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 | ![]() |
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5 | 8 | 5 | 4 | 1 | C4v, [4], (*44) | 8 | 2.72 | 0.24 | ||||
2 | Pentagonal pyramid | ![]() |
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6 | 10 | 6 | 5 | 1 | C5v, [5], (*55) | 10 | 3.87 | 0.30 | ||||
3 | Triangular cupola | ![]() |
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9 | 15 | 8 | 4 | 3 | 1 | C3v, [3], (*33) | 6 | 7.32 | 1.18 | |||
4 | Square cupola | ![]() |
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12 | 20 | 10 | 4 | 5 | 1 | C4v, [4], (*44) | 8 | 11.55 | 1.94 | |||
5 | Pentagonal cupola | ![]() |
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15 | 25 | 12 | 5 | 5 | 1 | 1 | C5v, [5], (*55) | 10 | 16.56 | 2.32 | ||
6 | Pentagonal rotunda | ![]() |
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20 | 35 | 17 | 10 | 6 | 1 | C5v, [5], (*55) | 10 | 22.31 | 6.92 | |||
7 | Elongated triangular pyramid | ![]() |
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7 | 12 | 7 | 4 | 3 | C3v, [3], (*33) | 6 | 4.72 | 0.55 | ||||
8 | Elongated square pyramid | ![]() |
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9 | 16 | 9 | 4 | 5 | C4v, [4], (*44) | 8 | 6.72 | 1.24 | ||||
9 | Elongated pentagonal pyramid | ![]() |
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11 | 20 | 11 | 5 | 5 | 1 | C5v, [5], (*55) | 10 | 8.87 | 2.02 | |||
10 | Gyroelongated square pyramid | ![]() |
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9 | 20 | 13 | 12 | 1 | C4v, [4], (*44) | 8 | 6.16 | 1.20 | ||||
11 | Gyroelongated pentagonal pyramid | ![]() |
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11 | 25 | 16 | 15 | 1 | C5v, [5], (*55) | 10 | 8.17 | 1.88 | ||||
12 | Triangular bipyramid | ![]() |
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5 | 9 | 6 | 6 | D3h, [3,2], (*223) | 12 | 2.58 | 0.24 | |||||
13 | Pentagonal bipyramid | ![]() |
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7 | 15 | 10 | 10 | D5h, [5,2], (*225) | 20 | 4.3 | 0.60 | |||||
14 | Elongated triangular bipyramid | ![]() |
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8 | 15 | 9 | 6 | 3 | D3h, [3,2], (*223) | 12 | 5.58 | 0.67 | ||||
15 | Elongated square bipyramid | ![]() |
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10 | 20 | 12 | 8 | 4 | D4h, [4,2], (*224) | 16 | 7.44 | 1.48 | ||||
16 | Elongated pentagonal bipyramid | ![]() |
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12 | 25 | 15 | 10 | 5 | D5h, [5,2], (*225) | 20 | 9.3 | 2.32 | ||||
17 | Gyroelongated square bipyramid | ![]() |
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10 | 24 | 16 | 16 | D4d, [2+,8], (2*4) | 16 | 6.88 | 1.44 | |||||
18 | Elongated triangular cupola | ![]() |
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15 | 27 | 14 | 4 | 9 | 1 | C3v, [3], (*33) | 6 | 13.32 | 3.78 | |||
19 | Elongated square cupola | ![]() |
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20 | 36 | 18 | 4 | 13 | 1 | C4v, [4], (*44) | 8 | 19.55 | 6.77 | |||
20 | Elongated pentagonal cupola | ![]() |
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25 | 45 | 22 | 5 | 15 | 1 | 1 | C5v, [5], (*55) | 10 | 26.56 | 10.01 | ||
21 | Elongated pentagonal rotunda | ![]() |
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30 | 55 | 27 | 10 | 10 | 6 | 1 | C5v, [5], (*55) | 10 | 32.31 | 14.61 | ||
22 | Gyroelongated triangular cupola | ![]() |
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15 | 33 | 20 | 16 | 3 | 1 | C3v, [3], (*33) | 6 | 12.48 | 3.52 | |||
23 | Gyroelongated square cupola | ![]() |
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20 | 44 | 26 | 20 | 5 | 1 | C4v, [4], (*44) | 8 | 18.43 | 6.21 | |||
24 | Gyroelongated pentagonal cupola | ![]() |
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25 | 55 | 32 | 25 | 5 | 1 | 1 | C5v, [5], (*55) | 10 | 25.16 | 9.07 | ||
25 | Gyroelongated pentagonal rotunda | ![]() |
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30 | 65 | 37 | 30 | 6 | 1 | C5v, [5], (*55) | 10 | 30.91 | 13.67 | |||
26 | Gyrobifastigium | ![]() |
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8 | 14 | 8 | 4 | 4 | D2d, [2+,4], (2*2) | 8 | 5.72 | 0.87 | ||||
27 | Triangular orthobicupola | ![]() |
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12 | 24 | 14 | 8 | 6 | D3h, [3,2], (*223) | 12 | 9.44 | 2.36 | ||||
28 | Square orthobicupola | ![]() |
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16 | 32 | 18 | 8 | 10 | D4h, [4,2], (*224) | 16 | 13.44 | 3.88 | ||||
29 | Square gyrobicupola | ![]() |
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16 | 32 | 18 | 8 | 10 | D4d, [2+,8], (2*4) | 16 | 13.44 | 3.88 | ||||
30 | Pentagonal orthobicupola | ![]() |
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20 | 40 | 22 | 10 | 10 | 2 | D5h, [5,2], (*225) | 20 | 17.74 | 4.64 | |||
31 | Pentagonal gyrobicupola | ![]() |
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20 | 40 | 22 | 10 | 10 | 2 | D5d, [2+,10], (2*5) | 20 | 17.74 | 4.64 | |||
32 | Pentagonal orthocupolarotunda | ![]() |
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25 | 50 | 27 | 15 | 5 | 7 | C5v, [5], (*55) | 10 | 23.49 | 9.24 | |||
33 | Pentagonal gyrocupolarotunda | ![]() |
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25 | 50 | 27 | 15 | 5 | 7 | C5v, [5], (*55) | 10 | 23.49 | 9.24 | |||
34 | Pentagonal orthobirotunda | ![]() |
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30 | 60 | 32 | 20 | 12 | D5h, [5,2], (*225) | 20 | 29.24 | 13.84 | ||||
35 | Elongated triangular orthobicupola | ![]() |
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18 | 36 | 20 | 8 | 12 | D3h, [3,2], (*223) | 12 | 15.44 | 4.96 | ||||
36 | Elongated triangular gyrobicupola | ![]() |
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18 | 36 | 20 | 8 | 12 | D3d, [2+,6], (2*3) | 12 | 15.44 | 4.96 | ||||
37 | Elongated square gyrobicupola | ![]() |
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24 | 48 | 26 | 8 | 18 | D4d, [2+,8], (2*4) | 16 | 21.44 | 8.71 | ||||
38 | Elongated pentagonal orthobicupola | ![]() |
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30 | 60 | 32 | 10 | 20 | 2 | D5h, [5,2], (*225) | 20 | 27.74 | 12.33 | |||
39 | Elongated pentagonal gyrobicupola | ![]() |
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30 | 60 | 32 | 10 | 20 | 2 | D5d, [2+,10], (2*5) | 20 | 27.74 | 12.33 | |||
40 | Elongated pentagonal orthocupolarotunda | ![]() |
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35 | 70 | 37 | 15 | 15 | 7 | C5v, [5], (*55) | 10 | 33.49 | 16.93 | |||
41 | Elongated pentagonal gyrocupolarotunda | ![]() |
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35 | 70 | 37 | 15 | 15 | 7 | C5v, [5], (*55) | 10 | 33.49 | 16.93 | |||
42 | Elongated pentagonal orthobirotunda | ![]() |
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40 | 80 | 42 | 20 | 10 | 12 | D5h, [5,2], (*225) | 20 | 39.24 | 21.53 | |||
43 | Elongated pentagonal gyrobirotunda | ![]() |
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40 | 80 | 42 | 20 | 10 | 12 | D5d, [2+,10], (2*5) | 20 | 39.24 | 21.53 | |||
44 | Gyroelongated triangular bicupola | ![]() |
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18 | 42 | 26 | 20 | 6 | D3, [3,2]+,(223) | 6 | 14.60 | 4.70 | ||||
45 | Gyroelongated square bicupola | ![]() |
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24 | 56 | 34 | 24 | 10 | D4, [4,2]+, (224) | 8 | 20.32 | 8.15 | ||||
46 | Gyroelongated pentagonal bicupola | ![]() |
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30 | 70 | 42 | 30 | 10 | 2 | D5, [5,2]+, (225) | 10 | 26.34 | 11.39 | |||
47 | Gyroelongated pentagonal cupolarotunda | ![]() |
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35 | 80 | 47 | 35 | 5 | 7 | C5, [5]+, (55) | 5 | 32.09 | 15.99 | |||
48 | Gyroelongated pentagonal birotunda | ![]() |
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40 | 90 | 52 | 40 | 12 | D5, [5,2]+, (225) | 10 | 37.84 | 20.59 | ||||
49 | Augmented triangular prism | ![]() |
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7 | 13 | 8 | 6 | 2 | C2v, [2], (*22) | 4 | 4.58 | 0.67 | ||||
50 | Biaugmented triangular prism | ![]() |
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8 | 17 | 11 | 10 | 1 | C2v, [2], (*22) | 4 | 5.30 | 0.91 | ||||
51 | Triaugmented triangular prism | ![]() |
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9 | 21 | 14 | 14 | D3h, [3,2], (*223) | 12 | 6.02 | 1.15 | |||||
52 | Augmented pentagonal prism | ![]() |
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11 | 19 | 10 | 4 | 4 | 2 | C2v, [2], (*22) | 4 | 9.16 | 1.96 | |||
53 | Biaugmented pentagonal prism | ![]() |
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12 | 23 | 13 | 8 | 3 | 2 | C2v, [2], (*22) | 4 | 9.88 | 2.20 | |||
54 | Augmented hexagonal prism | ![]() |
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13 | 22 | 11 | 4 | 5 | 2 | C2v, [2], (*22) | 4 | 11.92 | 2.56 | |||
55 | Parabiaugmented hexagonal prism | ![]() |
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14 | 26 | 14 | 8 | 4 | 2 | D2h, [2,2], (*222) | 8 | 12.64 | 2.80 | |||
56 | Metabiaugmented hexagonal prism | ![]() |
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14 | 26 | 14 | 8 | 4 | 2 | C2v, [2], (*22) | 4 | 12.64 | 2.80 | |||
57 | Triaugmented hexagonal prism | ![]() |
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15 | 30 | 17 | 12 | 3 | 2 | D3h, [3,2], (*223) | 12 | 13.36 | 3.04 | |||
58 | Augmented dodecahedron | ![]() |
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21 | 35 | 16 | 5 | 11 | C5v, [5], (*55) | 10 | 21.07 | 7.96 | ||||
59 | Parabiaugmented dodecahedron | ![]() |
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22 | 40 | 20 | 10 | 10 | D5d, [2+,10], (2*5) | 20 | 21.5 | 8.26 | ||||
60 | Metabiaugmented dodecahedron | ![]() |
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22 | 40 | 20 | 10 | 10 | C2v, [2], (*22) | 4 | 21.5 | 8.26 | ||||
61 | Triaugmented dodecahedron | ![]() |
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23 | 45 | 24 | 15 | 9 | C3v, [3], (*33) | 6 | 21.93 | 8.56 | ||||
62 | Metabidiminished icosahedron | ![]() |
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10 | 20 | 12 | 10 | 2 | C2v, [2], (*22) | 4 | 7.74 | 1.58 | ||||
63 | Tridiminished icosahedron | ![]() |
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9 | 15 | 8 | 5 | 3 | C3v, [3], (*33) | 6 | 7.31 | 1.28 | ||||
64 | Augmented tridiminished icosahedron | ![]() |
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10 | 18 | 10 | 7 | 3 | C3v, [3], (*33) | 6 | 8.17 | 1.40 | ||||
65 | Augmented truncated tetrahedron | ![]() |
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15 | 27 | 14 | 8 | 3 | 3 | C3v, [3], (*33) | 6 | 14.24 | 3.89 | |||
66 | Augmented truncated cube | ![]() |
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28 | 48 | 22 | 12 | 5 | 5 | C4v, [4], (*44) | 8 | 34.31 | 15.54 | |||
67 | Biaugmented truncated cube | ![]() |
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32 | 60 | 30 | 16 | 10 | 4 | D4h, [4,2], (*224) | 16 | 36.20 | 17.48 | |||
68 | Augmented truncated dodecahedron | ![]() |
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65 | 105 | 42 | 25 | 5 | 1 | 11 | C5v, [5], (*55) | 10 | 102.06 | 87.36 | ||
69 | Parabiaugmented truncated dodecahedron | ![]() |
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70 | 120 | 52 | 30 | 10 | 2 | 10 | D5d, [2+,10], (2*5) | 20 | 103.24 | 89.68 | ||
70 | Metabiaugmented truncated dodecahedron | ![]() |
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70 | 120 | 52 | 30 | 10 | 2 | 10 | C2v, [2], (*22) | 4 | 103.24 | 89.68 | ||
71 | Triaugmented truncated dodecahedron | ![]() |
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75 | 135 | 62 | 35 | 15 | 3 | 9 | C3v, [3], (*33) | 6 | 104.42 | 92.00 | ||
72 | Gyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | C5v, [5], (*55) | 10 | 59.24 | 41.62 | |||
73 | Parabigyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | D5d, [2+,10], (2*5) | 20 | 59.24 | 41.62 | |||
74 | Metabigyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | C2v, [2], (*22) | 4 | 59.24 | 41.62 | |||
75 | Trigyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | C3v, [3], (*33) | 6 | 59.24 | 41.62 | |||
76 | Diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | C5v, [5], (*55) | 10 | 58.06 | 39.30 | ||
77 | Paragyrate diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | C5v, [5], (*55) | 10 | 58.06 | 39.30 | ||
78 | Metagyrate diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | Cs, [ ], (*11) | 2 | 58.06 | 39.30 | ||
79 | Bigyrate diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | Cs, [ ], (*11) | 2 | 58.06 | 39.30 | ||
80 | Parabidiminished rhombicosidodecahedron | ![]() |
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50 | 90 | 42 | 10 | 20 | 10 | 2 | D5d, [2+,10], (2*5) | 20 | 56.88 | 36.98 | ||
81 | Metabidiminished rhombicosidodecahedron | ![]() |
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50 | 90 | 42 | 10 | 20 | 10 | 2 | C2v, [2], (*22) | 4 | 56.88 | 36.98 | ||
82 | Gyrate bidiminished rhombicosidodecahedron | ![]() |
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50 | 90 | 42 | 10 | 20 | 10 | 2 | Cs, [ ], (*11) | 2 | 56.88 | 36.98 | ||
83 | Tridiminished rhombicosidodecahedron | ![]() |
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45 | 75 | 32 | 5 | 15 | 9 | 3 | C3v, [3], (*33) | 6 | 55.70 | 34.66 | ||
84 | Snub disphenoid | ![]() |
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8 | 18 | 12 | 12 | D2d, [2+,4], (2*2) | 8 | 5.16 | 0.86 | |||||
85 | Snub square antiprism | ![]() |
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16 | 40 | 26 | 24 | 2 | D4d, [2+,8], (2*4) | 16 | 12.32 | 3.60 | ||||
86 | Sphenocorona | ![]() |
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10 | 22 | 14 | 12 | 2 | C2v, [2], (*22) | 4 | 7.16 | 1.52 | ||||
87 | Augmented sphenocorona | ![]() |
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11 | 26 | 17 | 16 | 1 | Cs, [ ], (*11) | 2 | 7.88 | 1.75 | ||||
88 | Sphenomegacorona | ![]() |
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12 | 28 | 18 | 16 | 2 | C2v, [2], (*22) | 4 | 8.88 | 1.95 | ||||
89 | Hebesphenomegacorona | ![]() |
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14 | 33 | 21 | 18 | 3 | C2v, [2], (*22) | 4 | 10.74 | 2.91 | ||||
90 | Disphenocingulum | ![]() |
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16 | 38 | 24 | 20 | 4 | D2d, [2+,4], (2*2) | 8 | 12.6 | 3.78 | ||||
91 | Bilunabirotunda | ![]() |
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14 | 26 | 14 | 8 | 2 | 4 | D2h, [2,2], (*222) | 8 | 12.32 | 3.09 | |||
92 | Triangular hebesphenorotunda | ![]() |
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18 | 36 | 20 | 13 | 3 | 3 | 1 | C3v, [3], (*33) | 6 | 16.35 | 5.11 |
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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.
External links
- Sylvain Gagnon, "Convex polyhedra with regular faces", Structural Topology, No. 6, 1982, 83-95.
- Johnson Solids by George W. Hart.
- Images of all 92 solids, categorized, on one page
- Weisstein, Eric W. "Johnson Solid". MathWorld.
- VRML models of Johnson Solids by Jim McNeill
- VRML models of Johnson Solids by Vladimir Bulatov
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