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 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.

Vertices, edges, faces, and symmetry

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.

Surface area

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.

Jn Solid name A/a2 (approx.) A/a2 (exact)
1 Square pyramid 2.732050808
2 Pentagonal pyramid 3.885540910
3 Triangular cupola 7.330127019
4 Square cupola 11.560477932
5 Pentagonal cupola 16.579749753
6 Pentagonal rotunda 22.347200265
7 Elongated triangular pyramid 4.732050808
8 Elongated square pyramid 6.732050808
9 Elongated pentagonal pyramid 8.885540910
10 Gyroelongated square pyramid 6.196152423
11 Gyroelongated pentagonal pyramid 8.215667929
12 Triangular dipyramid 2.598076211
13 Pentagonal dipyramid 4.330127019
14 Elongated triangular dipyramid 5.598076211
15 Elongated square dipyramid 7.464101615
16 Elongated pentagonal dipyramid 9.330127019
17 Gyroelongated square dipyramid 6.928203230
18 Elongated triangular cupola 13.330127019
19 Elongated square cupola 19.560477932
20 Elongated pentagonal cupola 26.579749753
21 Elongated pentagonal rotunda 32.347200265
22 Gyroelongated triangular cupola 12.526279442
23 Gyroelongated square cupola 18.488681163
24 Gyroelongated pentagonal cupola 25.240003791
25 Gyroelongated pentagonal rotunda 31.007454303
26 Gyrobifastigium 5.732050808
27 Triangular orthobicupola 9.464101615
28 Square orthobicupola 13.464101615
29 Square gyrobicupola 13.464101615
30 Pentagonal orthobicupola 17.771081820
31 Pentagonal gyrobicupola 17.771081820
32 Pentagonal orthocupolarotunda 23.538532333
33 Pentagonal gyrocupolarotunda 23.538532333
34 Pentagonal orthobirotunda 29.305982845
35 Elongated triangular orthobicupola 15.464101615
36 Elongated triangular gyrobicupola 15.464101615
37 Elongated square gyrobicupola 21.464101615
38 Elongated pentagonal orthobicupola 27.771081820
39 Elongated pentagonal gyrobicupola 27.771081820
40 Elongated pentagonal orthocupolarotunda 33.538532333
41 Elongated pentagonal gyrocupolarotunda 33.538532333
42 Elongated pentagonal orthobirotunda 39.305982845
43 Elongated pentagonal gyrobirotunda 39.305982845
44 Gyroelongated triangular bicupola 14.660254038
45 Gyroelongated square bicupola 20.392304845
46 Gyroelongated pentagonal bicupola 26.431335858
47 Gyroelongated pentagonal cupolarotunda 32.198786370
48 Gyroelongated pentagonal birotunda 37.966236883
49 Augmented triangular prism 4.598076211
50 Biaugmented triangular prism 5.330127019
51 Triaugmented triangular prism 6.062177826
52 Augmented pentagonal prism 9.173005609
53 Biaugmented pentagonal prism 9.905056416
54 Augmented hexagonal prism 11.928203230
55 Parabiaugmented hexagonal prism 12.660254038
56 Metabiaugmented hexagonal prism 12.660254038
57 Triaugmented hexagonal prism 13.392304845
58 Augmented dodecahedron 21.090314916
59 Parabiaugmented dodecahedron 21.534901025
60 Metabiaugmented dodecahedron 21.534901025
61 Triaugmented dodecahedron 21.979487134
62 Metabidiminished icosahedron 7.771081820
63 Tridiminished icosahedron 7.326495711
64 Augmented tridiminished icosahedron 8.192521115
65 Augmented truncated tetrahedron 14.258330249
66 Augmented truncated cube 34.338288046
67 Biaugmented truncated cube 36.241911729
68 Augmented truncated dodecahedron 102.182092220
69 Parabiaugmented truncated dodecahedron 103.373424287
70 Metabiaugmented truncated dodecahedron 103.373424287
71 Triaugmented truncated dodecahedron 104.564756354
72 Gyrate rhombicosidodecahedron 59.305982845
73 Parabigyrate rhombicosidodecahedron 59.305982845
74 Metabigyrate rhombicosidodecahedron 59.305982845
75 Trigyrate rhombicosidodecahedron 59.305982845
76 Diminished rhombicosidodecahedron 58.114650778
77 Paragyrate diminished rhombicosidodecahedron 58.114650778
78 Metagyrate diminished rhombicosidodecahedron 58.114650778
79 Bigyrate diminished rhombicosidodecahedron 58.114650778
80 Parabidiminished rhombicosidodecahedron 56.923318711
81 Metabidiminished rhombicosidodecahedron 56.923318711
82 Gyrate bidiminished rhombicosidodecahedron 56.923318711
83 Tridiminished rhombicosidodecahedron 55.731986644
84 Snub disphenoid 5.196152423
85 Snub square antiprism 12.392304845
86 Sphenocorona 7.196152423
87 Augmented sphenocorona 7.928203230
88 Sphenomegacorona 8.928203230
89 Hebesphenomegacorona 10.794228634
90 Disphenocingulum 12.660254038
91 Bilunabirotunda 12.346011217
92 Triangular hebesphenorotunda 16.388673538

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.

Volume

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.

Jn Solid name V/a3 (approx.) V/a3 (exact)
1 Square pyramid 0.235702260
2 Pentagonal pyramid 0.301502832
3 Triangular cupola 1.178511302
4 Square cupola 1.942809042
5 Pentagonal cupola 2.324045318
6 Pentagonal rotunda 6.917762968
7 Elongated triangular pyramid 0.550863832
8 Elongated square pyramid 1.235702260
9 Elongated pentagonal pyramid 2.021980233
10 Gyroelongated square pyramid 1.192702242
11 Gyroelongated pentagonal pyramid 1.880192158
12 Triangular dipyramid 0.235702260
13 Pentagonal dipyramid 0.603005665
14 Elongated triangular dipyramid 0.668714962
15 Elongated square dipyramid 1.471404521
16 Elongated pentagonal dipyramid 2.323483065
17 Gyroelongated square dipyramid 1.428404503
18 Elongated triangular cupola 3.776587513
19 Elongated square cupola 6.771236166
20 Elongated pentagonal cupola 10.018254161
21 Elongated pentagonal rotunda 14.611971811
22 Gyroelongated triangular cupola 3.516053091
23 Gyroelongated square cupola 6.210765792 root of polynomial of degree 8
24 Gyroelongated pentagonal cupola 9.073333194 root of polynomial of degree 8
25 Gyroelongated pentagonal rotunda 13.667050844 root of polynomial of degree 8
26 Gyrobifastigium 0.866025404
27 Triangular orthobicupola 2.357022604
28 Square orthobicupola 3.885618083
29 Square gyrobicupola 3.885618083
30 Pentagonal orthobicupola 4.648090637
31 Pentagonal gyrobicupola 4.648090637
32 Pentagonal orthocupolarotunda 9.241808286
33 Pentagonal gyrocupolarotunda 9.241808286
34 Pentagonal orthobirotunda 13.835525936
35 Elongated triangular orthobicupola 4.955098815
36 Elongated triangular gyrobicupola 4.955098815
37 Elongated square gyrobicupola 8.714045208
38 Elongated pentagonal orthobicupola 12.342299480
39 Elongated pentagonal gyrobicupola 12.342299480
40 Elongated pentagonal orthocupolarotunda 16.936017129
41 Elongated pentagonal gyrocupolarotunda 16.936017129
42 Elongated pentagonal orthobirotunda 21.529734779
43 Elongated pentagonal gyrobirotunda 21.529734779
44 Gyroelongated triangular bicupola 4.694564393
45 Gyroelongated square bicupola 8.153574834 root of polynomial of degree 8
46 Gyroelongated pentagonal bicupola 11.397378512 root of polynomial of degree 8
47 Gyroelongated pentagonal cupolarotunda 15.991096162 root of polynomial of degree 8
48 Gyroelongated pentagonal birotunda 20.584813812 root of polynomial of degree 8
49 Augmented triangular prism 0.668714962
50 Biaugmented triangular prism 0.904417223
51 Triaugmented triangular prism 1.140119483
52 Augmented pentagonal prism 1.956179661
53 Biaugmented pentagonal prism 2.191881921
54 Augmented hexagonal prism 2.833778472
55 Parabiaugmented hexagonal prism 3.069480732
56 Metabiaugmented hexagonal prism 3.069480732
57 Triaugmented hexagonal prism 3.305182993
58 Augmented dodecahedron 7.964621793
59 Parabiaugmented dodecahedron 8.266124625
60 Metabiaugmented dodecahedron 8.266124625
61 Triaugmented dodecahedron 8.567627458
62 Metabidiminished icosahedron 1.578689326
63 Tridiminished icosahedron 1.277186493
64 Augmented tridiminished icosahedron 1.395037624
65 Augmented truncated tetrahedron 3.889087297
66 Augmented truncated cube 15.542472333
67 Biaugmented truncated cube 17.485281374
68 Augmented truncated dodecahedron 87.363709878
69 Parabiaugmented truncated dodecahedron 89.687755196
70 Metabiaugmented truncated dodecahedron 89.687755196
71 Triaugmented truncated dodecahedron 92.011800514
72 Gyrate rhombicosidodecahedron 41.615323782
73 Parabigyrate rhombicosidodecahedron 41.615323782
74 Metabigyrate rhombicosidodecahedron 41.615323782
75 Trigyrate rhombicosidodecahedron 41.615323782
76 Diminished rhombicosidodecahedron 39.291278464
77 Paragyrate diminished rhombicosidodecahedron 39.291278464
78 Metagyrate diminished rhombicosidodecahedron 39.291278464
79 Bigyrate diminished rhombicosidodecahedron 39.291278464
80 Parabidiminished rhombicosidodecahedron 36.967233146
81 Metabidiminished rhombicosidodecahedron 36.967233146
82 Gyrate bidiminished rhombicosidodecahedron 36.967233146
83 Tridiminished rhombicosidodecahedron 34.643187827
84 Snub disphenoid 0.859493646 root of polynomial of degree 6
85 Snub square antiprism 3.601222010 root of polynomial of degree 12
86 Sphenocorona 1.515351640
87 Augmented sphenocorona 1.751053900 root of polynomial of degree 16
88 Sphenomegacorona 1.948108229 root of polynomial of degree 32
89 Hebesphenomegacorona 2.912910415 root of polynomial of degree 20
90 Disphenocingulum 3.777645342 root of polynomial of degree 24
91 Bilunabirotunda 3.093717650
92 Triangular hebesphenorotunda 5.108745974

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
1616922 x4 + 396360 x3 + 1537020 x2 178632 x 3391

24

1679616 x8 11197440 x7 + 27060480 x6 + 35769600 x5
4456749600 x4 10714248000 x3 + 3828402000 x2 + 13859430000 x + 5340175625

25

1679616 x8 50388480 x7 + 603262080 x6 3520972800 x5
+ 5215460400 x4 + 4128624000 x3 8894943000 x2 + 3881385000 x 424924375

45

6561 x8 104976 x7 + 594864 x6 1384128 x5
552096 x4 + 1569024 x3 + 246528 x2 119808 x + 4352

46

6561 x8 87480 x7 + 313470 x6 + 753300 x5
22424850 x4 84591000 x3 85909500 x2 + 8715000 x + 35547500

47

1679616 x8 61585920 x7 + 851472000 x6 5108832000 x5
+ 4745790000 x4 + 21346200000 x3 29019375000 x2 4576875000 x 405859375

48

6561 x8 393660 x7 + 9316620 x6 108207900 x5
+ 601832025 x4 1417189500 x3 + 965841750 x2 + 597667500 x 668786875

84

5832 x6 1377 x4 2160 x2 4

85

531441 x12 85726026 x8 48347280 x6
+ 11588832 x4 + 4759488 x2 892448

87

45137758519296 x16 110336743047168 x14 191069246324736 x12 + 209269081571328 x10
+ 364547659290624 x8 58793017190400 x6 + 3306865979520 x4 1275399855936 x2 + 1439671249

88

521578814501447328359509917696 x32 985204427391622731345740955648 x30
16645447351681991898880656015360 x28 + 79710816694053483249372512649216 x26
152195045391070538203422101864448 x24 + 156280253448056209478031589244928 x22
96188116617075838858708654227456 x20 + 30636368373570166303441645731840 x18
+ 5828527077458909552923002273792 x16 8060049780765551057159394951168 x14
+ 1018074792115156107372011716608 x12 + 35220131544370794950945931264 x10
+ 327511698517355918956755959808 x8 116978732884218191486738706432 x6
+ 10231563774949176791703149568 x4 366323949299263261553952192 x2
+ 3071435678740442112675625

89

47330370277129322496 x20 722445512980071186432 x18
+ 3596480447590271287296 x16 8432333285523990773760 x14
+ 8973584611317745975296 x12 3065290664181478981632 x10
+ 366229890219212144640 x8 8337259437908852736 x6
22211277300912896 x4 + 132615435213216 x2
+ 2693461945329

90

1213025622610333925376 x24 + 54451372392730545094656 x22
796837093078664749252608 x20 4133410366404688544268288 x18
+ 20902529024429842816303104 x16 133907540390420673677230080 x14
+ 246234688242991598853881856 x12 63327534106871321714442240 x10
+ 14389309497459555704164608 x8 + 48042947402464500749392128 x6
5891096640600351061013664 x4 3212114716816853362953264 x2 + 479556973248657693884401

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.
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