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

### From Hi.gher. Space

(Redirected from Johnson solids)

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 |
T_{d} |
||

2 | Cube | B-SZ | x4o3o | 6 | 12 | 8 | 3. Octahedron (below) |
O_{h} |
||

3 | Octahedron | B--- | x3o4o | 8 | 12 | 6 | 2. Cube (above) |
O_{h} |
||

4 | Dodecahedron | B-S- | x5o3o | 12 | 30 | 20 | 5. Icosahedron (below) |
I_{h} |
||

5 | Icosahedron | B--- | x3o5o | 20 | 30 | 12 | 4. Dodecahedron (above) |
I_{h} |
||

Archimedean solids (total 13) | ||||||||||

6 | Tetrahedral truncate | --S- | x3x3o | 8 | 18 | 12 | Triakis tetrahedron | ---- | m3m3o | T_{d} |

7 | Stauromesohedron | B--- | o4x3o | 14 | 24 | 12 | Rhombic dodecahedron | B--Z | o4m3o | O_{h} |

8 | Cubic truncate | B-S- | x4x3o | 14 | 36 | 24 | Triakis octahedron | B--- | m4m3o | O_{h} |

9 | Octahedral truncate | B-S- | x3x4o | 14 | 36 | 24 | Tetrakis hexahedron | B--- | m3m4o | O_{h} |

10 | Stauroperihedron | B--- | x4o3x | 26 | 48 | 24 | Deltoidal icositetrahedron | B--- | m4o3m | O_{h} |

11 | Stauropantohedron | B-SZ | x4x3x | 26 | 72 | 48 | Disdyakis dodecahedron | B--- | m4m3m | O_{h} |

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 | I_{h} |

14 | Dodecahedral truncate | B-S- | x5x3o | 32 | 90 | 60 | Triakis icosahedron | B--- | m5m3o | I_{h} |

15 | Icosahedral truncate | B-S- | x3x5o | 32 | 90 | 30 | Pentakis dodecahedron | B--- | m3m5o | I_{h} |

16 | Rhodoperihedron | B--- | x5o3x | 62 | 120 | 60 | Deltoidal hexecontahedron | B--- | m5o3m | I_{h} |

17 | Rhodopantohedron | B-SZ | x5x3x | 62 | 180 | 120 | Disdyakis triacontahedron | B--- | m3m5m | I_{h} |

18 | Snub rhodohedron | -C-- | s5s3s | 92 | 150 | 60 | Pentagonal hexecontahedron | -C-- | g5g3g | I |

Johnson solids (total 92) | ||||||||||

19 = J_{1} |
Square pyramid | ---- | xo4oo | 5 | 8 | 5 | Self-dual |
C_{4v} |
||

20 = J_{2} |
Pentagonal pyramid | ---- | xo5oo | 6 | 10 | 6 | Self-dual |
C_{5v} |
||

21 = J_{3} |
Triangular cupola | ---- | xx3xo | 8 | 15 | 9 | Unknown |
C_{3v} |
||

22 = J_{4} |
Square cupola | ---- | xx4xo | 10 | 20 | 12 | Unknown |
C_{4v} |
||

23 = J_{5} |
Pentagonal cupola | ---- | xx5xo | 12 | 25 | 15 | Unknown |
C_{5v} |
||

24 = J_{6} |
Pentagonal rotunda | ---- | xox5ofx | 17 | 35 | 20 | Unknown |
C_{5v} |
||

25 = J_{7} |
Elongated triangular pyramid (elongated tetrahedron) | ---- | oxx3ooo | 7 | 12 | 7 | Self-dual |
C_{3v} |
||

26 = J_{8} |
Elongated square pyramid (augmented cube) | ---- | oxx4ooo | 9 | 16 | 9 | Self-dual |
C_{4v} |
||

27 = J_{9} |
Elongated pentagonal pyramid | ---- | oxx5ooo | 11 | 20 | 11 | Self-dual |
C_{5v} |
||

28 = J_{10} |
Gyroelongated square pyramid | ---- | oxo4oox | 13 | 20 | 9 | Unknown |
C_{4v} |
||

29 = J_{11} |
Gyroelongated pentagonal pyramid (diminished icosahedron) | ---- | oxo5oox | 16 | 25 | 11 | Unknown |
C_{5v} |
||

30 = J_{12} |
Triangular bipyramid | ---- | oxo3ooo | 6 | 9 | 5 | 111. Triangular prism (below) | D_{3h} |
||

31 = J_{13} |
Pentagonal bipyramid | ---- | oxo5ooo | 10 | 15 | 7 | 115. Pentagonal prism (below) | D_{5h} |
||

32 = J_{14} |
Elongated triangular bipyramid | ---- | oxxo3oooo | 9 | 15 | 8 | Unknown |
D_{3h} |
||

33 = J_{15} |
Elongated square bipyramid (biaugmented cube) | B--- | oxxo4oooo | 12 | 20 | 10 | Unknown |
D_{4h} |
||

34 = J_{16} |
Elongated pentagonal bipyramid | ---- | oxxo5oooo | 15 | 25 | 12 | Unknown |
D_{5h} |
||

35 = J_{17} |
Gyroelongated square bipyramid | ---- | oxoo4oooo | 16 | 24 | 10 | Unknown |
D_{4d} |
||

36 = J_{18} |
Elongated triangular cupola | ---- | oxx3xxx | 14 | 27 | 15 | Unknown |
C_{3v} |
||

37 = J_{19} |
Elongated square cupola (diminished rhombicuboctahedron) | ---- | oxx4xxx | 18 | 36 | 20 | Unknown |
C_{4v} |
||

38 = J_{20} |
Elongated pentagonal cupola | ---- | oxx5xxx | 22 | 45 | 25 | Unknown |
C_{5v} |
||

39 = J_{21} |
Elongated pentagonal rotunda | ---- | xoxx5ofxx | 27 | 55 | 30 | Unknown |
C_{5v} |
||

40 = J_{22} |
Gyroelongated triangular cupola | ---- | Unknown |
20 | 33 | 15 | Unknown |
C_{3v} |
||

41 = J_{23} |
Gyroelongated square cupola | ---- | Unknown |
26 | 44 | 20 | Unknown |
C_{4v} |
||

42 = J_{24} |
Gyroelongated pentagonal cupola | ---- | Unknown |
32 | 55 | 25 | Unknown |
C_{5v} |
||

43 = J_{25} |
Gyroelongated pentagonal rotunda | ---- | Unknown |
37 | 65 | 30 | Unknown |
C_{5v} |
||

44 = J_{26} |
Gyrobifastigium | ---- | Unknown |
8 | 14 | 8 | Unknown |
D_{2d} |
||

45 = J_{27} |
Triangular orthobicupola (gyrate cuboctahedron) | ---- | oxo3xox | 14 | 24 | 12 | Unknown |
D_{3h} |
||

46 = J_{28} |
Square orthobicupola | B--- | oxo4xxx | 18 | 32 | 16 | Unknown |
D_{4h} |
||

47 = J_{29} |
Square gyrobicupola | ---- | xxo4oxx | 18 | 32 | 16 | Unknown |
D_{4d} |
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48 = J_{30} |
Pentagonal orthobicupola | ---- | oxo5xxx | 22 | 40 | 20 | Unknown |
D_{5h} |
||

49 = J_{31} |
Pentagonal gyrobicupola | ---- | oxx5xxo | 22 | 40 | 20 | Unknown |
D_{5d} |
||

50 = J_{32} |
Pentagonal orthocupolarotunda | ---- | oxfo5xxox | 27 | 50 | 25 | Unknown |
C_{5v} |
||

51 = J_{33} |
Pentagonal gyrocupolarotunda | ---- | xxfo5oxox | 27 | 50 | 25 | Unknown |
C_{5v} |
||

52 = J_{34} |
Pentagonal orthobirotunda (gyrate icosidodecahedron) | ---- | xoxox5ofofo | 32 | 60 | 30 | Unknown |
D_{5h} |
||

53 = J_{35} |
Elongated triangular orthobicupola | ---- | xxxxx3oxxxo | 20 | 36 | 18 | Unknown |
D_{3h} |
||

54 = J_{36} |
Elongated triangular gyrobicupola | ---- | oxxxx3xxxxo | 20 | 36 | 18 | Unknown |
D_{3d} |
||

55 = J_{37} |
Elongated square gyrobicupola (gyrate rhombicuboctahedron) | ---- | oxxx4xxxo | 26 | 48 | 24 | Unknown |
D_{4d} |
||

56 = J_{38} |
Elongated pentagonal orthobicupola | ---- | xxxx5oxxo | 32 | 60 | 30 | Unknown |
D_{5h} |
||

57 = J_{39} |
Elongated pentagonal gyrobicupola | ---- | oxxxx5xxxxo | 32 | 60 | 30 | Unknown |
D_{5d} |
||

58 = J_{40} |
Elongated pentagonal orthocupolarotunda | ---- | xxox5oxfo | 37 | 70 | 35 | Unknown |
C_{5v} |
||

59 = J_{41} |
Elongated pentagonal gyrocupolarotunda | ---- | oxox5xxfo | 37 | 70 | 35 | Unknown |
C_{5v} |
||

60 = J_{42} |
Elongated pentagonal orthobirotunda | ---- | xoxxox5ofxxfo | 42 | 80 | 40 | Unknown |
D_{5h} |
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61 = J_{43} |
Elongated pentagonal gyrobirotunda | ---- | xoxxfo5ofxxox | 42 | 80 | 40 | Unknown |
D_{5d} |
||

62 = J_{44} |
Gyroelongated triangular bicupola | -C-- | Unknown |
26 | 42 | 18 | Unknown |
D_{3} |
||

63 = J_{45} |
Gyroelongated square bicupola | -C-- | Unknown |
34 | 56 | 24 | Unknown |
D_{4} |
||

64 = J_{46} |
Gyroelongated pentagonal bicupola | -C-- | Unknown |
42 | 70 | 30 | Unknown |
D_{5} |
||

65 = J_{47} |
Gyroelongated pentagonal cupolarotunda | -C-- | Unknown |
47 | 80 | 35 | Unknown |
C_{5} |
||

66 = J_{48} |
Gyroelongated pentagonal birotunda | -C-- | Unknown |
52 | 90 | 40 | Unknown |
D_{5} |
||

67 = J_{49} |
Augmented triangular prism | ---- | oxo2oxx | 8 | 13 | 7 | Unknown |
C_{2v} |
||

68 = J_{50} |
Biaugmented triangular prism | ---- | Unknown |
11 | 17 | 8 | Unknown |
C_{2v} |
||

69 = J_{51} |
Triaugmented triangular prism | ---- | Unknown |
14 | 21 | 9 | Associahedron | D_{3h} |
||

70 = J_{52} |
Augmented pentagonal prism | ---- | oxfo2oxxx | 10 | 19 | 11 | Unknown |
C_{2v} |
||

71 = J_{53} |
Biaugmented pentagonal prism | ---- | Unknown |
13 | 23 | 12 | Unknown |
C_{2v} |
||

72 = J_{54} |
Augmented hexagonal prism | ---- | oxhx2oxxx | 11 | 22 | 13 | Unknown |
C_{2v} |
||

73 = J_{55} |
Parabiaugmented hexagonal prism | B--- | oxhxo2oxxxo | 14 | 26 | 14 | Unknown |
D_{2h} |
||

74 = J_{56} |
Metabiaugmented hexagonal prism | ---- | Unknown |
14 | 26 | 14 | Unknown |
C_{2v} |
||

75 = J_{57} |
Triaugmented hexagonal prism | ---- | Unknown |
17 | 30 | 15 | Unknown |
D_{3h} |
||

76 = J_{58} |
Augmented dodecahedron | ---- | oxfoo5oofx | 16 | 35 | 21 | Unknown |
C_{5v} |
||

77 = J_{59} |
Parabiaugmented dodecahedron | ---- | oxfooo5ooofxo | 20 | 40 | 22 | Unknown |
D_{5d} |
||

78 = J_{60} |
Metabiaugmented dodecahedron | ---- | Unknown |
20 | 40 | 22 | Unknown |
C_{2v} |
||

79 = J_{61} |
Triaugmented dodecahedron | ---- | Unknown |
24 | 45 | 23 | Unknown |
C_{3v} |
||

80 = J_{62} |
Metabidiminished icosahedron | ---- | Unknown |
12 | 20 | 10 | Unknown |
C_{2v} |
||

81 = J_{63} |
Tridiminished icosahedron | ---- | xfo3oox | 8 | 15 | 9 | Unknown |
C_{3v} |
||

82 = J_{64} |
Augmented tridiminished icosahedron | ---- | oxfo3ooox | 10 | 18 | 10 | Unknown |
C_{3v} |
||

83 = J_{65} |
Augmented truncated tetrahedron | ---- | Unknown |
14 | 27 | 15 | Unknown |
C_{3v} |
||

84 = J_{66} |
Augmented truncated cube | ---- | Unknown |
22 | 48 | 28 | Unknown |
C_{4v} |
||

85 = J_{67} |
Biaugmented truncated cube | B--- | Unknown |
30 | 60 | 32 | Unknown |
D_{4h} |
||

86 = J_{68} |
Augmented truncated dodecahedron | ---- | Unknown |
42 | 105 | 65 | Unknown |
C_{5v} |
||

87 = J_{69} |
Parabiaugmented truncated dodecahedron | ---- | Unknown |
52 | 120 | 70 | Unknown |
D_{5d} |
||

88 = J_{70} |
Metabiaugmented truncated dodecahedron | ---- | Unknown |
52 | 120 | 70 | Unknown |
C_{2v} |
||

89 = J_{71} |
Triaugmented truncated dodecahedron | ---- | Unknown |
62 | 135 | 75 | Unknown |
C_{3v} |
||

90 = J_{72} |
Gyrate rhodoperihedron | ---- | Unknown |
62 | 120 | 60 | Unknown |
C_{5v} |
||

91 = J_{73} |
Parabigyrate rhodoperihedron | ---- | Unknown |
62 | 120 | 60 | Unknown |
D_{5d} |
||

92 = J_{74} |
Metabigyrate rhodoperihedron | ---- | Unknown |
62 | 120 | 60 | Unknown |
C_{2v} |
||

93 = J_{75} |
Trigyrate rhodoperihedron | ---- | Unknown |
62 | 120 | 60 | Unknown |
C_{3v} |
||

94 = J_{76} |
Diminished rhodoperihedron | ---- | Unknown |
52 | 105 | 55 | Unknown |
C_{5v} |
||

95 = J_{77} |
Paragyrate diminished rhodoperihedron | ---- | Unknown |
52 | 105 | 55 | Unknown |
C_{5v} |
||

96 = J_{78} |
Metagyrate diminished rhodoperihedron | ---- | Unknown |
52 | 105 | 55 | Unknown |
C_{s} |
||

97 = J_{79} |
Bigyrate diminished rhodoperihedron | ---- | Unknown |
52 | 105 | 55 | Unknown |
C_{s} |
||

98 = J_{80} |
Parabidiminished rhodoperihedron | ---- | Unknown |
42 | 90 | 50 | Unknown |
D_{5d} |
||

99 = J_{81} |
Metabidiminished rhodoperihedron | ---- | Unknown |
42 | 90 | 50 | Unknown |
C_{2v} |
||

100 = J_{82} |
Gyrate bidiminished rhodoperihedron | ---- | Unknown |
42 | 90 | 50 | Unknown |
C_{s} |
||

101 = J_{83} |
Tridiminished rhodoperihedron | ---- | Unknown |
32 | 75 | 45 | Unknown |
C_{3v} |
||

102 = J_{84} |
Snub disphenoid (Siamese dodecahedron) | ---- | Unknown |
12 | 18 | 8 | Unknown |
D_{2d} |
||

103 = J_{85} |
Snub square antiprism | ---- | Unknown |
26 | 40 | 16 | Unknown |
D_{4d} |
||

104 = J_{86} |
Sphenocorona | ---- | Unknown |
14 | 22 | 10 | Unknown |
C_{2v} |
||

105 = J_{87} |
Augmented sphenocorona | ---- | Unknown |
17 | 26 | 11 | Unknown |
C_{s} |
||

106 = J_{88} |
Sphenomegacorona | ---- | Unknown |
18 | 28 | 12 | Unknown |
C_{2v} |
||

107 = J_{89} |
Hebesphenomegacorona | ---- | Unknown |
21 | 33 | 14 | Unknown |
C_{2v} |
||

108 = J_{90} |
Disphenocingulum | ---- | Unknown |
24 | 38 | 16 | Unknown |
D_{2d} |
||

109 = J_{91} |
Bilunabirotunda | B--- | Unknown |
14 | 26 | 14 | Unknown |
D_{2h} |
||

110 = J_{92} |
Triangular hebesphenorotunda | ---- | Unknown |
20 | 36 | 18 | Unknown |
C_{3v} |
||

Utility prisms (total 7) | ||||||||||

111 | Triangular prism | --S- | x3o2x | 5 | 9 | 6 | 30. Triangular bipyramid (above) |
D_{3h} |
||

112 | Hexagonal prism | B-SZ | x3x2x | 8 | 18 | 12 | Hexagonal bipyramid | B--- | oxo3oxo | D_{6h} |

113 | Octagonal prism | B-SZ | x4x2x | 10 | 24 | 16 | Octagonal bipyramid | B--- | oxo4oxo | D_{8h} |

114 | Square antiprism | ---- | xo4ox | 10 | 16 | 8 | Tetragonal trapezohedron | ---- | Unknown |
D_{4d} |

115 | Pentagonal prism | --S- | x5o2x | 7 | 15 | 10 | 31. Pentagonal bipyramid (above) |
D_{5h} |
||

116 | Decagonal prism | B-SZ | x5x2x | 12 | 30 | 20 | Decagonal bipyramid | B--- | oxo5oxo | D_{10h} |

117 | Pentagonal antiprism | ---- | xo5ox | 12 | 20 | 10 | Pentagonal trapezohedron | ---- | Unknown |
D_{5d} |

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.