قائمة مجسمات جونسون
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 proved in 1969 that Johnson's list was complete.
Other polyhedra can be constructed that have only approximately regular planar polygon faces, and are informally called near-miss Johnson solids; there can be no definitive count of them.
The various sections that follow have tables listing all 92 Johnson solids, and values for some of their most important properties. Each table allows sorting by column so that numerical values, or the names of the solids, can be sorted in order.
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الرؤوس والحواف والأوجه والتناظر
Jn | Solid name | Net | Image | V | E | F | F3 | F4 | F5 | F6 | F8 | F10 | Symmetry group | Order |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | Square pyramid | 5 | 8 | 5 | 4 | 1 | C4v, [4], (*44) | 8 | ||||||
2 | Pentagonal pyramid | 6 | 10 | 6 | 5 | 1 | C5v, [5], (*55) | 10 | ||||||
3 | Triangular cupola | 9 | 15 | 8 | 4 | 3 | 1 | C3v, [3], (*33) | 6 | |||||
4 | Square cupola | 12 | 20 | 10 | 4 | 5 | 1 | C4v, [4], (*44) | 8 | |||||
5 | Pentagonal cupola | 15 | 25 | 12 | 5 | 5 | 1 | 1 | C5v, [5], (*55) | 10 | ||||
6 | Pentagonal rotunda | 20 | 35 | 17 | 10 | 6 | 1 | C5v, [5], (*55) | 10 | |||||
7 | Elongated triangular pyramid | 7 | 12 | 7 | 4 | 3 | C3v, [3], (*33) | 6 | ||||||
8 | Elongated square pyramid | 9 | 16 | 9 | 4 | 5 | C4v, [4], (*44) | 8 | ||||||
9 | Elongated pentagonal pyramid | 11 | 20 | 11 | 5 | 5 | 1 | C5v, [5], (*55) | 10 | |||||
10 | Gyroelongated square pyramid | 9 | 20 | 13 | 12 | 1 | C4v, [4], (*44) | 8 | ||||||
11 | Gyroelongated pentagonal pyramid | 11 | 25 | 16 | 15 | 1 | C5v, [5], (*55) | 10 | ||||||
12 | Triangular bipyramid | 5 | 9 | 6 | 6 | D3h, [3,2], (*223) | 12 | |||||||
13 | Pentagonal bipyramid | 7 | 15 | 10 | 10 | D5h, [5,2], (*225) | 20 | |||||||
14 | Elongated triangular bipyramid | 8 | 15 | 9 | 6 | 3 | D3h, [3,2], (*223) | 12 | ||||||
15 | Elongated square bipyramid | 10 | 20 | 12 | 8 | 4 | D4h, [4,2], (*224) | 16 | ||||||
16 | Elongated pentagonal bipyramid | 12 | 25 | 15 | 10 | 5 | D5h, [5,2], (*225) | 20 | ||||||
17 | Gyroelongated square bipyramid | 10 | 24 | 16 | 16 | D4d, [2+,8], (2*4) | 16 | |||||||
18 | Elongated triangular cupola | 15 | 27 | 14 | 4 | 9 | 1 | C3v, [3], (*33) | 6 | |||||
19 | Elongated square cupola | 20 | 36 | 18 | 4 | 13 | 1 | C4v, [4], (*44) | 8 | |||||
20 | Elongated pentagonal cupola | 25 | 45 | 22 | 5 | 15 | 1 | 1 | C5v, [5], (*55) | 10 | ||||
21 | Elongated pentagonal rotunda | 30 | 55 | 27 | 10 | 10 | 6 | 1 | C5v, [5], (*55) | 10 | ||||
22 | Gyroelongated triangular cupola | 15 | 33 | 20 | 16 | 3 | 1 | C3v, [3], (*33) | 6 | |||||
23 | Gyroelongated square cupola | 20 | 44 | 26 | 20 | 5 | 1 | C4v, [4], (*44) | 8 | |||||
24 | Gyroelongated pentagonal cupola | 25 | 55 | 32 | 25 | 5 | 1 | 1 | C5v, [5], (*55) | 10 | ||||
25 | Gyroelongated pentagonal rotunda | 30 | 65 | 37 | 30 | 6 | 1 | C5v, [5], (*55) | 10 | |||||
26 | Gyrobifastigium | 8 | 14 | 8 | 4 | 4 | D2d, [2+,4], (2*2) | 8 | ||||||
27 | Triangular orthobicupola | 12 | 24 | 14 | 8 | 6 | D3h, [3,2], (*223) | 12 | ||||||
28 | Square orthobicupola | 16 | 32 | 18 | 8 | 10 | D4h, [4,2], (*224) | 16 | ||||||
29 | Square gyrobicupola | 16 | 32 | 18 | 8 | 10 | D4d, [2+,8], (2*4) | 16 | ||||||
30 | Pentagonal orthobicupola | 20 | 40 | 22 | 10 | 10 | 2 | D5h, [5,2], (*225) | 20 | |||||
31 | Pentagonal gyrobicupola | 20 | 40 | 22 | 10 | 10 | 2 | D5d, [2+,10], (2*5) | 20 | |||||
32 | Pentagonal orthocupolarotunda | 25 | 50 | 27 | 15 | 5 | 7 | C5v, [5], (*55) | 10 | |||||
33 | Pentagonal gyrocupolarotunda | 25 | 50 | 27 | 15 | 5 | 7 | C5v, [5], (*55) | 10 | |||||
34 | Pentagonal orthobirotunda | 30 | 60 | 32 | 20 | 12 | D5h, [5,2], (*225) | 20 | ||||||
35 | Elongated triangular orthobicupola | 18 | 36 | 20 | 8 | 12 | D3h, [3,2], (*223) | 12 | ||||||
36 | Elongated triangular gyrobicupola | 18 | 36 | 20 | 8 | 12 | D3d, [2+,6], (2*3) | 12 | ||||||
37 | Elongated square gyrobicupola | 24 | 48 | 26 | 8 | 18 | D4d, [2+,8], (2*4) | 16 | ||||||
38 | Elongated pentagonal orthobicupola | 30 | 60 | 32 | 10 | 20 | 2 | D5h, [5,2], (*225) | 20 | |||||
39 | Elongated pentagonal gyrobicupola | 30 | 60 | 32 | 10 | 20 | 2 | D5d, [2+,10], (2*5) | 20 | |||||
40 | Elongated pentagonal orthocupolarotunda | 35 | 70 | 37 | 15 | 15 | 7 | C5v, [5], (*55) | 10 | |||||
41 | Elongated pentagonal gyrocupolarotunda | 35 | 70 | 37 | 15 | 15 | 7 | C5v, [5], (*55) | 10 | |||||
42 | Elongated pentagonal orthobirotunda | 40 | 80 | 42 | 20 | 10 | 12 | D5h, [5,2], (*225) | 20 | |||||
43 | Elongated pentagonal gyrobirotunda | 40 | 80 | 42 | 20 | 10 | 12 | D5d, [2+,10], (2*5) | 20 | |||||
44 | Gyroelongated triangular bicupola | 18 | 42 | 26 | 20 | 6 | D3, [3,2]+,(223) | 6 | ||||||
45 | Gyroelongated square bicupola | 24 | 56 | 34 | 24 | 10 | D4, [4,2]+, (224) | 8 | ||||||
46 | Gyroelongated pentagonal bicupola | 30 | 70 | 42 | 30 | 10 | 2 | D5, [5,2]+, (225) | 10 | |||||
47 | Gyroelongated pentagonal cupolarotunda | 35 | 80 | 47 | 35 | 5 | 7 | C5, [5]+, (55) | 5 | |||||
48 | Gyroelongated pentagonal birotunda | 40 | 90 | 52 | 40 | 12 | D5, [5,2]+, (225) | 10 | ||||||
49 | Augmented triangular prism | 7 | 13 | 8 | 6 | 2 | C2v, [2], (*22) | 4 | ||||||
50 | Biaugmented triangular prism | 8 | 17 | 11 | 10 | 1 | C2v, [2], (*22) | 4 | ||||||
51 | Triaugmented triangular prism | 9 | 21 | 14 | 14 | D3h, [3,2], (*223) | 12 | |||||||
52 | Augmented pentagonal prism | 11 | 19 | 10 | 4 | 4 | 2 | C2v, [2], (*22) | 4 | |||||
53 | Biaugmented pentagonal prism | 12 | 23 | 13 | 8 | 3 | 2 | C2v, [2], (*22) | 4 | |||||
54 | Augmented hexagonal prism | 13 | 22 | 11 | 4 | 5 | 2 | C2v, [2], (*22) | 4 | |||||
55 | Parabiaugmented hexagonal prism | 14 | 26 | 14 | 8 | 4 | 2 | D2h, [2,2], (*222) | 8 | |||||
56 | Metabiaugmented hexagonal prism | 14 | 26 | 14 | 8 | 4 | 2 | C2v, [2], (*22) | 4 | |||||
57 | Triaugmented hexagonal prism | 15 | 30 | 17 | 12 | 3 | 2 | D3h, [3,2], (*223) | 12 | |||||
58 | Augmented dodecahedron | 21 | 35 | 16 | 5 | 11 | C5v, [5], (*55) | 10 | ||||||
59 | Parabiaugmented dodecahedron | 22 | 40 | 20 | 10 | 10 | D5d, [2+,10], (2*5) | 20 | ||||||
60 | Metabiaugmented dodecahedron | 22 | 40 | 20 | 10 | 10 | C2v, [2], (*22) | 4 | ||||||
61 | Triaugmented dodecahedron | 23 | 45 | 24 | 15 | 9 | C3v, [3], (*33) | 6 | ||||||
62 | Metabidiminished icosahedron | 10 | 20 | 12 | 10 | 2 | C2v, [2], (*22) | 4 | ||||||
63 | Tridiminished icosahedron | 9 | 15 | 8 | 5 | 3 | C3v, [3], (*33) | 6 | ||||||
64 | Augmented tridiminished icosahedron | 10 | 18 | 10 | 7 | 3 | C3v, [3], (*33) | 6 | ||||||
65 | Augmented truncated tetrahedron | 15 | 27 | 14 | 8 | 3 | 3 | C3v, [3], (*33) | 6 | |||||
66 | Augmented truncated cube | 28 | 48 | 22 | 12 | 5 | 5 | C4v, [4], (*44) | 8 | |||||
67 | Biaugmented truncated cube | 32 | 60 | 30 | 16 | 10 | 4 | D4h, [4,2], (*224) | 16 | |||||
68 | Augmented truncated dodecahedron | 65 | 105 | 42 | 25 | 5 | 1 | 11 | C5v, [5], (*55) | 10 | ||||
69 | Parabiaugmented truncated dodecahedron | 70 | 120 | 52 | 30 | 10 | 2 | 10 | D5d, [2+,10], (2*5) | 20 | ||||
70 | Metabiaugmented truncated dodecahedron | 70 | 120 | 52 | 30 | 10 | 2 | 10 | C2v, [2], (*22) | 4 | ||||
71 | Triaugmented truncated dodecahedron | 75 | 135 | 62 | 35 | 15 | 3 | 9 | C3v, [3], (*33) | 6 | ||||
72 | Gyrate rhombicosidodecahedron | 60 | 120 | 62 | 20 | 30 | 12 | C5v, [5], (*55) | 10 | |||||
73 | Parabigyrate rhombicosidodecahedron | 60 | 120 | 62 | 20 | 30 | 12 | D5d, [2+,10], (2*5) | 20 | |||||
74 | Metabigyrate rhombicosidodecahedron | 60 | 120 | 62 | 20 | 30 | 12 | C2v, [2], (*22) | 4 | |||||
75 | Trigyrate rhombicosidodecahedron | 60 | 120 | 62 | 20 | 30 | 12 | C3v, [3], (*33) | 6 | |||||
76 | Diminished rhombicosidodecahedron | 55 | 105 | 52 | 15 | 25 | 11 | 1 | C5v, [5], (*55) | 10 | ||||
77 | Paragyrate diminished rhombicosidodecahedron | 55 | 105 | 52 | 15 | 25 | 11 | 1 | C5v, [5], (*55) | 10 | ||||
78 | Metagyrate diminished rhombicosidodecahedron | 55 | 105 | 52 | 15 | 25 | 11 | 1 | Cs, [ ], (*11) | 2 | ||||
79 | Bigyrate diminished rhombicosidodecahedron | 55 | 105 | 52 | 15 | 25 | 11 | 1 | Cs, [ ], (*11) | 2 | ||||
80 | Parabidiminished rhombicosidodecahedron | 50 | 90 | 42 | 10 | 20 | 10 | 2 | D5d, [2+,10], (2*5) | 20 | ||||
81 | Metabidiminished rhombicosidodecahedron | 50 | 90 | 42 | 10 | 20 | 10 | 2 | C2v, [2], (*22) | 4 | ||||
82 | Gyrate bidiminished rhombicosidodecahedron | 50 | 90 | 42 | 10 | 20 | 10 | 2 | Cs, [ ], (*11) | 2 | ||||
83 | Tridiminished rhombicosidodecahedron | 45 | 75 | 32 | 5 | 15 | 9 | 3 | C3v, [3], (*33) | 6 | ||||
84 | Snub disphenoid | 8 | 18 | 12 | 12 | D2d, [2+,4], (2*2) | 8 | |||||||
85 | Snub square antiprism | 16 | 40 | 26 | 24 | 2 | D4d, [2+,8], (2*4) | 16 | ||||||
86 | Sphenocorona | 10 | 22 | 14 | 12 | 2 | C2v, [2], (*22) | 4 | ||||||
87 | Augmented sphenocorona | 11 | 26 | 17 | 16 | 1 | Cs, [ ], (*11) | 2 | ||||||
88 | Sphenomegacorona | 12 | 28 | 18 | 16 | 2 | C2v, [2], (*22) | 4 | ||||||
89 | Hebesphenomegacorona | 14 | 33 | 21 | 18 | 3 | C2v, [2], (*22) | 4 | ||||||
90 | Disphenocingulum | 16 | 38 | 24 | 20 | 4 | D2d, [2+,4], (2*2) | 8 | ||||||
91 | Bilunabirotunda | 14 | 26 | 14 | 8 | 2 | 4 | D2h, [2,2], (*222) | 8 | |||||
92 | Triangular hebesphenorotunda | 18 | 36 | 20 | 13 | 3 | 3 | 1 | C3v, [3], (*33) | 6 |
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
The square pyramid J1 has the fewest vertices (5), the fewest edges (8), and the fewest faces (5).
The triaugmented truncated dodecahedron J71 has the most vertices (75) and the most edges (135). It also has the highest number of faces (62), along with the gyrate rhombicosidodecahedron J72, the parabigyrate rhombicosidodecahedron J73, the metabigyrate rhombicosidodecahedron J74, and the trigyrate rhombicosidodecahedron J75.
المساحة السطحية
Since all faces of Johnson solids are regular polygons with 3, 4, 5, 6, 8, or 10 sides, and since all these polygons have the same edge length a, the surface area of a Johnson solid can be calculated as
where the Fn are the polygonal face counts in the previous table and
is the area of a regular polygon with n sides of length a. In terms of radicals, one has
resulting in the following table of surface areas.
For a fixed edge length, the triangular dipyramid J12 has the smallest surface area and the triaugmented truncated dodecahedron J71 has the largest, more than 40 times larger.
الحجم
The following table lists the volume of each Johnson solid. Here V is the volume (not the number of vertices, as in the first table) and a is the edge length.
The source for this table is the PolyhedronData[..., "Volume"] command in Wolfram Research's Mathematica.
These volumes can be calculated from a set of vertex coordinates; such coordinates are known for all 92 Johnson solids. A conceptually simple approach is to triangulate the surface of the solid (for example, by adding an extra point in the center of each non-triangular face) and choose some interior point as an "origin" so that the interior can be subdivided into irregular tetrahedra. Each tetrahedron has one vertex at the origin inside and three vertices on the surface. The volume of the solid is then the sum of the volumes of these tetrahedra. There is a simple formula for the volume of an irregular tetrahedron.
For a fixed edge length, the square pyramid J1 and the triangular dipyramid J12 have the smallest volume and the triaugmented truncated dodecahedron J71 has the largest, more than 390 times larger.
Thirteen of the 92 Johnson solids have volumes for which V/a3 is not a number expressible using radicals. These values are the greatest real root of the following polynomials.
Jn | Polynomial |
---|---|
23 |
6561 x8
− 52488 x7
+ 113724 x6
− 9720 x5
|
24 |
1679616 x8
− 11197440 x7
+ 27060480 x6
+ 35769600 x5
|
25 |
1679616 x8
− 50388480 x7
+ 603262080 x6
− 3520972800 x5
|
45 |
6561 x8
− 104976 x7
+ 594864 x6
− 1384128 x5
|
46 |
6561 x8
− 87480 x7
+ 313470 x6
+ 753300 x5
|
47 |
1679616 x8
− 61585920 x7
+ 851472000 x6
− 5108832000 x5
|
48 |
6561 x8
− 393660 x7
+ 9316620 x6
− 108207900 x5
|
84 |
5832 x6 − 1377 x4 − 2160 x2 − 4 |
85 |
531441 x12
− 85726026 x8
− 48347280 x6
|
87 |
45137758519296 x16
− 110336743047168 x14
− 191069246324736 x12
+ 209269081571328 x10
|
88 |
521578814501447328359509917696 x32
− 985204427391622731345740955648 x30
|
89 |
47330370277129322496 x20
− 722445512980071186432 x18
|
90 |
1213025622610333925376 x24
+ 54451372392730545094656 x22
|
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Inradius, midradius, and circumradius
The following table lists the radius Ri of the insphere, the radius Rm of the midsphere, and the radius Rc of the circumsphere, each divided by the edge length a, when these spheres exist.
A polyhedron does not necessarily have an insphere, or a midsphere, or a circumsphere. For example, it does not have a circumsphere unless all its vertices lie on some sphere. The Johnson solids, having less symmetry than, say, the Platonic solids, lack many of these spheres. Only J1 and J2 possess all three of these spheres.
The source for this table is the PolyhedronData[..., "Inradius"], PolyhedronData[..., "Midradius"], and PolyhedronData[..., "Circumradius"] commands in Wolfram Research's Mathematica. The output has been simplified to a consistent form in terms of radicals.
Jn | Ri/a (approx.) | Ri/a (exact) | Rm/a (approx.) | Rm/a (exact) | Rc/a (approx.) | Rc/a (exact) |
---|---|---|---|---|---|---|
1 | 0.258819045 | 0.500000000 | 0.707106781 | |||
2 | 0.232788309 | 0.809016994 | 0.951056516 | |||
3 | - | - | 0.866025404 | 1.000000000 | ||
4 | - | - | 1.306562965 | 1.398966326 | ||
5 | - | - | 2.176250899 | 2.232950509 | ||
6 | - | - | 1.538841769 | 1.618033989 | ||
7 | - | - | - | - | - | - |
8 | - | - | - | - | - | - |
9 | - | - | - | - | - | - |
10 | - | - | - | - | - | - |
11 | - | - | 0.809016994 | 0.951056516 | ||
12 | 0.272165527 | - | - | - | - | |
13 | 0.417774579 | - | - | - | - | |
14 | - | - | - | - | - | - |
15 | - | - | - | - | - | - |
16 | - | - | - | - | - | - |
17 | - | - | - | - | - | - |
18 | - | - | - | - | - | - |
19 | - | - | 1.306562965 | 1.398966326 | ||
20 | - | - | - | - | - | - |
21 | - | - | - | - | - | - |
22 | - | - | - | - | - | - |
23 | - | - | - | - | - | - |
24 | - | - | - | - | - | - |
25 | - | - | - | - | - | - |
26 | - | - | - | - | - | - |
27 | - | - | 0.866025404 | 1.000000000 | ||
28 | - | - | - | - | - | - |
29 | - | - | - | - | - | - |
30 | - | - | - | - | - | - |
31 | - | - | - | - | - | - |
32 | - | - | - | - | - | - |
33 | - | - | - | - | - | - |
34 | - | - | 1.538841769 | 1.618033989 | ||
35 | - | - | - | - | - | - |
36 | - | - | - | - | - | - |
37 | - | - | 1.306562965 | 1.398966326 | ||
38 | - | - | - | - | - | - |
39 | - | - | - | - | - | - |
40 | - | - | - | - | - | - |
41 | - | - | - | - | - | - |
42 | - | - | - | - | - | - |
43 | - | - | - | - | - | - |
44 | - | - | - | - | - | - |
45 | - | - | - | - | - | - |
46 | - | - | - | - | - | - |
47 | - | - | - | - | - | - |
48 | - | - | - | - | - | - |
49 | - | - | - | - | - | - |
50 | - | - | - | - | - | - |
51 | - | - | - | - | - | - |
52 | - | - | - | - | - | - |
53 | - | - | - | - | - | - |
54 | - | - | - | - | - | - |
55 | - | - | - | - | - | - |
56 | - | - | - | - | - | - |
57 | - | - | - | - | - | - |
58 | - | - | - | - | - | - |
59 | - | - | - | - | - | - |
60 | - | - | - | - | - | - |
61 | - | - | - | - | - | - |
62 | - | - | 0.809016994 | 0.951056516 | ||
63 | - | - | 0.809016994 | 0.951056516 | ||
64 | - | - | - | - | - | - |
65 | - | - | - | - | - | - |
66 | - | - | - | - | - | - |
67 | - | - | - | - | - | - |
68 | - | - | - | - | - | - |
69 | - | - | - | - | - | - |
70 | - | - | - | - | - | - |
71 | - | - | - | - | - | - |
72 | - | - | 2.176250899 | 2.232950509 | ||
73 | - | - | 2.176250899 | 2.232950509 | ||
74 | - | - | 2.176250899 | 2.232950509 | ||
75 | - | - | 2.176250899 | 2.232950509 | ||
76 | - | - | 2.176250899 | 2.232950509 | ||
77 | - | - | 2.176250899 | 2.232950509 | ||
78 | - | - | 2.176250899 | 2.232950509 | ||
79 | - | - | 2.176250899 | 2.232950509 | ||
80 | - | - | 2.176250899 | 2.232950509 | ||
81 | - | - | 2.176250899 | 2.232950509 | ||
82 | - | - | 2.176250899 | 2.232950509 | ||
83 | - | - | 2.176250899 | 2.232950509 | ||
84 | - | - | - | - | - | - |
85 | - | - | - | - | - | - |
86 | - | - | - | - | - | - |
87 | - | - | - | - | - | - |
88 | - | - | - | - | - | - |
89 | - | - | - | - | - | - |
90 | - | - | - | - | - | - |
91 | - | - | - | - | - | - |
92 | - | - | - | - | - | - |
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.
وصلات خارجية
- Sylvain Gagnon, "Convex polyhedra with regular faces[dead link]", Structural Topology, No. 6, 1982, 83-95.
- Johnson Solids by George W. Hart.
- Images of all 92 solids, categorized, on one page
- Eric W. Weisstein, Johnson Solid at MathWorld.
- VRML models of Johnson Solids by Jim McNeill
- VRML models of Johnson Solids by Vladimir Bulatov