Sólidos de Johnson

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Um sólido de Johnson é um Poliedro onde as faces são polígonos regulares e que não são sólidos de Platão, nem um sólido de Arquimedes, nem um prisma nem um antiprisma.

Norman Johnson elaborou uma lista de 92 sólidos em 1966, nomeou e numerou todos. Ele não provou que eram apenas 92, mas fazia idéia que eram apenas 92. Victor Zalgaller provou em 1969 que Johnson estava correto.

Lista completa[editar | editar código-fonte]

Jn Nome Planificação Imagem V A F F3 F4 F5 F6 F8 F10 Simetría
1 pirâmide quadrada Johnson solid 1 net.png Square pyramid.png 5 8 5 4 1 C4v
2 pirâmide pentagonal Johnson solid 2 net.png Pentagonal pyramid.png 6 10 6 5 1 C5v
3 cúpula triangular Johnson solid 3 net.png Triangular cupola.png 9 15 8 4 3 1 C3v
4 cúpula quadrada Johnson solid 4 net.png Square cupola.png 12 20 10 4 5 1 C4v
5 cúpula pentagonal Johnson solid 5 net.png Pentagonal cupola.png 15 25 12 5 5 1 1 C5v
6 rotunda pentagonal Johnson solid 6 net.png Pentagonal rotunda.png 20 35 17 10 6 1 C5v
7 pirâmide triangular alongada Johnson solid 7 net.png Elongated triangular pyramid.png 7 12 7 4 3 C3v
8 pirâmide quadrada alongada Johnson solid 8 net.png Elongated square pyramid.png 9 16 9 4 5 C4v
9 pirâmide pentagonal alongada Johnson solid 9 net.png Elongated pentagonal pyramid.png 11 20 11 5 5 1 C5v
10 pirâmide quadrada giralongada Johnson solid 10 net.png Gyroelongated square pyramid.png 9 20 13 12 1 C4v
11 pirâmide pentagonal giralongada Johnson solid 11 net.png Gyroelongated pentagonal pyramid.png 11 25 16 15 1 C5v
12 bipirâmide triangular Johnson solid 12 net.png Triangular dipyramid.png 5 9 6 6 D3h
13 bipirâmide pentagonal Johnson solid 13 net.png Pentagonal dipyramid.png 7 15 10 10 D5h
14 bipirâmide triangular alongada Johnson solid 14 net.png Elongated triangular dipyramid.png 8 15 9 6 3 D3h
15 bipirâmide quadrada alongada Johnson solid 15 net.png Elongated square dipyramid.png 10 20 12 8 4 D4h
16 bipirâmide pentagonal alongada Johnson solid 16 net.png Elongated pentagonal dipyramid.png 12 25 15 10 5 D5h
17 bipirâmide quadrada giralongada Johnson solid 17 net.png Gyroelongated square dipyramid.png 10 24 16 16 D4d
18 cúpula triangular alongada Johnson solid 18 net.png Elongated triangular cupola.png 15 27 14 4 9 1 C3v
19 cúpula quadrada alongada Johnson solid 19 net.png Elongated square cupola.png 20 36 18 4 13 1 C4v
20 cúpula pentagonal alongada Johnson solid 20 net.png Elongated pentagonal cupola.png 25 45 22 5 15 1 1 C5v
21 rotunda pentagonal alongada Johnson solid 21 net.png Elongated pentagonal rotunda.png 30 55 27 10 10 6 1 C5v
22 cúpula triangular giralongada Johnson solid 22 net.png Gyroelongated triangular cupola.png 15 33 20 16 3 1 C3v
23 cúpula quadrada giralongada Johnson solid 23 net.png Gyroelongated square cupola.png 20 44 26 20 5 1 C4v
24 cúpula pentagonal giralongada Johnson solid 24 net.png Gyroelongated pentagonal cupola.png 25 55 32 25 5 1 1 C5v
25 rotunda pentagonal giralongada Johnson solid 25 net.png Gyroelongated pentagonal rotunda.png 30 65 37 30 6 1 C5v
26 girobiprisma triangular Johnson solid 26 net.png Gyrobifastigium.png 8 14 8 4 4 D2d
27 ortobicúpula triangular Johnson solid 27 net.png Triangular orthobicupola.png 12 24 14 8 6 D3h
28 ortobicúpula quadrada Johnson solid 28 net.png Square orthobicupola.png 16 32 18 8 10 D4h
29 girobicúpula quadrada Johnson solid 29 net.png Square gyrobicupola.png 16 32 18 8 10 D4d
30 ortobicúpula pentagonal Johnson solid 30 net.png Pentagonal orthobicupola.png 20 40 22 10 10 2 D5h
31 girobicúpula pentagonal Johnson solid 31 net.png Pentagonal gyrobicupola.png 20 40 22 10 10 2 D5d
32 ortocúpula-rotonde pentagonal Johnson solid 32 net.png Pentagonal orthocupolarotunda.png 25 50 27 15 5 7 C5v
33 girocúpula-rotonde pentagonal Johnson solid 33 net.png Pentagonal gyrocupolarotunda.png 25 50 27 15 5 7 C5v
34 ortobirotonde pentagonal Johnson solid 34 net.png Pentagonal orthobirotunda.png 30 60 32 20 12 D5h
35 ortobicúpula triangular alongada Johnson solid 35 net.png Elongated triangular orthobicupola.png 18 36 20 8 12 D3h
36 girobicúpula triangular alongada Johnson solid 36 net.png Elongated triangular gyrobicupola.png 18 36 20 8 12 D3d
37 girobicúpula quadrada alongada Johnson solid 37 net.png Elongated square gyrobicupola.png 24 48 26 8 18 D4d
38 ortobicúpula pentagonal alongada Johnson solid 38 net.png Elongated pentagonal orthobicupola.png 30 60 32 10 20 2 D5h
39 girobicúpula pentagonal alongada Johnson solid 39 net.png Elongated pentagonal gyrobicupola.png 30 60 32 10 20 2 D5d
40 ortocúpula-rotonde pentagonal alongada Johnson solid 40 net.png Elongated pentagonal orthocupolarotunda.png 35 70 37 15 15 7 C5v
41 girocúpula-rotonde pentagonal alongada Johnson solid 41 net.png Elongated pentagonal gyrocupolarotunda.png 35 70 37 15 15 7 C5v
42 ortobirotonde pentagonal alongada Johnson solid 42 net.png Elongated pentagonal orthobirotunda.png 40 80 42 20 10 12 D5h
43 girobirotonde pentagonal alongada Johnson solid 43 net.png Elongated pentagonal gyrobirotunda.png 40 80 42 20 10 12 D5d
44 bicúpula triangular giralongada Johnson solid 44 net.png Gyroelongated triangular bicupola.png 18 42 26 20 6 D3
45 bicúpula quadrada giralongada Johnson solid 45 net.png Gyroelongated square bicupola.png 24 56 34 24 10 D4
46 bicúpula pentagonal giralongada Johnson solid 46 net.png Gyroelongated pentagonal bicupola.png 30 70 42 30 10 2 D5
47 cúpula-rotonde pentagonal giralongada Johnson solid 47 net.png Gyroelongated pentagonal cupolarotunda.png 35 80 47 35 5 7 C5
48 birotonde pentagonal giralongada Johnson solid 48 net.png Gyroelongated pentagonal birotunda.png 40 90 52 40 12 D5
49 prisma triangular aumentado Johnson solid 49 net.png Augmented triangular prism.png 7 13 8 6 2 C2v
50 prisma triangular biaumentado Johnson solid 50 net.png Biaugmented triangular prism.png 8 17 11 10 1 C2v
51 prisma triangular triaumentado Johnson solid 51 net.png Triaugmented triangular prism.png 9 21 14 14 D3h
52 prisma pentagonal aumentado Johnson solid 52 net.png Augmented pentagonal prism.png 11 19 10 4 4 2 C2v
53 prisma pentagonal biaumentado Johnson solid 53 net.png Biaugmented pentagonal prism.png 12 23 13 8 3 2 C2v
54 prisma hexagonal aumentado Johnson solid 54 net.png Augmented hexagonal prism.png 13 22 11 4 5 2 C2v
55 prisma hexagonal parabiaumentado Johnson solid 55 net.png Parabiaugmented hexagonal prism.png 14 26 14 8 4 2 D2h
56 prisma hexagonal metabiaumentado Johnson solid 56 net.png Metabiaugmented hexagonal prism.png 14 26 14 8 4 2 C2v
57 prisma hexagonal triaumentado Johnson solid 57 net.png Triaugmented hexagonal prism.png 15 30 17 12 3 2 D3h
58 dodecaedro aumentado Johnson solid 58 net.png Augmented dodecahedron.png 21 35 16 5 11 C5v
59 dodecaedro parabiaumentado Johnson solid 59 net.png Parabiaugmented dodecahedron.png 22 40 20 10 10 D5d
60 dodecaedro metabiaumentado Johnson solid 60 net.png Metabiaugmented dodecahedron.png 22 40 20 10 10 C2v
61 dodecaedro triaumentado Johnson solid 61 net.png Triaugmented dodecahedron.png 23 45 24 15 9 C3v
62 icosaedro metabidiminuído Johnson solid 62 net.png Metabidiminished icosahedron.png 10 20 12 10 2 C2v
63 icosaedro tridiminuído Johnson solid 63 net.png Tridiminished icosahedron.png 9 15 8 5 3 C3v
64 icosaedro tridiminuído aumentado Johnson solid 64 net.png Augmented tridiminished icosahedron.png 10 18 10 7 3 C3v
65 tetraedro truncado aumentado Johnson solid 65 net.png Augmented truncated tetrahedron.png 15 27 14 8 3 3 C3v
66 cubo truncado aumentado Johnson solid 66 net.png Augmented truncated cube.png 28 48 22 12 5 5 C4v
67 cubo truncado biaumentado Johnson solid 67 net.png Biaugmented truncated cube.png 32 60 30 16 10 4 D4h
68 dodecaedro truncado aumentado Johnson solid 68 net.png Augmented truncated dodecahedron.png 65 105 42 25 5 1 11 C5v
69 dodecaedro truncado parabiaumentado Johnson solid 69 net.png Parabiaugmented truncated dodecahedron.png 70 120 52 30 10 2 10 D5d
70 dodecaedro truncado metabiaumentado Johnson solid 70 net.png Metabiaugmented truncated dodecahedron.png 70 120 52 30 10 2 10 C2v
71 dodecaedro truncado triaumentado Johnson solid 71 net.png Triaugmented truncated dodecahedron.png 75 135 62 35 15 3 9 C3v
72 girorrombicosidodecaedro Johnson solid 72 net.png Gyrate rhombicosidodecahedron.png 60 120 62 20 30 12 C5v
73 parabigirorrombicosidodecaedro Johnson solid 73 net.png Parabigyrate rhombicosidodecahedron.png 60 120 62 20 30 12 D5d
74 metabigirorrombicosidodecaedro Johnson solid 74 net.png Metabigyrate rhombicosidodecahedron.png 60 120 62 20 30 12 C2v
75 trigirorrombicosidodecaedro Johnson solid 75 net.png Trigyrate rhombicosidodecahedron.png 60 120 62 20 30 12 C3v
76 rombicosidodecaedro diminuído Johnson solid 76 net.png Diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 C5v
77 rombicosidodecaedro paragirodiminuído Johnson solid 77 net.png Paragyrate diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 C5v
78 rombicosidodecaedro metagirodiminuído Johnson solid 78 net.png Metagyrate diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 Cs
79 rombicosidodecaedro bigirodiminuído Johnson solid 79 net.png Bigyrate diminished rhombicosidodecahedron.png 55 105 52 15 25 11 1 Cs
80 rombicosidodecaedro parabidiminuído Johnson solid 80 net.png Parabidiminished rhombicosidodecahedron.png 50 90 42 10 20 10 2 D5d
81 rombicosidodecaedro metabidiminuído Johnson solid 81 net.png Metabidiminished rhombicosidodecahedron.png 50 90 42 10 20 10 2 C2v
82 rombicosidodecaedro girobidiminuído Johnson solid 82 net.png Gyrate bidiminished rhombicosidodecahedron.png 50 90 42 10 20 10 2 Cs
83 rombicosidodecaedro tridiminuído Johnson solid 83 net.png Tridiminished rhombicosidodecahedron.png 45 75 32 5 15 9 3 C3v
84 disfenoide achatado Johnson solid 84 net.png Snub disphenoid.png 8 18 12 12 D2d
85 antiprisma quadrado achatado Johnson solid 85 net.png Snub square antiprism.png 16 40 26 24 2 D4d
86 esfenocorona Johnson solid 86 net.png Sphenocorona.png 10 22 14 12 2 C2v
87 esfenocorona aumentada Johnson solid 87 net.png Augmented sphenocorona.png 11 26 17 16 1 Cs
88 esfenomegacorona Johnson solid 88 net.png Sphenomegacorona.png 12 28 18 16 2 C2v
89 Hebesfenomegacorona Johnson solid 89 net.png Hebesphenomegacorona.png 14 33 21 18 3 C2v
90 disfenocingulum Johnson solid 90 net.png Disphenocingulum.png 16 38 24 20 4 D2d
91 birrotunda bilunar Johnson solid 91 net.png Bilunabirotunda.png 14 26 14 8 2 4 D2h
92 hebesfenorrotunda triangular Johnson solid 92 net.png Triangular hebesphenorotunda.png 18 36 20 13 3 3 1 C3v

Referências[editar | editar código-fonte]

  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contém a enumeração original dos 92 sólidos.
  • Victor A. Zalgaller, "Convex Polyhedra with Regular Faces", 1969.

Ligações externas[editar | editar código-fonte]