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

(Redirected from CRF polyhedra)

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
Name Classes Symbol Faces Edges Vertices Name Classes Symbol
Platonic solids (total 5)
1 Tetrahedron--S-x3o3o 464 Self-dual Td
2 CubeB-SZx4o3o 6128 3. Octahedron (below) Oh
3 OctahedronB---x3o4o 8126 2. Cube (above) Oh
4 DodecahedronB-S-x5o3o 123020 5. Icosahedron (below) Ih
5 IcosahedronB---x3o5o 203012 4. Dodecahedron (above) Ih
Archimedean solids (total 13)
6 Tetrahedral truncate--S-x3x3o 81812 Triakis tetrahedron----m3m3o Td
7 StauromesohedronB---o4x3o 142412 Rhombic dodecahedronB--Zo4m3o Oh
8 Cubic truncateB-S-x4x3o 143624 Triakis octahedronB---m4m3o Oh
9 Octahedral truncateB-S-x3x4o 143624 Tetrakis hexahedronB---m3m4o Oh
10 StauroperihedronB---x4o3x 264824 Deltoidal icositetrahedronB---m4o3m Oh
11 StauropantohedronB-SZx4x3x 267248 Disdyakis dodecahedronB---m4m3m Oh
12 Snub staurohedron-C--s4s3s 386024 Pentagonal icositetrahedron-C--g4g3g O
13 RhodomesohedronB---o5x3o 326030 Rhombic triacontahedronB--Zo5m3o Ih
14 Dodecahedral truncateB-S-x5x3o 329060 Triakis icosahedronB---m5m3o Ih
15 Icosahedral truncateB-S-x3x5o 329030 Pentakis dodecahedronB---m3m5o Ih
16 RhodoperihedronB---x5o3x 6212060 Deltoidal hexecontahedronB---m5o3m Ih
17 RhodopantohedronB-SZx5x3x 62180120 Disdyakis triacontahedronB---m3m5m Ih
18 Snub rhodohedron-C--s5s3s 9215060 Pentagonal hexecontahedron-C--g5g3g I
Johnson solids (total 92)
19 = J1 Square pyramid----xo4oo 585 Self-dual C4v
20 = J2 Pentagonal pyramid----xo5oo 6106 Self-dual C5v
21 = J3 Triangular cupola----xx3xo 8159 Unknown C3v
22 = J4 Square cupola----xx4xo 102012 Unknown C4v
23 = J5 Pentagonal cupola----xx5xo 122515 Unknown C5v
24 = J6 Pentagonal rotunda----xox5ofx 173520 Unknown C5v
25 = J7 Elongated triangular pyramid
(elongated tetrahedron)
----oxx3ooo 7127 Self-dual C3v
26 = J8 Elongated square pyramid
(augmented cube)
----oxx4ooo 9169 Self-dual C4v
27 = J9 Elongated pentagonal pyramid----oxx5ooo 112011 Self-dual C5v
28 = J10 Gyroelongated square pyramid----oxo4oox 13209 Unknown C4v
29 = J11 Gyroelongated pentagonal pyramid
(diminished icosahedron)
----oxo5oox 162511 Unknown C5v
30 = J12 Triangular bipyramid----oxo3ooo 695 111. Triangular prism (below) D3h
31 = J13 Pentagonal bipyramid----oxo5ooo 10157 115. Pentagonal prism (below) D5h
32 = J14 Elongated triangular bipyramid----oxxo3oooo 9158 Unknown D3h
33 = J15 Elongated square bipyramid
(biaugmented cube)
B---oxxo4oooo 122010 Unknown D4h
34 = J16 Elongated pentagonal bipyramid----oxxo5oooo 152512 Unknown D5h
35 = J17 Gyroelongated square bipyramid----oxoo4oooo 162410 Unknown D4d
36 = J18 Elongated triangular cupola----oxx3xxx 142715 Unknown C3v
37 = J19 Elongated square cupola
(diminished rhombicuboctahedron)
----oxx4xxx 183620 Unknown C4v
38 = J20 Elongated pentagonal cupola----oxx5xxx 224525 Unknown C5v
39 = J21 Elongated pentagonal rotunda----xoxx5ofxx 275530 Unknown C5v
40 = J22 Gyroelongated triangular cupola----Unknown 203315 Unknown C3v
41 = J23 Gyroelongated square cupola----Unknown 264420 Unknown C4v
42 = J24 Gyroelongated pentagonal cupola----Unknown 325525 Unknown C5v
43 = J25 Gyroelongated pentagonal rotunda----Unknown 376530 Unknown C5v
44 = J26 Gyrobifastigium----Unknown 8148 Unknown D2d
45 = J27 Triangular orthobicupola
(gyrate cuboctahedron)
----oxo3xox 142412 Unknown D3h
46 = J28 Square orthobicupolaB---oxo4xxx 183216 Unknown D4h
47 = J29 Square gyrobicupola----xxo4oxx 183216 Unknown D4d
48 = J30 Pentagonal orthobicupola----oxo5xxx 224020 Unknown D5h
49 = J31 Pentagonal gyrobicupola----oxx5xxo 224020 Unknown D5d
50 = J32 Pentagonal orthocupolarotunda----oxfo5xxox 275025 Unknown C5v
51 = J33 Pentagonal gyrocupolarotunda----xxfo5oxox 275025 Unknown C5v
52 = J34 Pentagonal orthobirotunda
(gyrate icosidodecahedron)
----xoxox5ofofo 326030 Unknown D5h
53 = J35 Elongated triangular orthobicupola----xxxxx3oxxxo 203618 Unknown D3h
54 = J36 Elongated triangular gyrobicupola----oxxxx3xxxxo 203618 Unknown D3d
55 = J37 Elongated square gyrobicupola
(gyrate rhombicuboctahedron)
----oxxx4xxxo 264824 Unknown D4d
56 = J38 Elongated pentagonal orthobicupola----xxxx5oxxo 326030 Unknown D5h
57 = J39 Elongated pentagonal gyrobicupola----oxxxx5xxxxo 326030 Unknown D5d
58 = J40 Elongated pentagonal orthocupolarotunda----xxox5oxfo 377035 Unknown C5v
59 = J41 Elongated pentagonal gyrocupolarotunda----oxox5xxfo 377035 Unknown C5v
60 = J42 Elongated pentagonal orthobirotunda----xoxxox5ofxxfo 428040 Unknown D5h
61 = J43 Elongated pentagonal gyrobirotunda----xoxxfo5ofxxox 428040 Unknown D5d
62 = J44 Gyroelongated triangular bicupola-C--Unknown 264218 Unknown D3
63 = J45 Gyroelongated square bicupola-C--Unknown 345624 Unknown D4
64 = J46 Gyroelongated pentagonal bicupola-C--Unknown 427030 Unknown D5
65 = J47 Gyroelongated pentagonal cupolarotunda-C--Unknown 478035 Unknown C5
66 = J48 Gyroelongated pentagonal birotunda-C--Unknown 529040 Unknown D5
67 = J49 Augmented triangular prism----oxo2oxx 8137 Unknown C2v
68 = J50 Biaugmented triangular prism----Unknown 11178 Unknown C2v
69 = J51 Triaugmented triangular prism----Unknown 14219 Associahedron D3h
70 = J52 Augmented pentagonal prism----oxfo2oxxx 101911 Unknown C2v
71 = J53 Biaugmented pentagonal prism----Unknown 132312 Unknown C2v
72 = J54 Augmented hexagonal prism----oxhx2oxxx 112213 Unknown C2v
73 = J55 Parabiaugmented hexagonal prismB---oxhxo2oxxxo 142614 Unknown D2h
74 = J56 Metabiaugmented hexagonal prism----Unknown 142614 Unknown C2v
75 = J57 Triaugmented hexagonal prism----Unknown 173015 Unknown D3h
76 = J58 Augmented dodecahedron----oxfoo5oofx 163521 Unknown C5v
77 = J59 Parabiaugmented dodecahedron----oxfooo5ooofxo 204022 Unknown D5d
78 = J60 Metabiaugmented dodecahedron----Unknown 204022 Unknown C2v
79 = J61 Triaugmented dodecahedron----Unknown 244523 Unknown C3v
80 = J62 Metabidiminished icosahedron----Unknown 122010 Unknown C2v
81 = J63 Tridiminished icosahedron----xfo3oox 8159 Unknown C3v
82 = J64 Augmented tridiminished icosahedron----oxfo3ooox 101810 Unknown C3v
83 = J65 Augmented truncated tetrahedron----Unknown 142715 Unknown C3v
84 = J66 Augmented truncated cube----Unknown 224828 Unknown C4v
85 = J67 Biaugmented truncated cubeB---Unknown 306032 Unknown D4h
86 = J68 Augmented truncated dodecahedron----Unknown 4210565 Unknown C5v
87 = J69 Parabiaugmented truncated dodecahedron----Unknown 5212070 Unknown D5d
88 = J70 Metabiaugmented truncated dodecahedron----Unknown 5212070 Unknown C2v
89 = J71 Triaugmented truncated dodecahedron----Unknown 6213575 Unknown C3v
90 = J72 Gyrate rhodoperihedron----Unknown 6212060 Unknown C5v
91 = J73 Parabigyrate rhodoperihedron----Unknown 6212060 Unknown D5d
92 = J74 Metabigyrate rhodoperihedron----Unknown 6212060 Unknown C2v
93 = J75 Trigyrate rhodoperihedron----Unknown 6212060 Unknown C3v
94 = J76 Diminished rhodoperihedron----Unknown 5210555 Unknown C5v
95 = J77 Paragyrate diminished rhodoperihedron----Unknown 5210555 Unknown C5v
96 = J78 Metagyrate diminished rhodoperihedron----Unknown 5210555 Unknown Cs
97 = J79 Bigyrate diminished rhodoperihedron----Unknown 5210555 Unknown Cs
98 = J80 Parabidiminished rhodoperihedron----Unknown 429050 Unknown D5d
99 = J81 Metabidiminished rhodoperihedron----Unknown 429050 Unknown C2v
100 = J82 Gyrate bidiminished rhodoperihedron----Unknown 429050 Unknown Cs
101 = J83 Tridiminished rhodoperihedron----Unknown 327545 Unknown C3v
102 = J84 Snub disphenoid
(Siamese dodecahedron)
----Unknown 12188 Unknown D2d
103 = J85 Snub square antiprism----Unknown 264016 Unknown D4d
104 = J86 Sphenocorona----Unknown 142210 Unknown C2v
105 = J87 Augmented sphenocorona----Unknown 172611 Unknown Cs
106 = J88 Sphenomegacorona----Unknown 182812 Unknown C2v
107 = J89 Hebesphenomegacorona----Unknown 213314 Unknown C2v
108 = J90 Disphenocingulum----Unknown 243816 Unknown D2d
109 = J91 BilunabirotundaB---Unknown 142614 Unknown D2h
110 = J92 Triangular hebesphenorotunda----Unknown 203618 Unknown C3v
Utility prisms (total 7)
111 Triangular prism--S-x3o2x 596 30. Triangular bipyramid (above) D3h
112 Hexagonal prismB-SZx3x2x 81812 Hexagonal bipyramidB---oxo3oxo D6h
113 Octagonal prismB-SZx4x2x 102416 Octagonal bipyramidB---oxo4oxo D8h
114 Square antiprism----xo4ox 10168 Tetragonal trapezohedron----Unknown D4d
115 Pentagonal prism--S-x5o2x 71510 31. Pentagonal bipyramid (above) D5h
116 Decagonal prismB-SZx5x2x 123020 Decagonal bipyramidB---oxo5oxo D10h
117 Pentagonal antiprism----xo5ox 122010 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.