List of Johnson solids

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

The complete list is here with sorting by column. Other polyhedra can be constructed that are only approximately regular planar polygon faces, and are informally called near-miss Johnson solid; there can be no definitive count of them.

Jn Solid name Net Image V E F F3 F4 F5 F6 F8 F10 Symmetry group Order Surface area Volume
1 Square pyramid 5 8 5 4 1 C4v, [4], (*44)8 2.72 0.24
2 Pentagonal pyramid 6 10 6 5 1 C5v, [5], (*55)10 3.87 0.30
3 Triangular cupola 9 15 8 4 3 1 C3v, [3], (*33)6 7.32 1.18
4 Square cupola 12 20 10 4 5 1 C4v, [4], (*44)8 11.55 1.94
5 Pentagonal cupola 15 25 12 5 5 1 1 C5v, [5], (*55)10 16.56 2.32
6 Pentagonal rotunda 20 35 17 10 6 1 C5v, [5], (*55)10 22.31 6.92
7 Elongated triangular pyramid 7 12 7 4 3 C3v, [3], (*33)6 4.72 0.55
8 Elongated square pyramid 9 16 9 4 5 C4v, [4], (*44)8 6.72 1.24
9 Elongated pentagonal pyramid 11 20 11 5 5 1 C5v, [5], (*55)10 8.87 2.02
10 Gyroelongated square pyramid 9 20 13 12 1 C4v, [4], (*44)8 6.16 1.20
11 Gyroelongated pentagonal pyramid 11 25 16 15 1 C5v, [5], (*55)10 8.17 1.88
12 Triangular bipyramid 5 9 6 6 D3h, [3,2], (*223)12 2.58 0.24
13 Pentagonal bipyramid 7 15 10 10 D5h, [5,2], (*225)20 4.3 0.60
14 Elongated triangular bipyramid 8 15 9 6 3 D3h, [3,2], (*223)12 5.58 0.67
15 Elongated square bipyramid 10 20 12 8 4 D4h, [4,2], (*224)16 7.44 1.48
16 Elongated pentagonal bipyramid 12 25 15 10 5 D5h, [5,2], (*225)20 9.3 2.32
17 Gyroelongated square bipyramid 10 24 16 16 D4d, [2+,8], (2*4)16 6.88 1.44
18 Elongated triangular cupola 15 27 14 4 9 1 C3v, [3], (*33)6 13.32 3.78
19 Elongated square cupola 20 36 18 4 13 1 C4v, [4], (*44)8 19.55 6.77
20 Elongated pentagonal cupola 25 45 22 5 15 1 1 C5v, [5], (*55)10 26.56 10.01
21 Elongated pentagonal rotunda 30 55 27 10 10 6 1 C5v, [5], (*55)10 32.31 14.61
22 Gyroelongated triangular cupola 15 33 20 16 3 1 C3v, [3], (*33)6 12.48 3.52
23 Gyroelongated square cupola 20 44 26 20 5 1 C4v, [4], (*44)8 18.43 6.21
24 Gyroelongated pentagonal cupola 25 55 32 25 5 1 1 C5v, [5], (*55)10 25.16 9.07
25 Gyroelongated pentagonal rotunda 30 65 37 30 6 1 C5v, [5], (*55)10 30.91 13.67
26 Gyrobifastigium 8 14 8 4 4 D2d, [2+,4], (2*2)8 5.72 0.87
27 Triangular orthobicupola 12 24 14 8 6 D3h, [3,2], (*223)12 9.44 2.36
28 Square orthobicupola 16 32 18 8 10 D4h, [4,2], (*224)16 13.44 3.88
29 Square gyrobicupola 16 32 18 8 10 D4d, [2+,8], (2*4)16 13.44 3.88
30 Pentagonal orthobicupola 20 40 22 10 10 2 D5h, [5,2], (*225)20 17.74 4.64
31 Pentagonal gyrobicupola 20 40 22 10 10 2 D5d, [2+,10], (2*5)20 17.74 4.64
32 Pentagonal orthocupolarotunda 25 50 27 15 5 7 C5v, [5], (*55)10 23.49 9.24
33 Pentagonal gyrocupolarotunda 25 50 27 15 5 7 C5v, [5], (*55)10 23.49 9.24
34 Pentagonal orthobirotunda 30 60 32 20 12 D5h, [5,2], (*225)20 29.24 13.84
35 Elongated triangular orthobicupola 18 36 20 8 12 D3h, [3,2], (*223)12 15.44 4.96
36 Elongated triangular gyrobicupola 18 36 20 8 12 D3d, [2+,6], (2*3)12 15.44 4.96
37 Elongated square gyrobicupola 24 48 26 8 18 D4d, [2+,8], (2*4)16 21.44 8.71
38 Elongated pentagonal orthobicupola 30 60 32 10 20 2 D5h, [5,2], (*225)20 27.74 12.33
39 Elongated pentagonal gyrobicupola 30 60 32 10 20 2 D5d, [2+,10], (2*5)20 27.74 12.33
40 Elongated pentagonal orthocupolarotunda 35 70 37 15 15 7 C5v, [5], (*55)10 33.49 16.93
41 Elongated pentagonal gyrocupolarotunda 35 70 37 15 15 7 C5v, [5], (*55)10 33.49 16.93
42 Elongated pentagonal orthobirotunda 40 80 42 20 10 12 D5h, [5,2], (*225)20 39.24 21.53
43 Elongated pentagonal gyrobirotunda 40 80 42 20 10 12 D5d, [2+,10], (2*5)20 39.24 21.53
44 Gyroelongated triangular bicupola 18 42 26 20 6 D3, [3,2]+,(223)6 14.60 4.70
45 Gyroelongated square bicupola 24 56 34 24 10 D4, [4,2]+, (224)8 20.32 8.15
46 Gyroelongated pentagonal bicupola 30 70 42 30 10 2 D5, [5,2]+, (225)10 26.34 11.39
47 Gyroelongated pentagonal cupolarotunda 35 80 47 35 5 7 C5, [5]+, (55)5 32.09 15.99
48 Gyroelongated pentagonal birotunda 40 90 52 40 12 D5, [5,2]+, (225)10 37.84 20.59
49 Augmented triangular prism 7 13 8 6 2 C2v, [2], (*22)4 4.58 0.67
50 Biaugmented triangular prism 8 17 11 10 1 C2v, [2], (*22)4 5.30 0.91
51 Triaugmented triangular prism 9 21 14 14 D3h, [3,2], (*223)12 6.02 1.15
52 Augmented pentagonal prism 11 19 10 4 4 2 C2v, [2], (*22)4 9.16 1.96
53 Biaugmented pentagonal prism 12 23 13 8 3 2 C2v, [2], (*22)4 9.88 2.20
54 Augmented hexagonal prism 13 22 11 4 5 2 C2v, [2], (*22)4 11.92 2.56
55 Parabiaugmented hexagonal prism 14 26 14 8 4 2 D2h, [2,2], (*222)8 12.64 2.80
56 Metabiaugmented hexagonal prism 14 26 14 8 4 2 C2v, [2], (*22)4 12.64 2.80
57 Triaugmented hexagonal prism 15 30 17 12 3 2 D3h, [3,2], (*223)12 13.36 3.04
58 Augmented dodecahedron 21 35 16 5 11 C5v, [5], (*55)10 21.07 7.96
59 Parabiaugmented dodecahedron 22 40 20 10 10 D5d, [2+,10], (2*5)20 21.5 8.26
60 Metabiaugmented dodecahedron 22 40 20 10 10 C2v, [2], (*22)4 21.5 8.26
61 Triaugmented dodecahedron 23 45 24 15 9 C3v, [3], (*33)6 21.93 8.56
62 Metabidiminished icosahedron 10 20 12 10 2 C2v, [2], (*22)4 7.74 1.58
63 Tridiminished icosahedron 9 15 8 5 3 C3v, [3], (*33)6 7.31 1.28
64 Augmented tridiminished icosahedron 10 18 10 7 3 C3v, [3], (*33)6 8.17 1.40
65 Augmented truncated tetrahedron 15 27 14 8 3 3 C3v, [3], (*33)6 14.24 3.89
66 Augmented truncated cube 28 48 22 12 5 5 C4v, [4], (*44)8 34.31 15.54
67 Biaugmented truncated cube 32 60 30 16 10 4 D4h, [4,2], (*224)16 36.20 17.48
68 Augmented truncated dodecahedron 65 105 42 25 5 1 11 C5v, [5], (*55)10 102.06 87.36
69 Parabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 D5d, [2+,10], (2*5)20 103.24 89.68
70 Metabiaugmented truncated dodecahedron 70 120 52 30 10 2 10 C2v, [2], (*22)4 103.24 89.68
71 Triaugmented truncated dodecahedron 75 135 62 35 15 3 9 C3v, [3], (*33)6 104.42 92.00
72 Gyrate rhombicosidodecahedron 60 120 62 20 30 12 C5v, [5], (*55)10 59.24 41.62
73 Parabigyrate rhombicosidodecahedron 60 120 62 20 30 12 D5d, [2+,10], (2*5)20 59.24 41.62
74 Metabigyrate rhombicosidodecahedron 60 120 62 20 30 12 C2v, [2], (*22)4 59.24 41.62
75 Trigyrate rhombicosidodecahedron 60 120 62 20 30 12 C3v, [3], (*33)6 59.24 41.62
76 Diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v, [5], (*55)10 58.06 39.30
77 Paragyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 C5v, [5], (*55)10 58.06 39.30
78 Metagyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs, [ ], (*11)2 58.06 39.30
79 Bigyrate diminished rhombicosidodecahedron 55 105 52 15 25 11 1 Cs, [ ], (*11)2 58.06 39.30
80 Parabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 D5d, [2+,10], (2*5)20 56.88 36.98
81 Metabidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 C2v, [2], (*22)4 56.88 36.98
82 Gyrate bidiminished rhombicosidodecahedron 50 90 42 10 20 10 2 Cs, [ ], (*11)2 56.88 36.98
83 Tridiminished rhombicosidodecahedron 45 75 32 5 15 9 3 C3v, [3], (*33)6 55.70 34.66
84 Snub disphenoid 8 18 12 12 D2d, [2+,4], (2*2)8 5.16 0.86
85 Snub square antiprism 16 40 26 24 2 D4d, [2+,8], (2*4)16 12.32 3.60
86 Sphenocorona 10 22 14 12 2 C2v, [2], (*22)4 7.16 1.52
87 Augmented sphenocorona 11 26 17 16 1 Cs, [ ], (*11)2 7.88 1.75
88 Sphenomegacorona 12 28 18 16 2 C2v, [2], (*22)4 8.88 1.95
89 Hebesphenomegacorona 14 33 21 18 3 C2v, [2], (*22)4 10.74 2.91
90 Disphenocingulum 16 38 24 20 4 D2d, [2+,4], (2*2)8 12.6 3.78
91 Bilunabirotunda 14 26 14 8 2 4 D2h, [2,2], (*222)8 12.32 3.09
92 Triangular hebesphenorotunda 18 36 20 13 3 3 1 C3v, [3], (*33)6 16.35 5.11

back to top Legend:

  • Jn – Johnson Solid Number
  • Net – Flattened (unfolded) image
  • V – Number of Vertices
  • E – Number of Edges
  • F – Number of Faces (total)
  • F3-F10 – Number of faces by side counts

References
  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.