List of convex regular-faced polyhedra (Meta, 13)

From Higher Dimensions Database

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Three classes used in the table below:
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Key to classes used in the table below:
*'''B''' = [[Brick]]: point (x, y, z) implies point (±x, ±y, ±z)
*'''B''' = [[Brick]]: point (x, y, z) implies point (±x, ±y, ±z)
*'''C''' = [[Chiral]]: there is no 3D rotation which can map the polyhedron onto its reflection in ''x'' = 0
*'''C''' = [[Chiral]]: there is no 3D rotation which can map the polyhedron onto its reflection in ''x'' = 0

Revision as of 22:10, 10 January 2012

Key to classes used in the table below:

  • B = Brick: point (x, y, z) implies point (±x, ±y, ±z)
  • C = Chiral: there is no 3D rotation which can map the polyhedron onto its reflection in x = 0
  • S = Simple: exactly three faces meet at each vertex
  • Z = Zonotope: each face has point symmetry


Index Parent Elements Dual
Name Classes SSC2 Faces Edges Vertices Name Classes SSC2
Platonic solids (total 5)
1 Tetrahedron--S-Kt1 464 Self-dual
2 CubeB-SZKo1 6128 3. Octahedron (below)
3 OctahedronB---Ko4 8126 2. Cube (above)
4 DodecahedronB-S-Ki1 123020 5. Icosahedron (below)
5 IcosahedronB---Ki4 203012 4. Dodecahedron (above)
Archimedean solids (total 13)
6 Tetrahedral truncate--S-Kt3 81812 Triakis tetrahedron----Unknown
7 CuboctahedronB---Ko2 142412 Rhombic dodecahedronB--ZUnknown
8 Cubic truncateB-S-Ko3 143624 Triakis octahedronB---Unknown
9 Octahedral truncateB-S-Ko6 143624 Tetrakis hexahedronB---Unknown
10 Cuboctahedral rectateB---Ko5 264824 Deltoidal icositetrahedronB---Unknown
11 Cuboctahedral truncateB-SZKo7 267248 Disdyakis dodecahedronB---Unknown
12 Cubic snub-C--Ko0 386024 Pentagonal icositetrahedron-C--Unknown
13 IcosidodecahedronB---Ki2 326030 Rhombic triacontahedronB--ZUnknown
14 Dodecahedral truncateB-S-Ki3 329060 Triakis icosahedronB---Unknown
15 Icosahedral truncateB-S-Ki6 329030 Pentakis dodecahedronB---Unknown
16 Icosidodecahedral rectateB---Ki5 6212060 Deltoidal hexecontahedronB---Unknown
17 Icosidodecahedral truncateB-SZKi7 62180120 Disdyakis triacontahedronB---Unknown
18 Dodecahedral snub-C--Ki0 9215060 Pentagonal hexecontahedron-C--Unknown
Johnson solids (total 92)
19 = J1 Square pyramid----&G4 585 Self-dual
20 = J2 Pentagonal pyramid----&G5 6106 Self-dual
21 = J3 Triangular cupola----G3$H6 8159 Unknown
22 = J4 Square cupola----G4$H8 102012 Unknown
23 = J5 Pentagonal cupola----G5$H10 122515 Unknown
24 = J6 Pentagonal rotunda----G5$H5$H10 173520 Unknown
25 = J7 Elongated triangular pyramid
(elongated tetrahedron)
----Unknown 7127 Self-dual
26 = J8 Elongated square pyramid
(augmented cube)
----Unknown 9169 Self-dual
27 = J9 Elongated pentagonal pyramid----Unknown 112011 Self-dual
28 = J10 Gyroelongated square pyramid----Unknown 13209 Unknown
29 = J11 Gyroelongated pentagonal pyramid
(diminished icosahedron)
----Unknown 162511 Unknown
30 = J12 Triangular bipyramid----Unknown 695 Unknown
31 = J13 Pentagonal bipyramid----Unknown 10157 Unknown
32 = J14 Elongated triangular bipyramid----Unknown 9158 Unknown
33 = J15 Elongated square bipyramid
(biaugmented cube)
B---Unknown 122010 Unknown
34 = J16 Elongated pentagonal bipyramid----Unknown 152512 Unknown
35 = J17 Gyroelongated square bipyramid----Unknown 162410 Unknown
36 = J18 Elongated triangular cupola----Unknown 142715 Unknown
37 = J19 Elongated square cupola
(diminished rhombicuboctahedron)
----Unknown 183620 Unknown
38 = J20 Elongated pentagonal cupola----Unknown 224525 Unknown
39 = J21 Elongated pentagonal rotunda----Unknown 275530 Unknown
40 = J22 Gyroelongated triangular cupola----Unknown 203315 Unknown
41 = J23 Gyroelongated square cupola----Unknown 264420 Unknown
42 = J24 Gyroelongated pentagonal cupola----Unknown 325525 Unknown
43 = J25 Gyroelongated pentagonal rotunda----Unknown 376530 Unknown
44 = J26 Gyrobifastigium----Unknown 8148 Unknown
45 = J27 Triangular orthobicupola
(gyrate cuboctahedron)
----Unknown 142412 Unknown
46 = J28 Square orthobicupolaB---Unknown 183216 Unknown
47 = J29 Square gyrobicupola----Unknown 183216 Unknown
48 = J30 Pentagonal orthobicupola----Unknown 224020 Unknown
49 = J31 Pentagonal gyrobicupola----Unknown 224020 Unknown
50 = J32 Pentagonal orthocupolarotunda----Unknown 275025 Unknown
51 = J33 Pentagonal gyrocupolarotunda----Unknown 275025 Unknown
52 = J34 Pentagonal orthobirotunda
(gyrate icosidodecahedron)
----Unknown 326030 Unknown
53 = J35 Elongated triangular orthobicupola----Unknown 203618 Unknown
54 = J36 Elongated triangular gyrobicupola----Unknown 203618 Unknown
55 = J37 Elongated square gyrobicupola
(gyrate rhombicuboctahedron)
----Unknown 264824 Unknown
56 = J38 Elongated pentagonal orthobicupola----Unknown 326030 Unknown
57 = J39 Elongated pentagonal gyrobicupola----Unknown 326030 Unknown
58 = J40 Elongated pentagonal orthocupolarotunda----Unknown 377035 Unknown
59 = J41 Elongated pentagonal gyrocupolarotunda----Unknown 377035 Unknown
60 = J42 Elongated pentagonal orthobirotunda----Unknown 428040 Unknown
61 = J43 Elongated pentagonal gyrobirotunda----Unknown 428040 Unknown
62 = J44 Gyroelongated triangular bicupola-C--Unknown 264218 Unknown
63 = J45 Gyroelongated square bicupola-C--Unknown 345624 Unknown
64 = J46 Gyroelongated pentagonal bicupola-C--Unknown 427030 Unknown
65 = J47 Gyroelongated pentagonal cupolarotunda-C--Unknown 478035 Unknown
66 = J48 Gyroelongated pentagonal birotunda-C--Unknown 529040 Unknown
67 = J49 Augmented triangular prism----Unknown 8137 Unknown
68 = J50 Biaugmented triangular prism----Unknown 11178 Unknown
69 = J51 Triaugmented triangular prism----Unknown 14219 Unknown
70 = J52 Augmented pentagonal prism----Unknown 101911 Unknown
71 = J53 Biaugmented pentagonal prism----Unknown 132312 Unknown
72 = J54 Augmented hexagonal prism----Unknown 112213 Unknown
73 = J55 Parabiaugmented hexagonal prismB---Unknown 142614 Unknown
74 = J56 Metabiaugmented hexagonal prism----Unknown 142614 Unknown
75 = J57 Triaugmented hexagonal prism----Unknown 173015 Unknown
76 = J58 Augmented dodecahedron----Unknown 163521 Unknown
77 = J59 Parabiaugmented dodecahedron----Unknown 204022 Unknown
78 = J60 Metabiaugmented dodecahedron----Unknown 204022 Unknown
79 = J61 Triaugmented dodecahedron----Unknown 244523 Unknown
80 = J62 Metabidiminished icosahedron----Unknown 122010 Unknown
81 = J63 Tridiminished icosahedron----Unknown 8159 Unknown
82 = J64 Augmented tridiminished icosahedron----Unknown 101810 Unknown
83 = J65 Augmented truncated tetrahedron----Unknown 142715 Unknown
84 = J66 Augmented truncated cube----Unknown 224828 Unknown
85 = J67 Biaugmented truncated cubeB---Unknown 306032 Unknown
86 = J68 Augmented truncated dodecahedron----Unknown 4210565 Unknown
87 = J69 Parabiaugmented truncated dodecahedron----Unknown 5212070 Unknown
88 = J70 Metabiaugmented truncated dodecahedron----Unknown 5212070 Unknown
89 = J71 Triaugmented truncated dodecahedron----Unknown 6213575 Unknown
90 = J72 Gyrate rhombicosidodecahedron----Unknown 6212060 Unknown
91 = J73 Parabigyrate rhombicosidodecahedron----Unknown 6212060 Unknown
92 = J74 Metabigyrate rhombicosidodecahedron----Unknown 6212060 Unknown
93 = J75 Trigyrate rhombicosidodecahedron----Unknown 6212060 Unknown
94 = J76 Diminished rhombicosidodecahedron----Unknown 5210555 Unknown
95 = J77 Paragyrate diminished rhombicosidodecahedron----Unknown 5210555 Unknown
96 = J78 Metagyrate diminished rhombicosidodecahedron----Unknown 5210555 Unknown
97 = J79 Bigyrate diminished rhombicosidodecahedron----Unknown 5210555 Unknown
98 = J80 Parabidiminished rhombicosidodecahedron----Unknown 429050 Unknown
99 = J81 Metabidiminished rhombicosidodecahedron----Unknown 429050 Unknown
100 = J82 Gyrate bidiminished rhombicosidodecahedron----Unknown 429050 Unknown
101 = J83 Tridiminished rhombicosidodecahedron----Unknown 327545 Unknown
102 = J84 Snub disphenoid
(Siamese dodecahedron)
----Unknown 12188 Unknown
103 = J85 Snub square antiprism----Unknown 264016 Unknown
104 = J86 Sphenocorona----Unknown 142210 Unknown
105 = J87 Augmented sphenocorona----Unknown 172611 Unknown
106 = J88 Sphenomegacorona----Unknown 182812 Unknown
107 = J89 Hebesphenomegacorona----Unknown 213314 Unknown
108 = J90 Disphenocingulum----Unknown 243816 Unknown
109 = J91 Bilunabirotunda----Unknown 142614 Unknown
110 = J92 Triangular hebesphenorotunda----Unknown 203618 Unknown
Utility prisms (total 7)
111 Triangular prism--S-+G3 596 30. Triangular bipyramid (above)
112 Hexagonal prismB-SZ+G6 81812 Hexagonal bipyramidB---G6vM1
113 Octagonal prismB-SZ+G8 102416 Octagonal bipyramidB---G8vM1
114 Square antiprism----G4$H4 10168 Tetragonal trapezohedron----Unknown
115 Pentagonal prism--S-+G5 71510 31. Pentagonal bipyramid (above)
116 Decagonal prismB-SZ+G10 123020 Decagonal bipyramidB---G10vM1
117 Pentagonal antiprism----G5$H5 122010 Pentagonal trapezohedron----Unknown

The utility prisms are so named because they are the only prisms which appear as cells in higher-dimensional uniform polytopes. The antiprisms of the same symmetry groups are included for completeness.