List of Johnson solids

Short description: none

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]

This article lists the 92 non-uniform Johnson solids, accompanied by images. They are listed alongside their basic elements (vertices, edges, and faces), and their most important general characteristics, including symmetry groups (Cn, Dn, Cnv, Dnh, Dnd, Cs), order, surface area, and volume; an overview of these follows first, before presenting the complete list of non-uniform Johnson solids.

## Contents

* 1 Characteristics
* 2 The solids
* 3 References
* 4 Bibliography
* 5 External links

## Characteristics

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.[4] A volume is a measurement of a region in three-dimensional space.[5] 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.[6]

A polygon that is rotated symmetrically by 360∘n is denoted by Cn, a cyclic group of order n; combining this with the reflection symmetry results in the symmetry of dihedral group Dn of order 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 Cnv of order 2n. The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry Dnh of order 4n. The antiprismatic symmetry Dnd of order 4n preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.[8] The symmetry group Cnh of order 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 C1h of order 2, often denoted as Cs.[9] Template:Sticky-header

## The 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: 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.[10] 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.[11]

The table below lists the 92 (non-uniform) Johnson solids. The table includes each solid's enumeration (denoted as Jn).[12] It also includes each solid's symmetry group and number of vertices, edges, and faces, as well as its surface area and volume when constructed with edge length 1.

Table of the 92 Johnson solids  Jn | Solid name  | Image  | Vertices  | Edges  | Faces  | Symmetry group and its order[13] | Surface area, exact, with edge length 1[14] | Surface area, approximate, with edge length 1[14] | Volume, exact, with edge length 1[14] | Volume, approximate, with edge length 1[14]

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1  | Square pyramid  | 100px  | 5  | 8  | 5  | C4v of order 8  | 1+3 | 2.7321  | 26 | 0.2357
2  | Pentagonal pyramid | 100px  | 6  | 10  | 6  | C5v of order 10  | 1252(10+5+75+305) | 3.8855  | 5+524 | 0.3015
3  | Triangular cupola | 100px  | 9  | 15  | 8  | C3v of order 6  | 3+532 | 7.3301  | 532 | 1.1785
4  | Square cupola | 100px  | 12  | 20  | 10  | C4v of order 8  | 7+22+3 | 11.5605  | 1+223 | 1.9428
5  | Pentagonal cupola | 100px  | 15  | 25  | 12  | C5v of order 10  | 20+53+725+31054 | 16.5798  | 5+456 | 2.3241
6  | Pentagonal rotunda | 100px  | 20  | 35  | 17  | C5v of order 10  | 53+650+29052 | 22.3472  | 45+17512 | 6.9178
7  | Elongated triangular pyramid | 100px  | 7  | 12  | 7  | C3v of order 6  | 3+3 | 4.7321  | 2+3312 | 0.5509
8  | Elongated square pyramid | 100px  | 9  | 16  | 9  | C4v of order 8  | 5+3 | 6.7321  | 1+26 | 1.2357
9  | Elongated pentagonal pyramid | 100px  | 11  | 20  | 11  | C5v of order 10  | 20+53+25+1054 | 8.8855  | 5+5+625+10524 | 2.022
10  | Gyroelongated square pyramid | 100px  | 9  | 20  | 13  | C4v of order 8  | 1+33 | 6.1962  | 2+24+326 | 1.1927
11  | Gyroelongated pentagonal pyramid | 100px  | 11  | 25  | 16  | C5v of order 10  | 153+25+1054 | 8.2157  | 25+9524 | 1.8802
12  | Triangular bipyramid | 100px  | 5  | 9  | 6  | D3h of order 12  | 332 | 2.5981  | 26 | 0.2357
13  | Pentagonal bipyramid | 100px  | 7  | 15  | 10  | D5h of order 20  | 532 | 4.3301  | 5+512 | 0.6030
14  | Elongated triangular bipyramid | 100px  | 8  | 15  | 9  | D3h of order 12  | 3+332 | 5.5981  | 26+34 | 0.6687
15  | Elongated square bipyramid | 100px  | 10  | 20  | 12  | D4h of order 16  | 4+23 | 7.4641  | 1+23 | 1.4714
16  | Elongated pentagonal bipyramid | 100px  | 12  | 25  | 15  | D5h of order 20  | 5+523 | 9.3301  | 5+5+325+10512 | 2.3235
17  | Gyroelongated square bipyramid | 100px  | 10  | 24  | 16  | D4d of order 16  | 43 | 6.9282  | 2+4+323 | 1.4284
18  | Elongated triangular cupola | 100px  | 15  | 27  | 14  | C3v of order 6  | 9+523 | 13.3301  | 52+936 | 3.7766
19  | Elongated square cupola | 100px  | 20  | 36  | 18  | C4v of order 8  | 15+22+3 | 19.5605  | 3+823 | 6.7712
20  | Elongated pentagonal cupola | 100px  | 25  | 45  | 22  | C5v of order 10  | 60+53+105+25+25+1054 | 26.5798  | 5+45+155+256 | 10.0183
21  | Elongated pentagonal rotunda | 100px  | 30  | 55  | 27  | C5v of order 10  | 20+53+55+25+325+1052 | 32.3472  | 45+175+305+2512 | 14.612
22  | Gyroelongated triangular cupola | 100px  | 15  | 33  | 20  | C3v of order 6  | 3+1123 | 12.5263  | 13612+183+301+3 | 3.5161
23  | Gyroelongated square cupola | 100px  | 20  | 44  | 26  | C4v of order 8  | 7+22+53 | 18.4887  | 1+232+234+22+2146+1032 | 6.2108
24  | Gyroelongated pentagonal cupola | 100px  | 25  | 55  | 32  | C5v of order 10  | 5+253+105+25+25+1054 | 25.2400  | 56+235+562650+2905−25−2 | 9.0733
25  | Gyroelongated pentagonal rotunda | 100px  | 30  | 65  | 37  | C5v of order 10  | 153+(5+35)5+252 | 31.0075  | 45+175+102650+2905−25−212 | 13.6671
26  | Gyrobifastigium | 100px  | 8  | 14  | 8  | D2d of order 8  | 4+3 | 5.7321  | 32 | 0.8660
27  | Triangular orthobicupola | 100px  | 12  | 24  | 14  | D3h of order 12  | 6+23 | 9.4641  | 523 | 2.3570
28  | Square orthobicupola | 100px  | 16  | 32  | 18  | D4h of order 16  | 10+23 | 13.4641  | 2+423 | 3.8856
29  | Square gyrobicupola | 100px  | 16  | 32  | 18  | D4d of order 16
30  | Pentagonal orthobicupola | 100px  | 20  | 40  | 22  | D5h of order 20  | 10+51+5+75+30510 | 17.7711  | 5+453 | 4.6481
31  | Pentagonal gyrobicupola | 100px  | 20  | 40  | 22  | D5d of order 20
32  | Pentagonal orthocupolarotunda | 100px  | 25  | 50  | 27  | C5v of order 10  | 5+141900+4905+21075+305 | 23.5385  | 55+25512 | 9.2418
33  | Pentagonal gyrocupolarotunda | 100px  | 25  | 50  | 27  | C5v of order 10  | 5+1543+7425+105 | 23.5385
34  | Pentagonal orthobirotunda | 100px  | 30  | 60  | 32  | D5h of order 20  | 53+325+105 | 29.306  | 45+1756 | 13.8355
35  | Elongated triangular orthobicupola | 100px  | 18  | 36  | 20  | D3h of order 12  | 12+23 | 15.4641  | 523+332 | 4.9551
36  | Elongated triangular gyrobicupola | 100px  | 18  | 36  | 20  | D3d of order 12
37  | Elongated square gyrobicupola | 100px  | 24  | 48  | 26  | D4d of order 16  | 18+23 | 21.4641  | 4+1023 | 8.714
38  | Elongated pentagonal orthobicupola | 100px  | 30  | 60  | 32  | D5h of order 20  | 20+51+5+75+30510 | 27.7711  | 10+85+155+256 | 12.3423
39  | Elongated pentagonal gyrobicupola | 100px  | 30  | 60  | 32  | D5d of order 20
40  | Elongated pentagonal orthocupolarotunda | 100px  | 35  | 70  | 37  | C5v of order 10  | 15+141900+4905+21075+305 | 33.5385  | 55+255+305+2512 | 16.936
41  | Elongated pentagonal gyrocupolarotunda | 100px  | 35  | 70  | 37  | C5v of order 10  |
42  | Elongated pentagonal orthobirotunda | 100px  | 40  | 80  | 42  | D5h of order 20  | 10+300+905+3075+305 | 39.306  | 45+175+155+256 | 21.5297
43  | Elongated pentagonal gyrobirotunda | 100px  | 40  | 80  | 42  | D5d of order 20
44  | Gyroelongated triangular bicupola | 100px  | 18  | 42  | 26  | D3 of order 6  | 6+53 | 14.6603  | 532+2+23 | 4.6946
45  | Gyroelongated square bicupola | 100px  | 24  | 56  | 34  | D4 of order 8  | 10+63 | 20.3923  | 2+432+234+22+2146+1032 | 8.1536
46  | Gyroelongated pentagonal bicupola | 100px  | 30  | 70  | 42  | D5 of order 10  | 10+1523+1225+105 | 26.4313  | 5+453+562650+2905−25−2 | 11.3974
47  | Gyroelongated pentagonal cupolarotunda | 100px  | 35  | 80  | 47  | C5 of order 5  | 5+74(53+25+105) | 32.1988  | 55+25512+562650+2905−25−2 | 15.9911
48  | Gyroelongated pentagonal birotunda | 100px  | 40  | 90  | 52  | D5 of order 10  | 103+325+105 | 37.9662  | 45+175+52650+2905−25−26 | 20.5848
49  | Augmented triangular prism | 100px  | 7  | 13  | 8  | C2v of order 4  | 2+323 | 4.5981  | 26+34 | 0.6687
50  | Biaugmented triangular prism | 100px  | 8  | 17  | 11  | C2v of order 4  | 1+523 | 5.3301  | 59144+16 | 0.9044
51  | Triaugmented triangular prism | 100px  | 9  | 21  | 14  | D3h of order 12  | 732 | 6.0622  | 22+34 | 1.1401
52  | Augmented pentagonal prism | 100px  | 11  | 19  | 10  | C2v of order 4  | 4+3+1225+105 | 9.173  | 112233+905+1250+205 | 1.9562
53  | Biaugmented pentagonal prism | 100px  | 12  | 23  | 13  | C2v of order 4  | 3+23+1225+105 | 9.9051  | 112257+905+2450+205 | 2.1919
54  | Augmented hexagonal prism | 100px  | 13  | 22  | 11  | C2v of order 4  | 5+43 | 11.9282  | 26+332 | 2.8338
55  | Parabiaugmented hexagonal prism | 100px  | 14  | 26  | 14  | D2h of order 8  | 4+53 | 12.6603  | 23+332 | 3.0695
56  | Metabiaugmented hexagonal prism | 100px  | 14  | 26  | 14  | C2v of order 4
57  | Triaugmented hexagonal prism | 100px  | 15  | 30  | 17  | D3h of order 12  | 3+63 | 13.3923  | 12+332 | 3.3052
58  | Augmented dodecahedron | 100px  | 21  | 35  | 16  | C5v of order 10  | 53+1125+1054 | 21.0903  | 95+43524 | 7.9646
59  | Parabiaugmented dodecahedron | 100px  | 22  | 40  | 20  | D5d of order 20  | 52(3+25+105) | 21.5349  | 25+1156 | 8.2661
60  | Metabiaugmented dodecahedron | 100px  | 22  | 40  | 20  | C2v of order 4
61  | Triaugmented dodecahedron | 100px  | 23  | 45  | 24  | C3v of order 6  | 34(53+325+105) | 21.9795  | 58(7+35) | 8.5676
62  | Metabidiminished icosahedron | 100px  | 10  | 20  | 12  | C2v of order 4  | 53+25+1052 | 7.7711  | 5+256 | 1.5787
63  | Tridiminished icosahedron | 100px  | 9  | 15  | 8  | C3v of order 6  | 53+325+1054 | 7.3265  | 58+7524 | 1.2772
64  | Augmented tridiminished icosahedron | 100px  | 10  | 18  | 10  | C3v of order 6  | 73+325+1054 | 8.1925  | 15+22+7524 | 1.3950
65  | Augmented truncated tetrahedron | 100px  | 15  | 27  | 14  | C3v of order 6  | 3+1323 | 14.2583  | 1122 | 3.8891
66  | Augmented truncated cube | 100px  | 28  | 48  | 22  | C4v of order 8  | 15+102+33 | 34.3383  | 8+1623 | 15.5425
67  | Biaugmented truncated cube | 100px  | 32  | 60  | 30  | D4h of order 16  | 18+82+43 | 36.2419  | 9+62 | 17.4853
68  | Augmented truncated dodecahedron | 100px  | 65  | 105  | 42  | C5v of order 10  | 5+253+1105+25+25+1054 | 102.1821  | 50512+8154 | 87.3637
69  | Parabiaugmented truncated dodecahedron | 100px  | 70  | 120  | 52  | D5d of order 20  | 10+255+25+153+25+1052 | 103.3734  | 515+251512 | 89.6878
70  | Metabiaugmented truncated dodecahedron | 100px  | 70  | 120  | 52  | C2v of order 4
71  | Triaugmented truncated dodecahedron | 100px  | 75  | 135  | 62  | C3v of order 6  | 15+353+905+25+325+1054 | 104.5648  | 712(75+375) | 92.0118
72  | Gyrate rhombicosidodecahedron | 100px  | 60  | 120  | 62  | C5v of order 10  | 30+53+325+105 | 59.306  | 20+2953 | 41.6153
73  | Parabigyrate rhombicosidodecahedron | 100px  | 60  | 120  | 62  | D5d of order 20
74  | Metabigyrate rhombicosidodecahedron | 100px  | 60  | 120  | 62  | C2v of order 4
75  | Trigyrate rhombicosidodecahedron | 100px  | 60  | 120  | 62  | C3v of order 6
76  | Diminished rhombicosidodecahedron | 100px  | 55  | 105  | 52  | C5v of order 10  | 25+153+105+25+1125+1054 | 58.1147  | 1156+95 | 39.2913
77  | Paragyrate diminished rhombicosidodecahedron | 100px  | 55  | 105  | 52  | C5v of order 10
78  | Metagyrate diminished rhombicosidodecahedron | 100px  | 55  | 105  | 52  | Cs of order 2
79  | Bigyrate diminished rhombicosidodecahedron | 100px  | 55  | 105  | 52  | Cs of order 2
80  | Parabidiminished rhombicosidodecahedron | 100px  | 50  | 90  | 42  | D5d of order 20  | 20+52(3+25+25+25+105) | 56.9233  | 55+2553 | 36.9672
81  | Metabidiminished rhombicosidodecahedron | 100px  | 50  | 90  | 42  | C2v of order 4
82  | Gyrate bidiminished rhombicosidodecahedron | 100px  | 50  | 90  | 42  | Cs of order 2
83  | Tridiminished rhombicosidodecahedron | 100px  | 45  | 75  | 32  | C3v of order 6  | 15+53+305+25+925+1054 | 55.732  | 352+2353 | 34.6432
84  | Snub disphenoid | 100px  | 8  | 18  | 12  | D2d of order 8  | 33 | 5.1962  |  | 0.8595
85  | Snub square antiprism | 100px  | 16  | 40  | 26  | D4d of order 16  | 2+63 | 12.3923  |  | 3.6012
86  | Sphenocorona | 100px  | 10  | 22  | 14  | C2v of order 4  | 2+33 | 7.1962  | 121+332+13+36 | 1.5154
87  | Augmented sphenocorona | 100px  | 11  | 26  | 17  | Cs of order 2  | 1+43 | 7.9282  | 121+332+13+36+132 | 1.7511
88  | Sphenomegacorona | 100px  | 12  | 28  | 18  | C2v of order 4  | 2+43 | 8.9282  |  | 1.9481
89  | Hebesphenomegacorona | 100px  | 14  | 33  | 21  | C2v of order 4  | 3+923 | 10.7942  |  | 2.9129
90  | Disphenocingulum | 100px  | 16  | 38  | 24  | D2d of order 8  | 4+53 | 12.6603  |  | 3.7776
91  | Bilunabirotunda | 100px  | 14  | 26  | 14  | D2h of order 8  | 2+23+25+105 | 12.346  | 17+9512 | 3.0937
92  | Triangular hebesphenorotunda | 100px  | 18  | 36  | 20  | C3v of order 6  | 3+193+325+1054 | 16.3887  | 52+756 | 5.1087

## References

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

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

3. ↑

* Johnson (1966) harv error: no target: CITEREFJohnson1966 (help)
* Zalgaller (1969) harv error: no target: CITEREFZalgaller1969 (help)

4. ↑ Walsh (2014), p. 284.
5. ↑ Parker (1997), p. 264.
6. ↑

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

7. ↑

* Powell (2010), p. 27
* Solomon (2003), p. 40 harv error: no target: CITEREFSolomon2003 (help)

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

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

10. ↑

* Cromwell (1997), p. 86–87, See the figure on p.89
* Johnson (1966) harv error: no target: CITEREFJohnson1966 (help)

11. ↑

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

12. ↑ Uehara (2020), p. 62.
13. ↑ Johnson (1966). harv error: no target: CITEREFJohnson1966 (help)
14. ↑ 14.0 14.1 14.2 14.3 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 (in en). WALCOM: Algorithms and Computation. Lecture Notes in Computer Science. 8973. Cham: Springer International Publishing. pp. 294–305. doi:10.1007/978-3-319-15612-5_26. ISBN 978-3-319-15612-5. https://link.springer.com/chapter/10.1007/978-3-319-15612-5_26.
* Berman, M. (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. Bibcode: 1971FrInJ.291..329B.
* Cromwell, P. R. (1997). Polyhedra. Cambridge University Press. ISBN 978-0-521-66405-9. https://archive.org/details/polyhedra0000crom.
* Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. 10. Springer. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2. https://books.google.com/books?id=p_06DwAAQBAJ.
* Flusser, J.; Suk, T.; Zitofa, B. (2017). 2D and 3D Image Analysis by Moments. John Wiley & Sons. ISBN 978-1-119-03935-8. https://books.google.com/books?id=jwKLDQAAQBAJ.
* 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. https://books.google.com/books?id=6mvpCgAAQBAJ.
* "Convex Solids with Regular Faces". Canadian Journal of Mathematics 18: 169–200. 1966. doi:10.4153/CJM-1966-021-8.
* Parker, S. P. (1997). Dictionary of Mathematics. McGraw Hill. ISBN 978-0-070-52433-0. https://books.google.com/books?id=IdlzQgAACAAJ.
* Powell, R. C. (2010). Symmetry, Group Theory, and the Physical Properties of Crystals. Lecture Notes in Physics. 824. Springer. doi:10.1007/978-1-4419-7598-0. ISBN 978-1-441-97598-0. https://books.google.com/books?id=ojq5BQAAQBAJ.
* 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. https://books.google.com/books?id=AGonAQAAIAAJ.
* Abstract Algebra. American Mathematical Society. 2003. ISBN 978-0-821-84795-4. https://books.google.com/books?id=ouvZKQiykf4C.
* Slobodan, M.; Obradović, M.; Ðukanović, G. (2015). "Composite Concave Cupolae as Geometric and Architectural Forms". Journal for Geometry and Graphics 19 (1): 79–91. https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf.
* 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". Imagine Math 7: Between Culture and Mathematics. Springer. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8. https://books.google.com/books?id=wtIBEAAAQBAJ.
* 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. https://books.google.com/books?id=51juDwAAQBAJ.
* Walsh, E. T. (2014). A First Course in Geometry. Dover. ISBN 978-0-486-78020-7. https://books.google.com/books?id=ZhDdAwAAQBAJ.
* 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. https://books.google.com/books?id=w5RBEAAAQBAJ.
* Convex Polyhedra with Regular Faces. Springer. 1969. ISBN 978-1-489-95671-2.

## External links

* "Johnson Solids". http://www.georgehart.com/virtual-polyhedra/johnson-info.html.
* Bulatov, Vladimir. "Johnson solids". http://bulatov.org/polyhedra/johnson/. - VRML models of Johnson solids
* Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières" (in fr). Topologie Structurale (6): 83–95. https://upcommons.upc.edu/bitstream/handle/2099/890/st6-11-a7.pdf.

* v
* t
* e

Johnson solids

Pyramids, cupolae and rotundae|

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

Modified pyramids|

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

Modified cupolae and rotundae|

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

Augmented prisms|

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

Modified Platonic solids|

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

Modified Archimedean solids|

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

Elementary solids|

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

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

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