16 |
D
4
d
{\displaystyle D_{4d}}
 of order 16 |
4
3
{\displaystyle 4{\sqrt {3}}}
 |
6.9282 |
2
+
4
+
3
2
3
{\displaystyle {\frac {{\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}}{3}}}
 |
1.4284 |
| 18 | Elongated triangular cupola |  |
15 | 27 | 14 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
9
+
5
2
3
{\displaystyle 9+{\frac {5}{2}}{\sqrt {3}}}
 |
13.3301 |
5
2
+
9
3
6
{\displaystyle {\frac {5{\sqrt {2}}+9{\sqrt {3}}}{6}}}
 |
3.7766 |
| 19 | Elongated square cupola | |
20 | 36 | 18 |
C
4
v
{\displaystyle C_{4v}}
of order 8 |
15
+
2
2
+
3
{\displaystyle 15+2{\sqrt {2}}+{\sqrt {3}}}
 |
19.5605 |
3
+
8
2
3
{\displaystyle 3+{\frac {8{\sqrt {2}}}{3}}}
 |
6.7712 |
| 20 | Elongated pentagonal cupola |  |
25 | 45 | 22 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
15
+
5
3
+
10
α
+
5
α
4
{\displaystyle 15+{\frac {5{\sqrt {3}}+10{\sqrt {\alpha }}+{\sqrt {5\alpha }}}{4}}}
 |
26.5798 |
5
+
4
5
+
15
α
6
{\displaystyle {\frac {5+4{\sqrt {5}}+15{\sqrt {\alpha }}}{6}}}
 |
10.0183 |
| 21 | Elongated pentagonal rotunda |  |
30 | 55 | 27 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
10
+
5
3
+
5
α
+
3
5
α
2
{\displaystyle 10+{\frac {5{\sqrt {3}}+5{\sqrt {\alpha }}+3{\sqrt {5\alpha }}}{2}}}
 |
32.3472 |
45
+
17
5
+
30
α
12
{\displaystyle {\frac {45+17{\sqrt {5}}+30{\sqrt {\alpha }}}{12}}}
 |
14.612 |
| 22 | Gyroelongated triangular cupola |  |
15 | 33 | 20 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
3
+
11
2
3
{\displaystyle 3+{\frac {11}{2}}{\sqrt {3}}}
 |
12.5263 |
1
3
61
2
+
18
3
+
30
1
+
3
{\displaystyle {\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}}
 |
3.5161 |
| 23 | Gyroelongated square cupola |  |
20 | 44 | 26 |
C
4
v
{\displaystyle C_{4v}}
of order 8 |
7
+
2
2
+
5
3
{\displaystyle 7+2{\sqrt {2}}+5{\sqrt {3}}}
 |
18.4887 |
1
+
2
3
2
+
2
3
4
+
2
2
+
2
146
+
103
2
{\displaystyle 1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}}
 |
6.2108 |
| 24 | Gyroelongated pentagonal cupola |  |
25 | 55 | 32 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
+
25
3
+
10
α
+
5
α
4
{\displaystyle 5+{\frac {25{\sqrt {3}}+10{\sqrt {\alpha }}+{\sqrt {5\alpha }}}{4}}}
 |
25.2400 |
5
6
+
2
3
5
+
5
6
2
145
α
−
75
−
2
5
−
2
{\displaystyle {\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {145\alpha -75}}-2{\sqrt {5}}-2}}}
 |
9.0733 |
| 25 | Gyroelongated pentagonal rotunda |  |
30 | 65 | 37 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
15
3
+
(
α
+
5
)
α
2
{\displaystyle {\frac {15{\sqrt {3}}+\left(\alpha +{\sqrt {5}}\right){\sqrt {\alpha }}}{2}}}
 |
31.0075 |
45
+
17
5
+
10
2
145
α
−
75
−
2
5
−
2
12
{\displaystyle {\frac {45+17{\sqrt {5}}+10{\sqrt {2{\sqrt {145\alpha -75}}-2{\sqrt {5}}-2}}}{12}}}
 |
13.6671 |
| 26 | Gyrobifastigium | |
8 | 14 | 8 |
D
2
d
{\displaystyle D_{2d}}
 of order 8 |
4
+
3
{\displaystyle 4+{\sqrt {3}}}
 |
5.7321 |
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
 |
0.8660 |
| 27 | Triangular orthobicupola | |
12 | 24 | 14 |
D
3
h
{\displaystyle D_{3h}}
of order 12 |
6
+
2
3
{\displaystyle 6+2{\sqrt {3}}}
 |
9.4641 |
5
2
3
{\displaystyle {\frac {5{\sqrt {2}}}{3}}}
 |
2.3570 |
| 28 | Square orthobicupola |  |
16 | 32 | 18 |
D
4
h
{\displaystyle D_{4h}}
of order 16 |
10
+
2
3
{\displaystyle 10+2{\sqrt {3}}}
 |
13.4641 |
2
+
4
2
3
{\displaystyle 2+{\frac {4{\sqrt {2}}}{3}}}
 |
3.8856 |
| 29 | Square gyrobicupola |  |
16 | 32 | 18 |
D
4
d
{\displaystyle D_{4d}}
of order 16 | ||||
| 30 | Pentagonal orthobicupola |  |
20 | 40 | 22 |
D
5
h
{\displaystyle D_{5h}}
of order 20 |
10
+
75
+
5
α
+
10
15
α
2
{\displaystyle 10+{\frac {\sqrt {75+5\alpha +10{\sqrt {15\alpha }}}}{2}}}
 |
17.7711 |
5
+
4
5
3
{\displaystyle {\frac {5+4{\sqrt {5}}}{3}}}
 |
4.6481 |
| 31 | Pentagonal gyrobicupola |  |
20 | 40 | 22 |
D
5
d
{\displaystyle D_{5d}}
 of order 20 | ||||
| 32 | Pentagonal orthocupolarotunda |  |
25 | 50 | 27 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
+
675
+
245
α
+
210
15
α
4
{\displaystyle 5+{\frac {\sqrt {675+245\alpha +210{\sqrt {15\alpha }}}}{4}}}
 |
23.5385 |
55
+
25
5
12
{\displaystyle {\frac {55+25{\sqrt {5}}}{12}}}
 |
9.2418 |
| 33 | Pentagonal gyrocupolarotunda |  |
25 | 50 | 27 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
+
15
4
3
+
7
4
5
α
{\displaystyle 5+{\frac {15}{4}}{\sqrt {3}}+{\frac {7}{4}}{\sqrt {5\alpha }}}
 |
23.5385 | ||
| 34 | Pentagonal orthobirotunda | |
30 | 60 | 32 |
D
5
h
{\displaystyle D_{5h}}
of order 20 |
5
3
+
3
5
α
{\displaystyle 5{\sqrt {3}}+3{\sqrt {5\alpha }}}
 |
29.306 |
45
+
17
5
6
{\displaystyle {\frac {45+17{\sqrt {5}}}{6}}}
 |
13.8355 |
| 35 | Elongated triangular orthobicupola |  |
18 | 36 | 20 |
D
3
h
{\displaystyle D_{3h}}
of order 12 |
12
+
2
3
{\displaystyle 12+2{\sqrt {3}}}
 |
15.4641 |
5
2
3
+
3
3
2
{\displaystyle {\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}}
 |
4.9551 |
| 36 | Elongated triangular gyrobicupola |  |
18 | 36 | 20 |
D
3
d
{\displaystyle D_{3d}}
 of order 12 | ||||
| 37 | Elongated square gyrobicupola |  |
24 | 48 | 26 |
D
4
d
{\displaystyle D_{4d}}
of order 16 |
18
+
2
3
{\displaystyle 18+2{\sqrt {3}}}
 |
21.4641 |
4
+
10
2
3
{\displaystyle 4+{\frac {10{\sqrt {2}}}{3}}}
 |
8.714 |
| 38 | Elongated pentagonal orthobicupola |  |
30 | 60 | 32 |
D
5
h
{\displaystyle D_{5h}}
of order 20 |
20
+
75
+
5
α
+
10
15
α
2
{\displaystyle 20+{\frac {\sqrt {75+5\alpha +10{\sqrt {15\alpha }}}}{2}}}
 |
27.7711 |
10
+
8
5
+
15
α
6
{\displaystyle {\frac {10+8{\sqrt {5}}+15{\sqrt {\alpha }}}{6}}}
 |
12.3423 |
| 39 | Elongated pentagonal gyrobicupola |  |
30 | 60 | 32 |
D
5
d
{\displaystyle D_{5d}}
of order 20 | ||||
| 40 | Elongated pentagonal orthocupolarotunda |  |
35 | 70 | 37 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
15
+
675
+
245
α
+
210
15
α
4
{\displaystyle 15+{\frac {\sqrt {675+245\alpha +210{\sqrt {15\alpha }}}}{4}}}
 |
33.5385 |
55
+
25
5
+
30
α
12
{\displaystyle {\frac {55+25{\sqrt {5}}+30{\sqrt {\alpha }}}{12}}}
 |
16.936 |
| 41 | Elongated pentagonal gyrocupolarotunda |  |
35 | 70 | 37 |
C
5
v
{\displaystyle C_{5v}}
of order 10 | ||||
| 42 | Elongated pentagonal orthobirotunda |  |
40 | 80 | 42 |
D
5
h
{\displaystyle D_{5h}}
of order 20 |
10
+
75
+
45
α
+
30
15
α
{\displaystyle 10+{\sqrt {75+45\alpha +30{\sqrt {15\alpha }}}}}
 |
39.306 |
45
+
17
5
+
15
α
6
{\displaystyle {\frac {45+17{\sqrt {5}}+15{\sqrt {\alpha }}}{6}}}
 |
21.5297 |
| 43 | Elongated pentagonal gyrobirotunda |  |
40 | 80 | 42 |
D
5
d
{\displaystyle D_{5d}}
of order 20 | ||||
| 44 | Gyroelongated triangular bicupola |  |
18 | 42 | 26 |
D
3
{\displaystyle D_{3}}
 of order 6 |
6
+
5
3
{\displaystyle 6+5{\sqrt {3}}}
 |
14.6603 |
5
3
2
+
2
+
2
3
{\displaystyle {\frac {5}{3}}{\sqrt {2}}+{\sqrt {2+2{\sqrt {3}}}}}
 |
4.6946 |
| 45 | Gyroelongated square bicupola |  |
24 | 56 | 34 |
D
4
{\displaystyle D_{4}}
 of order 8 |
10
+
6
3
{\displaystyle 10+6{\sqrt {3}}}
 |
20.3923 |
2
+
4
3
2
+
2
3
4
+
2
2
+
2
146
+
103
2
{\displaystyle 2+{\frac {4}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}}
 |
8.1536 |
| 46 | Gyroelongated pentagonal bicupola |  |
30 | 70 | 42 |
D
5
{\displaystyle D_{5}}
 of order 10 |
10
+
15
2
3
+
1
2
5
α
{\displaystyle 10+{\frac {15}{2}}{\sqrt {3}}+{\frac {1}{2}}{\sqrt {5\alpha }}}
 |
26.4313 |
5
+
4
5
3
+
5
6
2
145
α
−
75
−
α
+
3
{\displaystyle {\frac {5+4{\sqrt {5}}}{3}}+{\frac {5}{6}}{\sqrt {2{\sqrt {145\alpha -75}}-\alpha +3}}}
 |
11.3974 |
| 47 | Gyroelongated pentagonal cupolarotunda |  |
35 | 80 | 47 |
C
5
{\displaystyle C_{5}}
 of order 5 |
5
+
7
4
(
5
3
+
5
α
)
{\displaystyle 5+{\frac {7}{4}}\left(5{\sqrt {3}}+{\sqrt {5\alpha }}\right)}
 |
32.1988 |
55
+
25
5
12
+
5
6
2
145
α
−
75
−
α
+
3
{\displaystyle {\frac {55+25{\sqrt {5}}}{12}}+{\frac {5}{6}}{\sqrt {2{\sqrt {145\alpha -75}}-\alpha +3}}}
 |
15.9911 |
| 48 | Gyroelongated pentagonal birotunda |  |
40 | 90 | 52 |
D
5
{\displaystyle D_{5}}
of order 10 |
10
3
+
3
5
α
{\displaystyle 10{\sqrt {3}}+3{\sqrt {5\alpha }}}
 |
37.9662 |
45
+
17
5
+
5
2
145
α
−
75
−
α
+
3
6
{\displaystyle {\frac {45+17{\sqrt {5}}+5{\sqrt {2{\sqrt {145\alpha -75}}-\alpha +3}}}{6}}}
 |
20.5848 |
| 49 | Augmented triangular prism |  |
7 | 13 | 8 |
C
2
v
{\displaystyle C_{2v}}
 of order 4 |
2
+
3
2
3
{\displaystyle 2+{\frac {3}{2}}{\sqrt {3}}}
 |
4.5981 |
2
6
+
3
4
{\displaystyle {\frac {\sqrt {2}}{6}}+{\frac {\sqrt {3}}{4}}}
|
0.6687 |
| 50 | Biaugmented triangular prism |  |
8 | 17 | 11 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
1
+
5
2
3
{\displaystyle 1+{\frac {5}{2}}{\sqrt {3}}}
 |
5.3301 |
59
144
+
1
6
{\displaystyle {\sqrt {{\frac {59}{144}}+{\frac {1}{\sqrt {6}}}}}}
 |
0.9044 |
| 51 | Triaugmented triangular prism |  |
9 | 21 | 14 |
D
3
h
{\displaystyle D_{3h}}
of order 12 |
7
3
2
{\displaystyle {\frac {7{\sqrt {3}}}{2}}}
 |
6.0622 |
2
2
+
3
4
{\displaystyle {\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}}
 |
1.1401 |
| 52 | Augmented pentagonal prism |  |
11 | 19 | 10 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
4
+
3
+
1
2
5
α
{\displaystyle 4+{\sqrt {3}}+{\frac {1}{2}}{\sqrt {5\alpha }}}
 |
9.173 |
1
12
8
+
45
α
+
12
10
α
{\displaystyle {\frac {1}{12}}{\sqrt {8+45\alpha +12{\sqrt {10\alpha }}}}}
 |
1.9562 |
| 53 | Biaugmented pentagonal prism |  |
12 | 23 | 13 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
3
+
2
3
+
1
2
5
α
{\displaystyle 3+2{\sqrt {3}}+{\frac {1}{2}}{\sqrt {5\alpha }}}
 |
9.9051 |
1
12
32
+
45
α
+
24
10
α
{\displaystyle {\frac {1}{12}}{\sqrt {32+45\alpha +24{\sqrt {10\alpha }}}}}
 |
2.1919 |
| 54 | Augmented hexagonal prism |  |
13 | 22 | 11 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
5
+
4
3
{\displaystyle 5+4{\sqrt {3}}}
 |
11.9282 |
2
6
+
3
3
2
{\displaystyle {\frac {\sqrt {2}}{6}}+{\frac {3{\sqrt {3}}}{2}}}
 |
2.8338 |
| 55 | Parabiaugmented hexagonal prism |  |
14 | 26 | 14 |
D
2
h
{\displaystyle D_{2h}}
 of order 8 |
4
+
5
3
{\displaystyle 4+5{\sqrt {3}}}
 |
12.6603 |
2
3
+
3
3
2
{\displaystyle {\frac {\sqrt {2}}{3}}+{\frac {3{\sqrt {3}}}{2}}}
 |
3.0695 |
| 56 | Metabiaugmented hexagonal prism |  |
14 | 26 | 14 |
C
2
v
{\displaystyle C_{2v}}
of order 4 | ||||
| 57 | Triaugmented hexagonal prism |  |
15 | 30 | 17 |
D
3
h
{\displaystyle D_{3h}}
of order 12 |
3
+
6
3
{\displaystyle 3+6{\sqrt {3}}}
 |
13.3923 |
1
2
+
3
3
2
{\displaystyle {\frac {1}{\sqrt {2}}}+{\frac {3{\sqrt {3}}}{2}}}
 |
3.3052 |
| 58 | Augmented dodecahedron |  |
21 | 35 | 16 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
3
+
11
5
α
4
{\displaystyle {\frac {5{\sqrt {3}}+11{\sqrt {5\alpha }}}{4}}}
 |
21.0903 |
95
+
43
5
24
{\displaystyle {\frac {95+43{\sqrt {5}}}{24}}}
 |
7.9646 |
| 59 | Parabiaugmented dodecahedron |  |
22 | 40 | 20 |
D
5
d
{\displaystyle D_{5d}}
of order 20 |
5
2
(
3
+
5
α
)
{\displaystyle {\frac {5}{2}}\left({\sqrt {3}}+{\sqrt {5\alpha }}\right)}
 |
21.5349 |
25
+
11
5
6
{\displaystyle {\frac {25+11{\sqrt {5}}}{6}}}
 |
8.2661 |
| 60 | Metabiaugmented dodecahedron |  |
22 | 40 | 20 |
C
2
v
{\displaystyle C_{2v}}
of order 4 | ||||
| 61 | Triaugmented dodecahedron |  |
23 | 45 | 24 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
3
4
(
5
3
+
3
5
α
)
{\displaystyle {\frac {3}{4}}\left(5{\sqrt {3}}+3{\sqrt {5\alpha }}\right)}
 |
21.9795 |
5
8
(
7
+
3
5
)
{\displaystyle {\frac {5}{8}}\left(7+3{\sqrt {5}}\right)}
 |
8.5676 |
| 62 | Metabidiminished icosahedron | |
10 | 20 | 12 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
5
3
+
5
α
2
{\displaystyle {\frac {5{\sqrt {3}}+{\sqrt {5\alpha }}}{2}}}
 |
7.7711 |
α
6
{\displaystyle {\frac {\alpha }{6}}}
 |
1.5787 |
| 63 | Tridiminished icosahedron | |
9 | 15 | 8 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
5
3
+
3
5
α
4
{\displaystyle {\frac {5{\sqrt {3}}+3{\sqrt {5\alpha }}}{4}}}
 |
7.3265 |
5
8
+
7
5
24
{\displaystyle {\frac {5}{8}}+{\frac {7{\sqrt {5}}}{24}}}
 |
1.2772 |
| 64 | Augmented tridiminished icosahedron |  |
10 | 18 | 10 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
7
3
+
3
5
α
4
{\displaystyle {\frac {7{\sqrt {3}}+3{\sqrt {5\alpha }}}{4}}}
 |
8.1925 |
15
+
2
2
+
7
5
24
{\displaystyle {\frac {15+2{\sqrt {2}}+7{\sqrt {5}}}{24}}}
 |
1.3950 |
| 65 | Augmented truncated tetrahedron |  |
15 | 27 | 14 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
3
+
13
2
3
{\displaystyle 3+{\frac {13}{2}}{\sqrt {3}}}
 |
14.2583 |
11
2
2
{\displaystyle {\frac {11}{2{\sqrt {2}}}}}
 |
3.8891 |
| 66 | Augmented truncated cube |  |
28 | 48 | 22 |
C
4
v
{\displaystyle C_{4v}}
of order 8 |
15
+
10
2
+
3
3
{\displaystyle 15+10{\sqrt {2}}+3{\sqrt {3}}}
 |
34.3383 |
8
+
16
2
3
{\displaystyle 8+{\frac {16{\sqrt {2}}}{3}}}
 |
15.5425 |
| 67 | Biaugmented truncated cube |  |
32 | 60 | 30 |
D
4
h
{\displaystyle D_{4h}}
of order 16 |
18
+
8
2
+
4
3
{\displaystyle 18+8{\sqrt {2}}+4{\sqrt {3}}}
 |
36.2419 |
9
+
6
2
{\displaystyle 9+6{\sqrt {2}}}
 |
17.4853 |
| 68 | Augmented truncated dodecahedron |  |
65 | 105 | 42 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
5
+
25
3
+
110
α
+
5
α
4
{\displaystyle 5+{\frac {25{\sqrt {3}}+110{\sqrt {\alpha }}+{\sqrt {5\alpha }}}{4}}}
 |
102.1821 |
505
12
+
81
5
4
{\displaystyle {\frac {505}{12}}+{\frac {81{\sqrt {5}}}{4}}}
 |
87.3637 |
| 69 | Parabiaugmented truncated dodecahedron |  |
70 | 120 | 52 |
D
5
d
{\displaystyle D_{5d}}
of order 20 |
10
+
25
α
+
15
3
+
5
α
2
{\displaystyle 10+25{\sqrt {\alpha }}+{\frac {15{\sqrt {3}}+{\sqrt {5\alpha }}}{2}}}
 |
103.3734 |
515
+
251
5
12
{\displaystyle {\frac {515+251{\sqrt {5}}}{12}}}
 |
89.6878 |
| 70 | Metabiaugmented truncated dodecahedron |  |
70 | 120 | 52 |
C
2
v
{\displaystyle C_{2v}}
of order 4 | ||||
| 71 | Triaugmented truncated dodecahedron |  |
75 | 135 | 62 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
15
+
35
3
+
90
α
+
3
5
α
4
{\displaystyle 15+{\frac {35{\sqrt {3}}+90{\sqrt {\alpha }}+3{\sqrt {5\alpha }}}{4}}}
 |
104.5648 |
7
12
(
75
+
37
5
)
{\displaystyle {\frac {7}{12}}\left(75+37{\sqrt {5}}\right)}
 |
92.0118 |
| 72 | Gyrate rhombicosidodecahedron | |
60 | 120 | 62 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
30
+
5
3
+
3
5
α
{\displaystyle 30+5{\sqrt {3}}+3{\sqrt {5\alpha }}}
 |
59.306 |
20
+
29
5
3
{\displaystyle 20+{\frac {29{\sqrt {5}}}{3}}}
 |
41.6153 |
| 73 | Parabigyrate rhombicosidodecahedron | |
60 | 120 | 62 |
D
5
d
{\displaystyle D_{5d}}
of order 20 | ||||
| 74 | Metabigyrate rhombicosidodecahedron | |
60 | 120 | 62 |
C
2
v
{\displaystyle C_{2v}}
of order 4 | ||||
| 75 | Trigyrate rhombicosidodecahedron | |
60 | 120 | 62 |
C
3
v
{\displaystyle C_{3v}}
of order 6 | ||||
| 76 | Diminished rhombicosidodecahedron | |
55 | 105 | 52 |
C
5
v
{\displaystyle C_{5v}}
of order 10 |
25
+
15
3
+
10
α
+
11
5
α
4
{\displaystyle 25+{\frac {15{\sqrt {3}}+10{\sqrt {\alpha }}+11{\sqrt {5\alpha }}}{4}}}
 |
58.1147 |
115
6
+
9
5
{\displaystyle {\frac {115}{6}}+9{\sqrt {5}}}
 |
39.2913 |
| 77 | Paragyrate diminished rhombicosidodecahedron | |
55 | 105 | 52 |
C
5
v
{\displaystyle C_{5v}}
of order 10 | ||||
| 78 | Metagyrate diminished rhombicosidodecahedron | |
55 | 105 | 52 |
C
s
{\displaystyle C_{s}}
of order 2 | ||||
| 79 | Bigyrate diminished rhombicosidodecahedron | |
55 | 105 | 52 |
C
s
{\displaystyle C_{s}}
of order 2 | ||||
| 80 | Parabidiminished rhombicosidodecahedron | |
50 | 90 | 42 |
D
5
d
{\displaystyle D_{5d}}
of order 20 |
20
+
5
2
(
3
+
2
α
+
5
α
)
{\displaystyle 20+{\frac {5}{2}}\left({\sqrt {3}}+2{\sqrt {\alpha }}+{\sqrt {5\alpha }}\right)}
 |
56.9233 |
55
+
25
5
3
{\displaystyle {\frac {55+25{\sqrt {5}}}{3}}}
 |
36.9672 |
| 81 | Metabidiminished rhombicosidodecahedron | |
50 | 90 | 42 |
C
2
v
{\displaystyle C_{2v}}
of order 4 | ||||
| 82 | Gyrate bidiminished rhombicosidodecahedron | |
50 | 90 | 42 |
C
s
{\displaystyle C_{s}}
of order 2 | ||||
| 83 | Tridiminished rhombicosidodecahedron | |
45 | 75 | 32 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
15
+
5
3
+
30
α
+
9
5
α
4
{\displaystyle 15+{\frac {5{\sqrt {3}}+30{\sqrt {\alpha }}+9{\sqrt {5\alpha }}}{4}}}
 |
55.732 |
35
2
+
23
5
3
{\displaystyle {\frac {35}{2}}+{\frac {23{\sqrt {5}}}{3}}}
 |
34.6432 |
| 84 | Snub disphenoid | |
8 | 18 | 12 |
D
2
d
{\displaystyle D_{2d}}
of order 8 |
3
3
{\displaystyle 3{\sqrt {3}}}
 |
5.1962 | 0.8595 | |
| 85 | Snub square antiprism | |
16 | 40 | 26 |
D
4
d
{\displaystyle D_{4d}}
of order 16 |
2
+
6
3
{\displaystyle 2+6{\sqrt {3}}}
 |
12.3923 | 3.6012 | |
| 86 | Sphenocorona | |
10 | 22 | 14 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
2
+
3
3
{\displaystyle 2+3{\sqrt {3}}}
 |
7.1962 |
1
2
1
+
3
3
2
+
13
+
3
6
{\displaystyle {\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}}
 |
1.5154 |
| 87 | Augmented sphenocorona | |
11 | 26 | 17 |
C
s
{\displaystyle C_{s}}
of order 2 |
1
+
4
3
{\displaystyle 1+4{\sqrt {3}}}
 |
7.9282 |
1
2
1
+
3
3
2
+
13
+
3
6
+
1
3
2
{\displaystyle {\frac {1}{2}}{\sqrt {1+3{\sqrt {\frac {3}{2}}}+{\sqrt {13+3{\sqrt {6}}}}}}+{\frac {1}{3{\sqrt {2}}}}}
 |
1.7511 |
| 88 | Sphenomegacorona | |
12 | 28 | 18 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
2
+
4
3
{\displaystyle 2+4{\sqrt {3}}}
 |
8.9282 | 1.9481 | |
| 89 | Hebesphenomegacorona | |
14 | 33 | 21 |
C
2
v
{\displaystyle C_{2v}}
of order 4 |
3
+
9
2
3
{\displaystyle 3+{\frac {9}{2}}{\sqrt {3}}}
 |
10.7942 | 2.9129 | |
| 90 | Disphenocingulum | |
16 | 38 | 24 |
D
2
d
{\displaystyle D_{2d}}
of order 8 |
4
+
5
3
{\displaystyle 4+5{\sqrt {3}}}
|
12.6603 | 3.7776 | |
| 91 | Bilunabirotunda | |
14 | 26 | 14 |
D
2
h
{\displaystyle D_{2h}}
of order 8 |
2
+
2
3
+
5
α
{\displaystyle 2+2{\sqrt {3}}+{\sqrt {5\alpha }}}
 |
12.346 |
17
+
9
5
12
{\displaystyle {\frac {17+9{\sqrt {5}}}{12}}}
 |
3.0937 |
| 92 | Triangular hebesphenorotunda | |
18 | 36 | 20 |
C
3
v
{\displaystyle C_{3v}}
of order 6 |
3
+
19
3
+
3
5
α
4
{\displaystyle 3+{\frac {19{\sqrt {3}}+3{\sqrt {5\alpha }}}{4}}}
 |
16.3887 |
5
2
+
7
5
6
{\displaystyle {\frac {5}{2}}+{\frac {7{\sqrt {5}}}{6}}}
 |
5.1087 |
See also
References
- 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.
- By definition, each face is the intersection of the convex polyhedron with a different bounding plane, so no two faces are coplanar — any two adjacent faces form an angle less than 180 degrees. If instead a convex polyhedron is presented by giving a collection of polygons that a priori may be coplanar (e.g., by subdividing a face), one could write "strictly convex polyhedron" here to indicate the condition that no two of the polygons are coplanar, that no two meet in a 180-degree angle. This notion of "strictly convex" for polyhedra is not the same as the standard notion used for general convex sets: no convex polyhedra are strictly convex in the latter sense; see p. 263 of A. G. Khovanskii, Geometry of generalized virtual polyhedra, J. Math. Sciences 269 (2023), 256–269.
- Todesco, Gian Marco (2020). "Hyperbolic Honeycomb". In Emmer, Michele; Abate, Marco (eds.). Imagine Math 7: Between Culture and Mathematics. Springer. p. 282. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8.
- Williams, Kim; Monteleone, Cosino (2021). Daniele Barbaro's Perspective of 1568. Springer. p. 23. doi:10.1007/978-3-030-76687-0. ISBN 978-3-030-76687-0.
- Buldygin, V. V.; Kharazishvili, A. B. (2000). Geometric Aspects of Probability Theory and Mathematical Statistics. Springer. p. 2. doi:10.1007/978-94-017-1687-1. ISBN 978-94-017-1687-1.
- Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
- Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5.
- Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
- Kaplan, Craig S.; Hart, George W. (2001). "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons" (PDF). Bridges: Mathematical Connections in Art, Music and Science: 21–28.
- Berman, Martin (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 (1997), p. 86–87, See the figure on p.89
- Johnson (1966)
- Klitzing, Dr. Richard. "Johnson solids et al". bendwavy.org. Retrieved 17 April 2018.
- Walsh (2014), p. 284.
- Parker (1997), p. 264.
-
- Powell (2010), p. 27
- Solomon (2003), p. 40
- Flusser, Suk & Zitofa (2017), p. 126.
- Uehara (2020), p. 62.
- Johnson (1966).
- Berman (1971).
Bibliography
- 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.
- 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.
- Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières" [Convex polyhedra with regular faces] (PDF). Structural Topology (6): 83–95.
- 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.
- 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.
- 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, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.
External links
- Paper Models of Polyhedra Archived 2013-02-26 at the Wayback Machine Many links
- Hart, George W. "Johnson Solids".
- Visual Polyhedra, with 3D models and data for all 92 solids, by David I. McCooey.
- Images of all 92 solids, categorized, on one page
- Weisstein, Eric W. "Johnson Solid". MathWorld.
- VRML models of Johnson Solids by Jim McNeill
- Bulatov, Vladimir. "Johnson solids". – VRML models of Johnson solids
- CRF polychora discovery project attempts to discover CRF polychora Archived 2020-10-31 at the Wayback Machine (Convex 4-dimensional polytopes with Regular polygons as 2-dimensional Faces), a generalization of the Johnson solids to 4-dimensional space
- https://levskaya.github.io/polyhedronisme/ a generator of polyhedrons and Conway operations applied to them, including Johnson solids.