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, equalatial 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.

Here they are grouped 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, [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 , [3,3], (*332) 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, [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, [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, [4,3], (*432) Td, [3,3], (*332) 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 , [4,3], (*432) 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, [4,3], (*432) Th, [3,3] and (*332) 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, [4,3], (*432) 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, [4,3], (*432) 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, [4,3]+, (432) | 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, [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, [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, [5,3], (*532) 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, [5,3], (*532) 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, [5,3], (*532) 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, [5,3], (*532) 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, [5,3], (*532) 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, [5,3]+, (532) | 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 - 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 - 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 - 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 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 | - 4.6.8/3 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, [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, [5,3], (*532) 3 | 25/2 - (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 - 3.5/2.3.5/2 W94, U54, K59, C70	Great stellated truncated dodecahedron Quitgissid 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 - 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 - 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{5/2} χ=-6, group=Ih, [5,3], (*532) 5 | 25/2 - (5/2)5 W20, U34, K39, C43	Great dodecahedron Gad V 12,E 30,F 12=12{5} χ=-6, group=Ih, [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 - 5.5/2.5.5/2 W73, U36, K41, C45	Small stellated truncated dodecahedron Quitsissid V 60,E 90,F 24=12{5}+12{10/3} χ=−6, group=Ih, [5,3], *532 2 5 | 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 - 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 | - 4.10.10/3 W98, U59, K64, C75 Quasitruncated dodecahedron

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 Td, [3,3], *332 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 - ((3.5)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 3 | 5/3 - 3.10/3.3.10/3 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - (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 - 5.6.5/3.6 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 - 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 - 5/2.10/3.5/3.10/3 W107, U70, K75, C86