List of convex regular-faced polyhedra (Meta, 13)
From Hi.gher. Space
See also: CRF polychora discovery project
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 | Symmetry group |
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---|---|---|---|---|---|---|---|---|---|---|
Name | Classes | Symbol | Faces | Edges | Vertices | Name | Classes | Symbol | ||
Platonic solids (total 5) | ||||||||||
1 | Tetrahedron | --S- | x3o3o | 4 | 6 | 4 | Self-dual | Td | ||
2 | Cube | B-SZ | x4o3o | 6 | 12 | 8 | 3. Octahedron (below) | Oh | ||
3 | Octahedron | B--- | x3o4o | 8 | 12 | 6 | 2. Cube (above) | Oh | ||
4 | Dodecahedron | B-S- | x5o3o | 12 | 30 | 20 | 5. Icosahedron (below) | Ih | ||
5 | Icosahedron | B--- | x3o5o | 20 | 30 | 12 | 4. Dodecahedron (above) | Ih | ||
Archimedean solids (total 13) | ||||||||||
6 | Tetrahedral truncate | --S- | x3x3o | 8 | 18 | 12 | Triakis tetrahedron | ---- | m3m3o | Td |
7 | Stauromesohedron | B--- | o4x3o | 14 | 24 | 12 | Rhombic dodecahedron | B--Z | o4m3o | Oh |
8 | Cubic truncate | B-S- | x4x3o | 14 | 36 | 24 | Triakis octahedron | B--- | m4m3o | Oh |
9 | Octahedral truncate | B-S- | x3x4o | 14 | 36 | 24 | Tetrakis hexahedron | B--- | m3m4o | Oh |
10 | Stauroperihedron | B--- | x4o3x | 26 | 48 | 24 | Deltoidal icositetrahedron | B--- | m4o3m | Oh |
11 | Stauropantohedron | B-SZ | x4x3x | 26 | 72 | 48 | Disdyakis dodecahedron | B--- | m4m3m | Oh |
12 | Snub staurohedron | -C-- | s4s3s | 38 | 60 | 24 | Pentagonal icositetrahedron | -C-- | g4g3g | O |
13 | Rhodomesohedron | B--- | o5x3o | 32 | 60 | 30 | Rhombic triacontahedron | B--Z | o5m3o | Ih |
14 | Dodecahedral truncate | B-S- | x5x3o | 32 | 90 | 60 | Triakis icosahedron | B--- | m5m3o | Ih |
15 | Icosahedral truncate | B-S- | x3x5o | 32 | 90 | 30 | Pentakis dodecahedron | B--- | m3m5o | Ih |
16 | Rhodoperihedron | B--- | x5o3x | 62 | 120 | 60 | Deltoidal hexecontahedron | B--- | m5o3m | Ih |
17 | Rhodopantohedron | B-SZ | x5x3x | 62 | 180 | 120 | Disdyakis triacontahedron | B--- | m3m5m | Ih |
18 | Snub rhodohedron | -C-- | s5s3s | 92 | 150 | 60 | Pentagonal hexecontahedron | -C-- | g5g3g | I |
Johnson solids (total 92) | ||||||||||
19 = J1 | Square pyramid | ---- | xo4oo | 5 | 8 | 5 | Self-dual | C4v | ||
20 = J2 | Pentagonal pyramid | ---- | xo5oo | 6 | 10 | 6 | Self-dual | C5v | ||
21 = J3 | Triangular cupola | ---- | xx3xo | 8 | 15 | 9 | Unknown | C3v | ||
22 = J4 | Square cupola | ---- | xx4xo | 10 | 20 | 12 | Unknown | C4v | ||
23 = J5 | Pentagonal cupola | ---- | xx5xo | 12 | 25 | 15 | Unknown | C5v | ||
24 = J6 | Pentagonal rotunda | ---- | xox5ofx | 17 | 35 | 20 | Unknown | C5v | ||
25 = J7 | Elongated triangular pyramid (elongated tetrahedron) | ---- | oxx3ooo | 7 | 12 | 7 | Self-dual | C3v | ||
26 = J8 | Elongated square pyramid (augmented cube) | ---- | oxx4ooo | 9 | 16 | 9 | Self-dual | C4v | ||
27 = J9 | Elongated pentagonal pyramid | ---- | oxx5ooo | 11 | 20 | 11 | Self-dual | C5v | ||
28 = J10 | Gyroelongated square pyramid | ---- | oxo4oox | 13 | 20 | 9 | Unknown | C4v | ||
29 = J11 | Gyroelongated pentagonal pyramid (diminished icosahedron) | ---- | oxo5oox | 16 | 25 | 11 | Unknown | C5v | ||
30 = J12 | Triangular bipyramid | ---- | oxo3ooo | 6 | 9 | 5 | 111. Triangular prism (below) | D3h | ||
31 = J13 | Pentagonal bipyramid | ---- | oxo5ooo | 10 | 15 | 7 | 115. Pentagonal prism (below) | D5h | ||
32 = J14 | Elongated triangular bipyramid | ---- | oxxo3oooo | 9 | 15 | 8 | Unknown | D3h | ||
33 = J15 | Elongated square bipyramid (biaugmented cube) | B--- | oxxo4oooo | 12 | 20 | 10 | Unknown | D4h | ||
34 = J16 | Elongated pentagonal bipyramid | ---- | oxxo5oooo | 15 | 25 | 12 | Unknown | D5h | ||
35 = J17 | Gyroelongated square bipyramid | ---- | oxoo4oooo | 16 | 24 | 10 | Unknown | D4d | ||
36 = J18 | Elongated triangular cupola | ---- | oxx3xxx | 14 | 27 | 15 | Unknown | C3v | ||
37 = J19 | Elongated square cupola (diminished rhombicuboctahedron) | ---- | oxx4xxx | 18 | 36 | 20 | Unknown | C4v | ||
38 = J20 | Elongated pentagonal cupola | ---- | oxx5xxx | 22 | 45 | 25 | Unknown | C5v | ||
39 = J21 | Elongated pentagonal rotunda | ---- | xoxx5ofxx | 27 | 55 | 30 | Unknown | C5v | ||
40 = J22 | Gyroelongated triangular cupola | ---- | Unknown | 20 | 33 | 15 | Unknown | C3v | ||
41 = J23 | Gyroelongated square cupola | ---- | Unknown | 26 | 44 | 20 | Unknown | C4v | ||
42 = J24 | Gyroelongated pentagonal cupola | ---- | Unknown | 32 | 55 | 25 | Unknown | C5v | ||
43 = J25 | Gyroelongated pentagonal rotunda | ---- | Unknown | 37 | 65 | 30 | Unknown | C5v | ||
44 = J26 | Gyrobifastigium | ---- | Unknown | 8 | 14 | 8 | Unknown | D2d | ||
45 = J27 | Triangular orthobicupola (gyrate cuboctahedron) | ---- | oxo3xox | 14 | 24 | 12 | Unknown | D3h | ||
46 = J28 | Square orthobicupola | B--- | oxo4xxx | 18 | 32 | 16 | Unknown | D4h | ||
47 = J29 | Square gyrobicupola | ---- | xxo4oxx | 18 | 32 | 16 | Unknown | D4d | ||
48 = J30 | Pentagonal orthobicupola | ---- | oxo5xxx | 22 | 40 | 20 | Unknown | D5h | ||
49 = J31 | Pentagonal gyrobicupola | ---- | oxx5xxo | 22 | 40 | 20 | Unknown | D5d | ||
50 = J32 | Pentagonal orthocupolarotunda | ---- | oxfo5xxox | 27 | 50 | 25 | Unknown | C5v | ||
51 = J33 | Pentagonal gyrocupolarotunda | ---- | xxfo5oxox | 27 | 50 | 25 | Unknown | C5v | ||
52 = J34 | Pentagonal orthobirotunda (gyrate icosidodecahedron) | ---- | xoxox5ofofo | 32 | 60 | 30 | Unknown | D5h | ||
53 = J35 | Elongated triangular orthobicupola | ---- | xxxxx3oxxxo | 20 | 36 | 18 | Unknown | D3h | ||
54 = J36 | Elongated triangular gyrobicupola | ---- | oxxxx3xxxxo | 20 | 36 | 18 | Unknown | D3d | ||
55 = J37 | Elongated square gyrobicupola (gyrate rhombicuboctahedron) | ---- | oxxx4xxxo | 26 | 48 | 24 | Unknown | D4d | ||
56 = J38 | Elongated pentagonal orthobicupola | ---- | xxxx5oxxo | 32 | 60 | 30 | Unknown | D5h | ||
57 = J39 | Elongated pentagonal gyrobicupola | ---- | oxxxx5xxxxo | 32 | 60 | 30 | Unknown | D5d | ||
58 = J40 | Elongated pentagonal orthocupolarotunda | ---- | xxox5oxfo | 37 | 70 | 35 | Unknown | C5v | ||
59 = J41 | Elongated pentagonal gyrocupolarotunda | ---- | oxox5xxfo | 37 | 70 | 35 | Unknown | C5v | ||
60 = J42 | Elongated pentagonal orthobirotunda | ---- | xoxxox5ofxxfo | 42 | 80 | 40 | Unknown | D5h | ||
61 = J43 | Elongated pentagonal gyrobirotunda | ---- | xoxxfo5ofxxox | 42 | 80 | 40 | Unknown | D5d | ||
62 = J44 | Gyroelongated triangular bicupola | -C-- | Unknown | 26 | 42 | 18 | Unknown | D3 | ||
63 = J45 | Gyroelongated square bicupola | -C-- | Unknown | 34 | 56 | 24 | Unknown | D4 | ||
64 = J46 | Gyroelongated pentagonal bicupola | -C-- | Unknown | 42 | 70 | 30 | Unknown | D5 | ||
65 = J47 | Gyroelongated pentagonal cupolarotunda | -C-- | Unknown | 47 | 80 | 35 | Unknown | C5 | ||
66 = J48 | Gyroelongated pentagonal birotunda | -C-- | Unknown | 52 | 90 | 40 | Unknown | D5 | ||
67 = J49 | Augmented triangular prism | ---- | oxo2oxx | 8 | 13 | 7 | Unknown | C2v | ||
68 = J50 | Biaugmented triangular prism | ---- | Unknown | 11 | 17 | 8 | Unknown | C2v | ||
69 = J51 | Triaugmented triangular prism | ---- | Unknown | 14 | 21 | 9 | Associahedron | D3h | ||
70 = J52 | Augmented pentagonal prism | ---- | oxfo2oxxx | 10 | 19 | 11 | Unknown | C2v | ||
71 = J53 | Biaugmented pentagonal prism | ---- | Unknown | 13 | 23 | 12 | Unknown | C2v | ||
72 = J54 | Augmented hexagonal prism | ---- | oxhx2oxxx | 11 | 22 | 13 | Unknown | C2v | ||
73 = J55 | Parabiaugmented hexagonal prism | B--- | oxhxo2oxxxo | 14 | 26 | 14 | Unknown | D2h | ||
74 = J56 | Metabiaugmented hexagonal prism | ---- | Unknown | 14 | 26 | 14 | Unknown | C2v | ||
75 = J57 | Triaugmented hexagonal prism | ---- | Unknown | 17 | 30 | 15 | Unknown | D3h | ||
76 = J58 | Augmented dodecahedron | ---- | oxfoo5oofx | 16 | 35 | 21 | Unknown | C5v | ||
77 = J59 | Parabiaugmented dodecahedron | ---- | oxfooo5ooofxo | 20 | 40 | 22 | Unknown | D5d | ||
78 = J60 | Metabiaugmented dodecahedron | ---- | Unknown | 20 | 40 | 22 | Unknown | C2v | ||
79 = J61 | Triaugmented dodecahedron | ---- | Unknown | 24 | 45 | 23 | Unknown | C3v | ||
80 = J62 | Metabidiminished icosahedron | ---- | Unknown | 12 | 20 | 10 | Unknown | C2v | ||
81 = J63 | Tridiminished icosahedron | ---- | xfo3oox | 8 | 15 | 9 | Unknown | C3v | ||
82 = J64 | Augmented tridiminished icosahedron | ---- | oxfo3ooox | 10 | 18 | 10 | Unknown | C3v | ||
83 = J65 | Augmented truncated tetrahedron | ---- | Unknown | 14 | 27 | 15 | Unknown | C3v | ||
84 = J66 | Augmented truncated cube | ---- | Unknown | 22 | 48 | 28 | Unknown | C4v | ||
85 = J67 | Biaugmented truncated cube | B--- | Unknown | 30 | 60 | 32 | Unknown | D4h | ||
86 = J68 | Augmented truncated dodecahedron | ---- | Unknown | 42 | 105 | 65 | Unknown | C5v | ||
87 = J69 | Parabiaugmented truncated dodecahedron | ---- | Unknown | 52 | 120 | 70 | Unknown | D5d | ||
88 = J70 | Metabiaugmented truncated dodecahedron | ---- | Unknown | 52 | 120 | 70 | Unknown | C2v | ||
89 = J71 | Triaugmented truncated dodecahedron | ---- | Unknown | 62 | 135 | 75 | Unknown | C3v | ||
90 = J72 | Gyrate rhodoperihedron | ---- | Unknown | 62 | 120 | 60 | Unknown | C5v | ||
91 = J73 | Parabigyrate rhodoperihedron | ---- | Unknown | 62 | 120 | 60 | Unknown | D5d | ||
92 = J74 | Metabigyrate rhodoperihedron | ---- | Unknown | 62 | 120 | 60 | Unknown | C2v | ||
93 = J75 | Trigyrate rhodoperihedron | ---- | Unknown | 62 | 120 | 60 | Unknown | C3v | ||
94 = J76 | Diminished rhodoperihedron | ---- | Unknown | 52 | 105 | 55 | Unknown | C5v | ||
95 = J77 | Paragyrate diminished rhodoperihedron | ---- | Unknown | 52 | 105 | 55 | Unknown | C5v | ||
96 = J78 | Metagyrate diminished rhodoperihedron | ---- | Unknown | 52 | 105 | 55 | Unknown | Cs | ||
97 = J79 | Bigyrate diminished rhodoperihedron | ---- | Unknown | 52 | 105 | 55 | Unknown | Cs | ||
98 = J80 | Parabidiminished rhodoperihedron | ---- | Unknown | 42 | 90 | 50 | Unknown | D5d | ||
99 = J81 | Metabidiminished rhodoperihedron | ---- | Unknown | 42 | 90 | 50 | Unknown | C2v | ||
100 = J82 | Gyrate bidiminished rhodoperihedron | ---- | Unknown | 42 | 90 | 50 | Unknown | Cs | ||
101 = J83 | Tridiminished rhodoperihedron | ---- | Unknown | 32 | 75 | 45 | Unknown | C3v | ||
102 = J84 | Snub disphenoid (Siamese dodecahedron) | ---- | Unknown | 12 | 18 | 8 | Unknown | D2d | ||
103 = J85 | Snub square antiprism | ---- | Unknown | 26 | 40 | 16 | Unknown | D4d | ||
104 = J86 | Sphenocorona | ---- | Unknown | 14 | 22 | 10 | Unknown | C2v | ||
105 = J87 | Augmented sphenocorona | ---- | Unknown | 17 | 26 | 11 | Unknown | Cs | ||
106 = J88 | Sphenomegacorona | ---- | Unknown | 18 | 28 | 12 | Unknown | C2v | ||
107 = J89 | Hebesphenomegacorona | ---- | Unknown | 21 | 33 | 14 | Unknown | C2v | ||
108 = J90 | Disphenocingulum | ---- | Unknown | 24 | 38 | 16 | Unknown | D2d | ||
109 = J91 | Bilunabirotunda | B--- | Unknown | 14 | 26 | 14 | Unknown | D2h | ||
110 = J92 | Triangular hebesphenorotunda | ---- | Unknown | 20 | 36 | 18 | Unknown | C3v | ||
Utility prisms (total 7) | ||||||||||
111 | Triangular prism | --S- | x3o2x | 5 | 9 | 6 | 30. Triangular bipyramid (above) | D3h | ||
112 | Hexagonal prism | B-SZ | x3x2x | 8 | 18 | 12 | Hexagonal bipyramid | B--- | oxo3oxo | D6h |
113 | Octagonal prism | B-SZ | x4x2x | 10 | 24 | 16 | Octagonal bipyramid | B--- | oxo4oxo | D8h |
114 | Square antiprism | ---- | xo4ox | 10 | 16 | 8 | Tetragonal trapezohedron | ---- | Unknown | D4d |
115 | Pentagonal prism | --S- | x5o2x | 7 | 15 | 10 | 31. Pentagonal bipyramid (above) | D5h | ||
116 | Decagonal prism | B-SZ | x5x2x | 12 | 30 | 20 | Decagonal bipyramid | B--- | oxo5oxo | D10h |
117 | Pentagonal antiprism | ---- | xo5ox | 12 | 20 | 10 | Pentagonal trapezohedron | ---- | Unknown | D5d |
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.