List of uniform polyhedra by spherical triangle

Polyhedron
Class	Number and properties
Platonic solids	(5, convex, regular)
Archimedean solids	(13, convex, uniform)
Kepler–Poinsot polyhedra	(4, regular, non-convex)
Uniform polyhedra	(75, uniform)
Prismatoid: prisms, antiprisms etc.	(4 infinite uniform classes)
Polyhedra tilings	(11 regular, in the plane)
Quasi-regular polyhedra	(8)
Johnson solids	(92, convex, non-uniform)
Pyramids and Bipyramids	(infinite)
Stellations	Stellations
Polyhedral compounds	(5 regular)
Deltahedra	(Deltahedra, equilateral triangle faces)
Snub polyhedra	(12 uniform, not mirror image)
Zonohedron	(Zonohedra, faces have 180°symmetry)
Dual polyhedron
Self-dual polyhedron	(infinite)
Catalan solid	(13, Archimedean dual)

There are many relations among the uniform polyhedra. This List of uniform polyhedra by spherical triangle groups them by the Wythoff symbol.

Key

Image Name Bowers pet name V Number of vertices,E Number of edges,F Number of faces=Face configuration ?=Euler characteristic, group=Symmetry group Wythoff symbol - Vertex figure W - Wenninger number, U - Uniform number, K- Kalido number, C -Coxeter number alternative name second alternative name

The vertex figure can be discovered by considering the Wythoff symbol:

• p|q r - 2p edges, alternating q-gons and r-gons. Vertex figure (q.r)^p.
• p|q 2 - p edges, q-gons (here r=2 so the r-gons are degenerate lines).
• 2|q r - 4 edges, alternating q-gons and r-gons
• q r|p - 4 edges, 2p-gons, q-gons, 2p-gons r-gons, Vertex figure 2p.q.2p.r.
• q 2|p - 3 edges, 2p-gons, q-gons, 2p-gons, Vertex figure 2p.q.2p.
• p q r|- 3 edges, 2p-gons, 2q-gons, 2r-gons, vertex figure 2p.2q.2r

Convex

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
${\pi \over 3}\ {\pi \over 3}\ {\pi \over 2}$	Tetrahedron Tet V 4,E 6,F 4=4{3} χ=2, group=Td, A3, [3,3], (*332) 3 | 2 3 | 2 2 2 - 3.3.3 W1, U01, K06, C15		Octahedron	Truncated tetrahedron Tut V 12,E 18,F 8=4{3}+4{6} χ=2, group=Td, A3, [3,3], (*332), order 24 2 3 | 3 - 3.6.6 W6, U02, K07, C16		Cuboctahedron	Truncated octahedron	Icosahedron
${\pi \over 4}\ {\pi \over 3}\ {\pi \over 2}$	Octahedron Oct V 6,E 12,F 8=8{3} χ=2, group=Oh, BC3, [4,3], (*432) 4 | 2 3 - 3.3.3.3 W2, U05, K10, C17	Hexahedron Cube V 8,E 12,F 6=6{4} χ=2, group=Oh, B3, [4,3], (*432) 3 | 2 4 - 4.4.4 W3, U06, K11, C18	Cuboctahedron Co V 12,E 24,F 14=8{3}+6{4} χ=2, group=Oh, B3, [4,3], (*432), order 48 Td, [3,3], (*332), order 24 2 | 3 4 3 3 | 2 - 3.4.3.4 W11, U07, K12, C19	Truncated cube Tic V 24,E 36,F 14=8{3}+6{8} χ=2, group=Oh, B3, [4,3], (*432), order 48 2 3 | 4 - 3.8.8 W8, U09, K14, C21 Truncated hexahedron	Truncated octahedron Toe V 24,E 36,F 14=6{4}+8{6} χ=2, group=Oh, B3, [4,3], (*432), order 48 Th, [3,3] and (*332), order 24 2 4 | 3 3 3 2 | - 4.6.6 W7, U08, K13, C20	Rhombicuboctahedron Sirco V 24,E 48,F 26=8{3}+(6+12){4} χ=2, group=Oh, B3, [4,3], (*432), order 48 3 4 | 2 - 3.4.4.4 W13, U10, K15, C22 Rhombicuboctahedron	Truncated cuboctahedron Girco V 48,E 72,F 26=12{4}+8{6}+6{8} χ=2, group=Oh, B3, [4,3], (*432), order 48 2 3 4 | - 4.6.8 W15, U11, K16, C23 Rhombitruncated cuboctahedron Truncated cuboctahedron	Snub cube Snic V 24,E 60,F 38=(8+24){3}+6{4} χ=2, group=O, 1/2B3, [4,3]+, (432), order 24 | 2 3 4 - 3.3.3.3.4 W17, U12, K17, C24
${\pi \over 5}\ {\pi \over 3}\ {\pi \over 2}$	Icosahedron Ike V 12,E 30,F 20=20{3} χ=2, group=Ih, H3, [5,3], (*532) 5 | 2 3 - 3.3.3.3.3 W4, U22, K27, C25	Dodecahedron Doe V 20,E 30,F 12=12{5} χ=2, group=Ih, H3, [5,3], (*532) 3 | 2 5 - 5.5.5 W5, U23, K28, C26	Icosidodecahedron Id V 30,E 60,F 32=20{3}+12{5} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 | 3 5 - 3.5.3.5 W12, U24, K29, C28	Truncated dodecahedron Tid V 60,E 90,F 32=20{3}+12{10} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 3 | 5 - 3.10.10 W10, U26, K31, C29	Truncated icosahedron Ti V 60,E 90,F 32=12{5}+20{6} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 5 | 3 - 5.6.6 W9, U25, K30, C27	Rhombicosidodecahedron Srid V 60,E 120,F 62=20{3}+30{4}+12{5} χ=2, group=Ih, H3, [5,3], (*532), order 120 3 5 | 2 - 3.4.5.4 W14, U27, K32, C30 Rhombicosidodecahedron	Truncated icosidodecahedron Grid V 120,E 180,F 62=30{4}+20{6}+12{10} χ=2, group=Ih, H3, [5,3], (*532), order 120 2 3 5 | - 4.6.10 W16, U28, K33, C31 Rhombitruncated icosidodecahedron Truncated icosidodecahedron	Snub dodecahedron Snid V 60,E 150,F 92=(20+60){3}+12{5} χ=2, group=I, 1/2H3, [5,3]+, (532), order 60 | 2 3 5 - 3.3.3.3.5 W18, U29, K34, C32

Non-convex

a b 2

3 3 2

${a\pi \over 3}\ {b\pi \over 3}\ {c\pi \over 2}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
${\pi \over 3}\ {\pi \over 2}\ {2\pi \over 3}$					Tetrahemihexahedron Thah V 6,E 12,F 7=4{3}+3{4} χ=1, group=Td, [3,3], *332 3/2 3 | 2 (double-covering) - 3.4.3/2.4 W67, U04, K09, C36

4 3 2

${a\pi \over 4}\ {b\pi \over 3}\ {c\pi \over 2}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
${\pi \over 4}\ {2\pi \over 3}\ {\pi \over 2}$	octahedron	cube		Stellated truncated hexahedron Quith V 24,E 36,F 14=8{3}+6{8/3} χ=2, group=Oh, [4,3], *432 2 3 | 4/3 2 3/2 | 4/3 - 3.8/3.8/3 W92, U19, K24, C66 Quasitruncated hexahedron stellatruncated cube		Nonconvex great rhombicuboctahedron Querco V 24,E 48,F 26=8{3}+(6+12){4} χ=2, group=Oh, [4,3], *432 3/2 4 | 2 3 4/3 | 2 - 4.4.4.3/2 W85, U17, K22, C59 Quasirhombicuboctahedron	Small rhombihexahedron Sroh V 24,E 48,F 18=12{4}+6{8} χ=−6, group=Oh, [4,3], *432 2 4 (3/2 4/2) | - 4.8.4/3.8/7 W86, U18, K23, C60
${3\pi \over 4}\ {\pi \over 3}\ {\pi \over 2}$							Great truncated cuboctahedron Quitco V 48,E 72,F 26=12{4}+8{6}+6{8/3} χ=2, group=Oh, [4,3], *432 2 3 4/3 | - 4.6/5.8/3 W93, U20, K25, C67 Quasitruncated cuboctahedron
${3\pi \over 4}\ {2\pi \over 3}\ {\pi \over 2}$							Great rhombihexahedron Groh V 24,E 48,F 18=12{4}+6{8/3} χ=−6, group=Oh, [4,3], *432 2 4/3 (3/2 4/2) | - 4.8/3.4/3.8/5 W103, U21, K26, C82

5 3 2

${a\pi \over 5}\ {b\pi \over 3}\ {c\pi \over 2}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r
${2\pi \over 5}\ {\pi \over 3}\ {\pi \over 2}$	Great icosahedron Gike V 12,E 30,F 20=20{3} χ=2, group=Ih, H3, [5,3], (*532) 5⁄2 | 2 3 - (35)/2 W41, U53, K58, C69	Great stellated dodecahedron Gissid V 20,E 30,F 12=12 { 5⁄2 } χ=2, group=Ih, H3, [5,3], (*532) 3 | 2 5⁄2 - (5⁄2)3 W22, U52, K57, C68	Great icosidodecahedron Gid V 30,E 60,F 32=20{3}+12{5/2} χ=2, group=Ih, [5,3], *532 2 | 3 5/2 2 | 3 5/3 2 | 3/2 5/2 2 | 3/2 5/3 - 3.5/2.3.5/2 W94, U54, K59, C70	Great stellated truncated dodecahedron Quit Gissid V 60,E 90,F 32=20{3}+12{10/3} χ=2, group=Ih, [5,3], *532 2 3 | 5/3 - 3.10/3.10/3 W104, U66, K71, C83 Quasitruncated great stellated dodecahedron Great stellatruncated dodecahedron	Truncated great icosahedron Tiggy V 60,E 90,F 32=12{5/2}+20{6} χ=2, group=Ih, [5,3], *532 2 5/2 | 3 2 5/3 | 3 - 6.6.5/2 W95, U55, K60, C71	Nonconvex great rhombicosidodecahedron Qrid V 60,E 120,F 62=20{3}+30{4}+12{5/2} χ=2, group=Ih, [5,3], *532 5/3 3 | 2 5/2 3/2 | 2 - 3.4.5/3.4 W105, U67, K72, C84 Quasirhombicosidodecahedron
	p q r|	p q r|	p q r|	|p q r
${3\pi \over 5}\ {\pi \over 3}\ {\pi \over 2}$	Rhombicosahedron Ri V 60,E 120,F 50=30{4}+20{6} χ=−10, group=Ih, [5,3], *532 2 3 (5/4 5/2) | - 4.6.4/3.6/5 W96, U56, K61, C72	Great truncated icosidodecahedron Gaquatid V 120,E 180,F 62=30{4}+20{6}+12{10/3} χ=2, group=Ih, [5,3], *532 2 3 5/3 | - 4.6.10/3 W108, U68, K73, C87 Great quasitruncated icosidodecahedron	Great rhombidodecahedron Gird V 60,E 120,F 42=30{4}+12{10/3} χ=−18, group=Ih, [5,3], *532 2 5/3 (3/2 5/4) | - 4.10/3.4/3.10/7 W109, U73, K78, C89

5 5 2

${a\pi \over 5}\ {b\pi \over 5}\ {c\pi \over 2}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r
${\pi \over 5}\ {2\pi \over 5}\ {\pi \over 2}$	Small stellated dodecahedron Sissid V 12,E 30,F 12=12 5 χ=-6, group=Ih, H3, [5,3], (*532) 5 | 2 5⁄2 - (5⁄2)5 W20, U34, K39, C43	Great dodecahedron Gad V 12,E 30,F 12=12{5} χ=-6, group=Ih, H3, [5,3], (*532) 5⁄2 | 2 5 - (55)/2 W21, U35, K40, C44	Dodecadodecahedron Did V 30,E 60,F 24=12{5}+12{5/2} χ=−6, group=Ih, [5,3], *532 2 | 5 5/2 2 | 5 5/3 2 | 5/2 5/4 2 | 5/3 5/4 - 5.5/2.5.5/2 W73, U36, K41, C45	Small stellated truncated dodecahedron Quit Sissid V 60,E 90,F 24=12{5}+12{10/3} χ=−6, group=Ih, [5,3], *532 2 5 | 5/3 2 5/4 | 5/3 - 5.10/3.10/3 W97, U58, K63, C74 Quasitruncated small stellated dodecahedron Small stellatruncated dodecahedron	Truncated great dodecahedron Tigid V 60,E 90,F 24=12{5/2}+12{10} χ=−6, group=Ih, [5,3], *532 2 5/2 | 5 2 5/3 | 5 - 10.10.5/2 W75, U37, K42, C47	Rhombidodecadodecahedron Raded V 60,E 120,F 54=30{4}+12{5}+12{5/2} χ=−6, group=Ih, [5,3], *532 5/2 5 | 2 - 4.5/2.4.5 W76, U38, K43, C48
	p q r|	p q r|	|p q r
${\pi \over 5}\ {3\pi \over 5}\ {\pi \over 2}$	Small rhombidodecahedron Sird V 60,E 120,F 42=30{4}+12{10} χ=−18, group=Ih, [5,3], *532 2 5 (3/2 5/2) | - 4.10.4/3.10/9 W74, U39, K44, C46	Truncated dodecadodecahedron Quitdid V 120,E 180,F 54=30{4}+12{10}+12{10/3} χ=−6, group=Ih, [5,3], *532 2 5 5/3 | - 4.10/9.10/3 W98, U59, K64, C75 Quasitruncated dodecadodecahedron

a b 3

3 3 3

${a\pi \over 3}\ {b\pi \over 3}\ {c\pi \over 3}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
${\pi \over 3}\ {\pi \over 3}\ {2\pi \over 3}$				Octahemioctahedron Oho V 12,E 24,F 12=8{3}+4{6} χ=0, group=Oh, [4,3], *432 3/2 3 | 3 - 3.6.3/2.6 W68, U03, K08, C37

4 3 3

${a\pi \over 4}\ {b\pi \over 3}\ {c\pi \over 3}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r

5 3 3

${a\pi \over 5}\ {b\pi \over 3}\ {c\pi \over 3}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r
${3\pi \over 5}\ {\pi \over 3}\ {\pi \over 3}$	Great ditrigonal icosidodecahedron Gidtid V 20,E 60,F 32=20{3}+12{5} χ=−8, group=Ih, [5,3], *532 3/2 | 3 5 3 | 3/2 5 3 | 3 5/4 3/2 | 3/2 5/4 - ((3.5)3)/2 W87, U47, K52, C61	Small ditrigonal icosidodecahedron Sidtid V 20,E 60,F 32=20{3}+12{5/2} χ=−8, group=Ih, [5,3], *532 3 | 5/2 3 - (3.5/2)3 W70, U30, K35, C39		Great icosihemidodecahedron Geihid V 30,E 60,F 26=20{3}+6{10/3} χ=−4, group=Ih, [5,3], *532 3/2 3 | 5/3 - 3.10/3.3/2.10/3 W106, U71, K76, C85	Small icosihemidodecahedron Seihid V 30,E 60,F 26=20{3}+6{10} χ=−4, group=Ih, [5,3], *532 3/2 3 | 5 (double covering) - 3.10.3/2.10 W89, U49, K54, C63	Great icosicosidodecahedron Giid V 60,E 120,F 52=20{3}+12{5}+20{6} χ=−8, group=Ih, [5,3], *532 3/2 5 | 3 3 5/4 | 3 - 5.6.3/2.6 W88, U48, K53, C62
	p q r|	p q r|	|p q r
${\pi \over 5}\ {2\pi \over 3}\ {\pi \over 3}$	Small icosicosidodecahedron Siid V 60,E 120,F 52=20{3}+12{5/2}+20{6} χ=−8, group=Ih, [5,3], *532 5/2 3 | 3 - 6.5/2.6.3 W71, U31, K36, C40	Small dodecicosahedron Siddy V 60,E 120,F 32=20{6}+12{10} χ=−28, group=Ih, [5,3], *532 3 5 (3/2 5/4) | - 6.10.6/5.10/9 W90, U50, K55, C64

4 4 3

${a\pi \over 4}\ {b\pi \over 4}\ {c\pi \over 3}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
${\pi \over 4}\ {\pi \over 3}\ {3\pi \over 4}$					Cubohemioctahedron Cho V 12,E 24,F 10=6{4}+4{6} χ=−2, group=Oh, [4,3], *432 4/3 4 | 3 (double-covering) - 4.6.4/3.6 W78, U15, K20, C51	Great cubicuboctahedron Gocco V 24,E 48,F 20=8{3}+6{4}+6{8/3} χ=−4, group=Oh, [4,3], *432 3 4 | 4/3 4 3/2 | 4 - 3.8/3.4.8/3 W77, U14, K19, C50	Cubitruncated cuboctahedron Cotco V 48,E 72,F 20=8{6}+6{8}+6{8/3} χ=−4, group=Oh, [4,3], *432 3 4 4/3 | - 6.8.8/3 W79, U16, K21, C52 Cuboctatruncated cuboctahedron
${\pi \over 4}\ {\pi \over 4}\ {2\pi \over 3}$				Small cubicuboctahedron Socco V 24,E 48,F 20=8{3}+6{4}+6{8} χ=−4, group=Oh, [4,3], *432 3/2 4 | 4 3 4/3 | 4 - 4.8.3/2.8 W69, U13, K18, C38

5 5 3

${a\pi \over 5}\ {b\pi \over 5}\ {c\pi \over 3}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
${\pi \over 3}\ {2\pi \over 5}\ {3\pi \over 5}$				Small dodecahemicosahedron Sidhei V 30,E 60,F 22=12{5/2}+10{6} χ=−8, group=Ih, [5,3], *532 5/3 5/2 | 3 (double covering) - 6.5/2.6.5/3 W100, U62, K67, C78	Great dodecicosahedron Giddy V 60,E 120,F 32=20{6}+12{10/3} χ=−28, group=Ih, [5,3], *532 3 5/3 (3/2 5/2) | - 6.10/3.6/5.10/7 W101, U63, K68, C79	Small dodecicosidodecahedron Saddid V 60,E 120,F 44=20{3}+12{5}+12{10} χ=−16, group=Ih, [5,3], *532 3/2 5 | 5 3 5/4 | 5 - 5.10.3/2.10 W72, U33, K38, C42
${\pi \over 3}\ {\pi \over 5}\ {4\pi \over 5}$				Great dodecahemicosahedron Gidhei V 30,E 60,F 22=12{5}+10{6} χ=−8, group=Ih, [5,3], *532 5/4 5 | 3 (double covering) - 5.6.5/4.6 W102, U65, K70, C81	Small ditrigonal dodecicosidodecahedron Sidditdid V 60,E 120,F 44=20{3}+12{5/2}+12{10} χ=−16, group=Ih, [5,3], *532 5/3 3 | 5 5/2 3/2 | 5 - 3.10.5/3.10 W82, U43, K48, C55	Great ditrigonal dodecicosidodecahedron Gidditdid V 60,E 120,F 44=20{3}+12{5}+12{10/3} χ=−16, group=Ih, [5,3], *532 3 5 | 5/3 5/4 3/2 | 5/3 - 3.10/3.5.10/3 W81, U42, K47, C54
${\pi \over 5}\ {\pi \over 5}\ {2\pi \over 3}$				Small dodecicosidodecahedron Saddid V 60,E 120,F 44=20{3}+12{5}+12{10} χ=−16, group=Ih, [5,3], *532 3/2 5 | 5 3 5/4 | 5 - 5.10.3/2.10 W72, U33, K38, C42	Great dodecicosidodecahedron Gaddid V 60,E 120,F 44=20{3}+12{5/2}+12{10/3} χ=−16, group=Ih, [5,3], *532 5/2 3 | 5/3 5/3 3/2 | 5/3 - 3.10/3.5/2.10/7 W99, U61, K66, C77
${\pi \over 5}\ {\pi \over 3}\ {3\pi \over 5}$		Ditrigonal dodecadodecahedron Ditdid V 20,E 60,F 24=12{5}+12{5/2} χ=−16, group=Ih, [5,3], *532 3 | 5/3 5 3/2 | 5 5/2 3/2 | 5/3 5/4 3 | 5/2 5/4 - (5.5/3)3 W80, U41, K46, C53		Icosidodecadodecahedron Ided V 60,E 120,F 44=12{5}+12{5/2}+20{6} χ=−16, group=Ih, [5,3], *532 5/3 5 | 3 5/2 5/4 | 3 - 5.6.5/3.6 W83, U44, K49, C56	Small ditrigonal dodecicosidodecahedron Sidditdid V 60,E 120,F 44=20{3}+12{5/2}+12{10} χ=−16, group=Ih, [5,3], *532 5/3 3 | 5 5/2 3/2 | 5 - 3.10.5/3.10 W82, U43, K48, C55	Icositruncated dodecadodecahedron Idtid V 120,E 180,F 44=20{6}+12{10}+12{10/3} χ=−16, group=Ih, [5,3], *532 3 5 5/3 | - 6.10.10/3 W84, U45, K50, C57 Icosidodecatruncated icosidodecahedron

a b 5

5 5 5

${a\pi \over 5}\ {b\pi \over 5}\ {c\pi \over 5}$ Group

Spherical triangle ${\pi \over p}\ {\pi \over q}\ {\pi \over r}$	p|q r	q|p r	r|p q	q r|p	p r|q	p q|r	p q r|	|p q r
${2\pi \over 5}\ {3\pi \over 5}\ {3\pi \over 5}$					Great dodecahemidodecahedron Gidhid V 30,E 60,F 18=12{5/2}+6{10/3} χ=−12, group=Ih, [5,3], *532 5/3 5/2 | 5/3 (double covering) - 5/2.10/3.5/3.10/3 W107, U70, K75, C86