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Een johnsonlichaam is een convex veelvlak waarvan elk zijvlak een regelmatige veelhoek is, en dat niet een regelmatig veelvlak (platonisch lichaam), archimedisch lichaam, prisma of antiprisma is.
Het is niet verplicht dat elk zijvlak uit eenzelfde type veelhoek moet bestaan of dat steeds de dezelfde configuratie van veelhoeken samenkomt in de hoekpunten. Een voorbeeld van een Johnson-lichaam is een vierkante piramide met gelijke zijden (J1). Het heeft een vierkante basis en vier driehoekige zijvlakken.
De zijvlakken van johnsonlichamen hebben allemaal 3, 4, 5, 6, 8 of 10 zijden.
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 | H | R | Z | Z3 | Z4 | Z5 | Z6 | Z8 | Z10 | Symmetrie | orde |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | Square pyramid | ![]() |
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5 | 8 | 5 | 4 | 1 | C4v, [4], (*44) | 8 | ||||
2 | Pentagonal pyramid | ![]() |
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6 | 10 | 6 | 5 | 1 | C5v, [5], (*55) | 10 | ||||
3 | Triangular cupola | ![]() |
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9 | 15 | 8 | 4 | 3 | 1 | C3v, [3], (*33) | 6 | |||
4 | Square cupola | ![]() |
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12 | 20 | 10 | 4 | 5 | 1 | C4v, [4], (*44) | 8 | |||
5 | Pentagonal cupola | ![]() |
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15 | 25 | 12 | 5 | 5 | 1 | 1 | C5v, [5], (*55) | 10 | ||
6 | Pentagonal rotunda | ![]() |
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20 | 35 | 17 | 10 | 6 | 1 | C5v, [5], (*55) | 10 | |||
7 | Elongated triangular pyramid | ![]() |
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7 | 12 | 7 | 4 | 3 | C3v, [3], (*33) | 6 | ||||
8 | Elongated square pyramid | ![]() |
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9 | 16 | 9 | 4 | 5 | C4v, [4], (*44) | 8 | ||||
9 | Elongated pentagonal pyramid | ![]() |
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11 | 20 | 11 | 5 | 5 | 1 | C5v, [5], (*55) | 10 | |||
10 | Gyroelongated square pyramid | ![]() |
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9 | 20 | 13 | 12 | 1 | C4v, [4], (*44) | 8 | ||||
11 | Gyroelongated pentagonal pyramid | ![]() |
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11 | 25 | 16 | 15 | 1 | C5v, [5], (*55) | 10 | ||||
12 | Triangular bipyramid | ![]() |
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5 | 9 | 6 | 6 | D3h, [3,2], (*223) | 12 | |||||
13 | Pentagonal bipyramid | ![]() |
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7 | 15 | 10 | 10 | D5h, [5,2], (*225) | 20 | |||||
14 | Elongated triangular bipyramid | ![]() |
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8 | 15 | 9 | 6 | 3 | D3h, [3,2], (*223) | 12 | ||||
15 | Elongated square bipyramid | ![]() |
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10 | 20 | 12 | 8 | 4 | D4h, [4,2], (*224) | 16 | ||||
16 | Elongated pentagonal bipyramid | ![]() |
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12 | 25 | 15 | 10 | 5 | D5h, [5,2], (*225) | 20 | ||||
17 | Gyroelongated square bipyramid | ![]() |
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10 | 24 | 16 | 16 | D4d, [2+,8], (2*4) | 16 | |||||
18 | Elongated triangular cupola | ![]() |
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15 | 27 | 14 | 4 | 9 | 1 | C3v, [3], (*33) | 6 | |||
19 | Elongated square cupola | ![]() |
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20 | 36 | 18 | 4 | 13 | 1 | C4v, [4], (*44) | 8 | |||
20 | Elongated pentagonal cupola | ![]() |
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25 | 45 | 22 | 5 | 15 | 1 | 1 | C5v, [5], (*55) | 10 | ||
21 | Elongated pentagonal rotunda | ![]() |
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30 | 55 | 27 | 10 | 10 | 6 | 1 | C5v, [5], (*55) | 10 | ||
22 | Gyroelongated triangular cupola | ![]() |
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15 | 33 | 20 | 16 | 3 | 1 | C3v, [3], (*33) | 6 | |||
23 | Gyroelongated square cupola | ![]() |
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20 | 44 | 26 | 20 | 5 | 1 | C4v, [4], (*44) | 8 | |||
24 | Gyroelongated pentagonal cupola | ![]() |
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25 | 55 | 32 | 25 | 5 | 1 | 1 | C5v, [5], (*55) | 10 | ||
25 | Gyroelongated pentagonal rotunda | ![]() |
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30 | 65 | 37 | 30 | 6 | 1 | C5v, [5], (*55) | 10 | |||
26 | Gyrobifastigium | ![]() |
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8 | 14 | 8 | 4 | 4 | D2d, [2+,4], (2*2) | 8 | ||||
27 | Triangular orthobicupola | ![]() |
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12 | 24 | 14 | 8 | 6 | D3h, [3,2], (*223) | 12 | ||||
28 | Square orthobicupola | ![]() |
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16 | 32 | 18 | 8 | 10 | D4h, [4,2], (*224) | 16 | ||||
29 | Square gyrobicupola | ![]() |
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16 | 32 | 18 | 8 | 10 | D4d, [2+,8], (2*4) | 16 | ||||
30 | Pentagonal orthobicupola | ![]() |
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20 | 40 | 22 | 10 | 10 | 2 | D5h, [5,2], (*225) | 20 | |||
31 | Pentagonal gyrobicupola | ![]() |
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20 | 40 | 22 | 10 | 10 | 2 | D5d, [2+,10], (2*5) | 20 | |||
32 | Pentagonal orthocupolarotunda | ![]() |
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25 | 50 | 27 | 15 | 5 | 7 | C5v, [5], (*55) | 10 | |||
33 | Pentagonal gyrocupolarotunda | ![]() |
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25 | 50 | 27 | 15 | 5 | 7 | C5v, [5], (*55) | 10 | |||
34 | Pentagonal orthobirotunda | ![]() |
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30 | 60 | 32 | 20 | 12 | D5h, [5,2], (*225) | 20 | ||||
35 | Elongated triangular orthobicupola | ![]() |
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18 | 36 | 20 | 8 | 12 | D3h, [3,2], (*223) | 12 | ||||
36 | Elongated triangular gyrobicupola | ![]() |
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18 | 36 | 20 | 8 | 12 | D3d, [2+,6], (2*3) | 12 | ||||
37 | Elongated square gyrobicupola | ![]() |
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24 | 48 | 26 | 8 | 18 | D4d, [2+,8], (2*4) | 16 | ||||
38 | Elongated pentagonal orthobicupola | ![]() |
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30 | 60 | 32 | 10 | 20 | 2 | D5h, [5,2], (*225) | 20 | |||
39 | Elongated pentagonal gyrobicupola | ![]() |
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30 | 60 | 32 | 10 | 20 | 2 | D5d, [2+,10], (2*5) | 20 | |||
40 | Elongated pentagonal orthocupolarotunda | ![]() |
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35 | 70 | 37 | 15 | 15 | 7 | C5v, [5], (*55) | 10 | |||
41 | Elongated pentagonal gyrocupolarotunda | ![]() |
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35 | 70 | 37 | 15 | 15 | 7 | C5v, [5], (*55) | 10 | |||
42 | Elongated pentagonal orthobirotunda | ![]() |
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40 | 80 | 42 | 20 | 10 | 12 | D5h, [5,2], (*225) | 20 | |||
43 | Elongated pentagonal gyrobirotunda | ![]() |
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40 | 80 | 42 | 20 | 10 | 12 | D5d, [2+,10], (2*5) | 20 | |||
44 | Gyroelongated triangular bicupola | ![]() |
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18 | 42 | 26 | 20 | 6 | D3, [3,2]+,(223) | 6 | ||||
45 | Gyroelongated square bicupola | ![]() |
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24 | 56 | 34 | 24 | 10 | D4, [4,2]+, (224) | 8 | ||||
46 | Gyroelongated pentagonal bicupola | ![]() |
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30 | 70 | 42 | 30 | 10 | 2 | D5, [5,2]+, (225) | 10 | |||
47 | Gyroelongated pentagonal cupolarotunda | ![]() |
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35 | 80 | 47 | 35 | 5 | 7 | C5, [5]+, (55) | 5 | |||
48 | Gyroelongated pentagonal birotunda | ![]() |
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40 | 90 | 52 | 40 | 12 | D5, [5,2]+, (225) | 10 | ||||
49 | Augmented triangular prism | ![]() |
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7 | 13 | 8 | 6 | 2 | C2v, [2], (*22) | 4 | ||||
50 | Biaugmented triangular prism | ![]() |
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8 | 17 | 11 | 10 | 1 | C2v, [2], (*22) | 4 | ||||
51 | Triaugmented triangular prism | ![]() |
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9 | 21 | 14 | 14 | D3h, [3,2], (*223) | 12 | |||||
52 | Augmented pentagonal prism | ![]() |
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11 | 19 | 10 | 4 | 4 | 2 | C2v, [2], (*22) | 4 | |||
53 | Biaugmented pentagonal prism | ![]() |
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12 | 23 | 13 | 8 | 3 | 2 | C2v, [2], (*22) | 4 | |||
54 | Augmented hexagonal prism | ![]() |
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13 | 22 | 11 | 4 | 5 | 2 | C2v, [2], (*22) | 4 | |||
55 | Parabiaugmented hexagonal prism | ![]() |
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14 | 26 | 14 | 8 | 4 | 2 | D2h, [2,2], (*222) | 8 | |||
56 | Metabiaugmented hexagonal prism | ![]() |
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14 | 26 | 14 | 8 | 4 | 2 | C2v, [2], (*22) | 4 | |||
57 | Triaugmented hexagonal prism | ![]() |
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15 | 30 | 17 | 12 | 3 | 2 | D3h, [3,2], (*223) | 12 | |||
58 | Augmented dodecahedron | ![]() |
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21 | 35 | 16 | 5 | 11 | C5v, [5], (*55) | 10 | ||||
59 | Parabiaugmented dodecahedron | ![]() |
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22 | 40 | 20 | 10 | 10 | D5d, [2+,10], (2*5) | 20 | ||||
60 | Metabiaugmented dodecahedron | ![]() |
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22 | 40 | 20 | 10 | 10 | C2v, [2], (*22) | 4 | ||||
61 | Triaugmented dodecahedron | ![]() |
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23 | 45 | 24 | 15 | 9 | C3v, [3], (*33) | 6 | ||||
62 | Metabidiminished icosahedron | ![]() |
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10 | 20 | 12 | 10 | 2 | C2v, [2], (*22) | 4 | ||||
63 | Tridiminished icosahedron | ![]() |
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9 | 15 | 8 | 5 | 3 | C3v, [3], (*33) | 6 | ||||
64 | Augmented tridiminished icosahedron | ![]() |
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10 | 18 | 10 | 7 | 3 | C3v, [3], (*33) | 6 | ||||
65 | Augmented truncated tetrahedron | ![]() |
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15 | 27 | 14 | 8 | 3 | 3 | C3v, [3], (*33) | 6 | |||
66 | Augmented truncated cube | ![]() |
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28 | 48 | 22 | 12 | 5 | 5 | C4v, [4], (*44) | 8 | |||
67 | Biaugmented truncated cube | ![]() |
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32 | 60 | 30 | 16 | 10 | 4 | D4h, [4,2], (*224) | 16 | |||
68 | Augmented truncated dodecahedron | ![]() |
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65 | 105 | 42 | 25 | 5 | 1 | 11 | C5v, [5], (*55) | 10 | ||
69 | Parabiaugmented truncated dodecahedron | ![]() |
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70 | 120 | 52 | 30 | 10 | 2 | 10 | D5d, [2+,10], (2*5) | 20 | ||
70 | Metabiaugmented truncated dodecahedron | ![]() |
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70 | 120 | 52 | 30 | 10 | 2 | 10 | C2v, [2], (*22) | 4 | ||
71 | Triaugmented truncated dodecahedron | ![]() |
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75 | 135 | 62 | 35 | 15 | 3 | 9 | C3v, [3], (*33) | 6 | ||
72 | Gyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | C5v, [5], (*55) | 10 | |||
73 | Parabigyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | D5d, [2+,10], (2*5) | 20 | |||
74 | Metabigyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | C2v, [2], (*22) | 4 | |||
75 | Trigyrate rhombicosidodecahedron | ![]() |
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60 | 120 | 62 | 20 | 30 | 12 | C3v, [3], (*33) | 6 | |||
76 | Diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | C5v, [5], (*55) | 10 | ||
77 | Paragyrate diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | C5v, [5], (*55) | 10 | ||
78 | Metagyrate diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | Cs, [ ], (*11) | 2 | ||
79 | Bigyrate diminished rhombicosidodecahedron | ![]() |
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55 | 105 | 52 | 15 | 25 | 11 | 1 | Cs, [ ], (*11) | 2 | ||
80 | Parabidiminished rhombicosidodecahedron | ![]() |
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50 | 90 | 42 | 10 | 20 | 10 | 2 | D5d, [2+,10], (2*5) | 20 | ||
81 | Metabidiminished rhombicosidodecahedron | ![]() |
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50 | 90 | 42 | 10 | 20 | 10 | 2 | C2v, [2], (*22) | 4 | ||
82 | Gyrate bidiminished rhombicosidodecahedron | ![]() |
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50 | 90 | 42 | 10 | 20 | 10 | 2 | Cs, [ ], (*11) | 2 | ||
83 | Tridiminished rhombicosidodecahedron | ![]() |
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45 | 75 | 32 | 5 | 15 | 9 | 3 | C3v, [3], (*33) | 6 | ||
84 | Snub disphenoid | ![]() |
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8 | 18 | 12 | 12 | D2d, [2+,4], (2*2) | 8 | |||||
85 | Snub square antiprism | ![]() |
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16 | 40 | 26 | 24 | 2 | D4d, [2+,8], (2*4) | 16 | ||||
86 | Sphenocorona | ![]() |
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10 | 22 | 14 | 12 | 2 | C2v, [2], (*22) | 4 | ||||
87 | Augmented sphenocorona | ![]() |
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11 | 26 | 17 | 16 | 1 | Cs, [ ], (*11) | 2 | ||||
88 | Sphenomegacorona | ![]() |
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12 | 28 | 18 | 16 | 2 | C2v, [2], (*22) | 4 | ||||
89 | Hebesphenomegacorona | ![]() |
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14 | 33 | 21 | 18 | 3 | C2v, [2], (*22) | 4 | ||||
90 | Disphenocingulum | ![]() |
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16 | 38 | 24 | 20 | 4 | D2d, [2+,4], (2*2) | 8 | ||||
91 | Bilunabirotunda | ![]() |
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14 | 26 | 14 | 8 | 2 | 4 | D2h, [2,2], (*222) | 8 | |||
92 | Triangular hebesphenorotunda | ![]() |
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18 | 36 | 20 | 13 | 3 | 3 | 1 | C3v, [3], (*33) | 6 |
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Legend:
Hieronder volgt de lijst van johnsonlichamen, maar sorteerbaar, zie daar voor de indeling.
Nummer | Naam (hoekpuntconfiguratie) |
Afbeelding | Openvouwing | Vlakken | Soort vlakken |
Ribben | Knooppunten | Symmetriegroep |
---|---|---|---|---|---|---|---|---|
1 | Afgeknotte tetraëder (3.6.6) |
![]() (Animatie) |
![]() |
8 | 4 driehoeken 4 zeshoeken |
18 | 12 | Td |
2 | Kuboctaëder (3.4.3.4) |
![]() (Animatie) |
![]() |
14 | 8 driehoeken 6 vierkanten |
24 | 12 | Oh |
3 | Afgeknotte kubus of afgeknotte hexaëder (3.8.8) |
![]() (Animatie) |
![]() |
14 | 8 driehoeken 6 achthoeken |
36 | 24 | Oh |
4 | Afgeknotte octaëder (4.6.6) |
![]() (Animatie) |
![]() |
14 | 6 vierkanten 8 zeshoeken |
36 | 24 | Oh |
5 | Rombische kuboctaëder of kleine rombische kuboctaëder (3.4.4.4) |
![]() (Animatie) |
![]() |
26 | 8 driehoeken 18 vierkanten |
48 | 24 | Oh |
6 | Afgeknotte kuboctaëder of grote rombische kuboctaëder (4.6.8) |
![]() (Animatie) |
![]() |
26 | 12 vierkanten 8 zeshoeken 6 achthoeken |
72 | 48 | Oh |
7 | Stompe kubus of stompe hexaëder (2 chirale vormen) (3.3.3.3.4) |
![]() (Animatie) ![]() (Animatie) |
![]() |
38 | 32 driehoeken 6 vierkanten |
60 | 24 | O |
8 | Icosidodecaëder (3.5.3.5) |
![]() (Animatie) |
![]() |
32 | 20 driehoeken 12 vijfhoeken |
60 | 30 | Ih |
9 | Afgeknotte dodecaëder (3.10.10) |
![]() (Animatie) |
![]() |
32 | 20 driehoeken 12 tienhoeken |
90 | 60 | Ih |
10 | Afgeknotte icosaëder (5.6.6) |
![]() (Animatie) |
![]() |
32 | 12 vijfhoeken 20 zeshoeken |
90 | 60 | Ih |
11 | Rombische icosidodecaëder of kleine rombische icosidodecaëder (3.4.5.4) |
![]() (Animatie) |
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62 | 20 driehoeken 30 vierkanten 12 vijfhoeken |
120 | 60 | Ih |
12 | Afgeknotte icosidodecaëder of grote rombische icosidodecaëder (4.6.10) |
![]() (Animatie) |
![]() |
62 | 30 vierkanten 20 zeshoeken 12 tienhoeken |
180 | 120 | Ih |
13 | Stompe dodecaëder of afgeknotte icosidodecaëder (2 chirale vormen) (3.3.3.3.5) |
![]() (Animatie) ![]() (Animatie) |
![]() |
92 | 80 driehoeken 12 vijfhoeken |
150 | 60 | I |
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