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
Index	Name	Classes	Symbol	Faces	Edges	Vertices	Name	Classes	Symbol	Symmetry group
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 = J₁	Square pyramid	`----`	xo4oo	5	8	5	Self-dual			C_4v
20 = J₂	Pentagonal pyramid	`----`	xo5oo	6	10	6	Self-dual			C_5v
21 = J₃	Triangular cupola	`----`	xx3xo	8	15	9	Unknown			C_3v
22 = J₄	Square cupola	`----`	xx4xo	10	20	12	Unknown			C_4v
23 = J₅	Pentagonal cupola	`----`	xx5xo	12	25	15	Unknown			C_5v
24 = J₆	Pentagonal rotunda	`----`	xox5ofx	17	35	20	Unknown			C_5v
25 = J₇	Elongated triangular pyramid (elongated tetrahedron)	`----`	oxx3ooo	7	12	7	Self-dual			C_3v
26 = J₈	Elongated square pyramid (augmented cube)	`----`	oxx4ooo	9	16	9	Self-dual			C_4v
27 = J₉	Elongated pentagonal pyramid	`----`	oxx5ooo	11	20	11	Self-dual			C_5v
28 = J₁₀	Gyroelongated square pyramid	`----`	oxo4oox	13	20	9	Unknown			C_4v
29 = J₁₁	Gyroelongated pentagonal pyramid (diminished icosahedron)	`----`	oxo5oox	16	25	11	Unknown			C_5v
30 = J₁₂	Triangular bipyramid	`----`	oxo3ooo	6	9	5	111. Triangular prism (below)			D_3h
31 = J₁₃	Pentagonal bipyramid	`----`	oxo5ooo	10	15	7	115. Pentagonal prism (below)			D_5h
32 = J₁₄	Elongated triangular bipyramid	`----`	oxxo3oooo	9	15	8	Unknown			D_3h
33 = J₁₅	Elongated square bipyramid (biaugmented cube)	`B---`	oxxo4oooo	12	20	10	Unknown			D_4h
34 = J₁₆	Elongated pentagonal bipyramid	`----`	oxxo5oooo	15	25	12	Unknown			D_5h
35 = J₁₇	Gyroelongated square bipyramid	`----`	oxoo4oooo	16	24	10	Unknown			D_4d
36 = J₁₈	Elongated triangular cupola	`----`	oxx3xxx	14	27	15	Unknown			C_3v
37 = J₁₉	Elongated square cupola (diminished rhombicuboctahedron)	`----`	oxx4xxx	18	36	20	Unknown			C_4v
38 = J₂₀	Elongated pentagonal cupola	`----`	oxx5xxx	22	45	25	Unknown			C_5v
39 = J₂₁	Elongated pentagonal rotunda	`----`	xoxx5ofxx	27	55	30	Unknown			C_5v
40 = J₂₂	Gyroelongated triangular cupola	`----`	Unknown	20	33	15	Unknown			C_3v
41 = J₂₃	Gyroelongated square cupola	`----`	Unknown	26	44	20	Unknown			C_4v
42 = J₂₄	Gyroelongated pentagonal cupola	`----`	Unknown	32	55	25	Unknown			C_5v
43 = J₂₅	Gyroelongated pentagonal rotunda	`----`	Unknown	37	65	30	Unknown			C_5v
44 = J₂₆	Gyrobifastigium	`----`	Unknown	8	14	8	Unknown			D_2d
45 = J₂₇	Triangular orthobicupola (gyrate cuboctahedron)	`----`	oxo3xox	14	24	12	Unknown			D_3h
46 = J₂₈	Square orthobicupola	`B---`	oxo4xxx	18	32	16	Unknown			D_4h
47 = J₂₉	Square gyrobicupola	`----`	xxo4oxx	18	32	16	Unknown			D_4d
48 = J₃₀	Pentagonal orthobicupola	`----`	oxo5xxx	22	40	20	Unknown			D_5h
49 = J₃₁	Pentagonal gyrobicupola	`----`	oxx5xxo	22	40	20	Unknown			D_5d
50 = J₃₂	Pentagonal orthocupolarotunda	`----`	oxfo5xxox	27	50	25	Unknown			C_5v
51 = J₃₃	Pentagonal gyrocupolarotunda	`----`	xxfo5oxox	27	50	25	Unknown			C_5v
52 = J₃₄	Pentagonal orthobirotunda (gyrate icosidodecahedron)	`----`	xoxox5ofofo	32	60	30	Unknown			D_5h
53 = J₃₅	Elongated triangular orthobicupola	`----`	xxxxx3oxxxo	20	36	18	Unknown			D_3h
54 = J₃₆	Elongated triangular gyrobicupola	`----`	oxxxx3xxxxo	20	36	18	Unknown			D_3d
55 = J₃₇	Elongated square gyrobicupola (gyrate rhombicuboctahedron)	`----`	oxxx4xxxo	26	48	24	Unknown			D_4d
56 = J₃₈	Elongated pentagonal orthobicupola	`----`	xxxx5oxxo	32	60	30	Unknown			D_5h
57 = J₃₉	Elongated pentagonal gyrobicupola	`----`	oxxxx5xxxxo	32	60	30	Unknown			D_5d
58 = J₄₀	Elongated pentagonal orthocupolarotunda	`----`	xxox5oxfo	37	70	35	Unknown			C_5v
59 = J₄₁	Elongated pentagonal gyrocupolarotunda	`----`	oxox5xxfo	37	70	35	Unknown			C_5v
60 = J₄₂	Elongated pentagonal orthobirotunda	`----`	xoxxox5ofxxfo	42	80	40	Unknown			D_5h
61 = J₄₃	Elongated pentagonal gyrobirotunda	`----`	xoxxfo5ofxxox	42	80	40	Unknown			D_5d
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			C_2v
68 = J₅₀	Biaugmented triangular prism	`----`	Unknown	11	17	8	Unknown			C_2v
69 = J₅₁	Triaugmented triangular prism	`----`	Unknown	14	21	9	Associahedron			D_3h
70 = J₅₂	Augmented pentagonal prism	`----`	oxfo2oxxx	10	19	11	Unknown			C_2v
71 = J₅₃	Biaugmented pentagonal prism	`----`	Unknown	13	23	12	Unknown			C_2v
72 = J₅₄	Augmented hexagonal prism	`----`	oxhx2oxxx	11	22	13	Unknown			C_2v
73 = J₅₅	Parabiaugmented hexagonal prism	`B---`	oxhxo2oxxxo	14	26	14	Unknown			D_2h
74 = J₅₆	Metabiaugmented hexagonal prism	`----`	Unknown	14	26	14	Unknown			C_2v
75 = J₅₇	Triaugmented hexagonal prism	`----`	Unknown	17	30	15	Unknown			D_3h
76 = J₅₈	Augmented dodecahedron	`----`	oxfoo5oofx	16	35	21	Unknown			C_5v
77 = J₅₉	Parabiaugmented dodecahedron	`----`	oxfooo5ooofxo	20	40	22	Unknown			D_5d
78 = J₆₀	Metabiaugmented dodecahedron	`----`	Unknown	20	40	22	Unknown			C_2v
79 = J₆₁	Triaugmented dodecahedron	`----`	Unknown	24	45	23	Unknown			C_3v
80 = J₆₂	Metabidiminished icosahedron	`----`	Unknown	12	20	10	Unknown			C_2v
81 = J₆₃	Tridiminished icosahedron	`----`	xfo3oox	8	15	9	Unknown			C_3v
82 = J₆₄	Augmented tridiminished icosahedron	`----`	oxfo3ooox	10	18	10	Unknown			C_3v
83 = J₆₅	Augmented truncated tetrahedron	`----`	Unknown	14	27	15	Unknown			C_3v
84 = J₆₆	Augmented truncated cube	`----`	Unknown	22	48	28	Unknown			C_4v
85 = J₆₇	Biaugmented truncated cube	`B---`	Unknown	30	60	32	Unknown			D_4h
86 = J₆₈	Augmented truncated dodecahedron	`----`	Unknown	42	105	65	Unknown			C_5v
87 = J₆₉	Parabiaugmented truncated dodecahedron	`----`	Unknown	52	120	70	Unknown			D_5d
88 = J₇₀	Metabiaugmented truncated dodecahedron	`----`	Unknown	52	120	70	Unknown			C_2v
89 = J₇₁	Triaugmented truncated dodecahedron	`----`	Unknown	62	135	75	Unknown			C_3v
90 = J₇₂	Gyrate rhodoperihedron	`----`	Unknown	62	120	60	Unknown			C_5v
91 = J₇₃	Parabigyrate rhodoperihedron	`----`	Unknown	62	120	60	Unknown			D_5d
92 = J₇₄	Metabigyrate rhodoperihedron	`----`	Unknown	62	120	60	Unknown			C_2v
93 = J₇₅	Trigyrate rhodoperihedron	`----`	Unknown	62	120	60	Unknown			C_3v
94 = J₇₆	Diminished rhodoperihedron	`----`	Unknown	52	105	55	Unknown			C_5v
95 = J₇₇	Paragyrate diminished rhodoperihedron	`----`	Unknown	52	105	55	Unknown			C_5v
96 = J₇₈	Metagyrate diminished rhodoperihedron	`----`	Unknown	52	105	55	Unknown			C_s
97 = J₇₉	Bigyrate diminished rhodoperihedron	`----`	Unknown	52	105	55	Unknown			C_s
98 = J₈₀	Parabidiminished rhodoperihedron	`----`	Unknown	42	90	50	Unknown			D_5d
99 = J₈₁	Metabidiminished rhodoperihedron	`----`	Unknown	42	90	50	Unknown			C_2v
100 = J₈₂	Gyrate bidiminished rhodoperihedron	`----`	Unknown	42	90	50	Unknown			C_s
101 = J₈₃	Tridiminished rhodoperihedron	`----`	Unknown	32	75	45	Unknown			C_3v
102 = J₈₄	Snub disphenoid (Siamese dodecahedron)	`----`	Unknown	12	18	8	Unknown			D_2d
103 = J₈₅	Snub square antiprism	`----`	Unknown	26	40	16	Unknown			D_4d
104 = J₈₆	Sphenocorona	`----`	Unknown	14	22	10	Unknown			C_2v
105 = J₈₇	Augmented sphenocorona	`----`	Unknown	17	26	11	Unknown			C_s
106 = J₈₈	Sphenomegacorona	`----`	Unknown	18	28	12	Unknown			C_2v
107 = J₈₉	Hebesphenomegacorona	`----`	Unknown	21	33	14	Unknown			C_2v
108 = J₉₀	Disphenocingulum	`----`	Unknown	24	38	16	Unknown			D_2d
109 = J₉₁	Bilunabirotunda	`B---`	Unknown	14	26	14	Unknown			D_2h
110 = J₉₂	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.