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)
Bipyramids	(infinite)
Pyramids	(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 $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r
$\frac{\pi }{3}\frac{\pi }{3}\frac{\pi }{2}$	File:Tetrahedron.png 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	File:Polyhedron truncated 4a max.png 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
$\frac{\pi }{4}\frac{\pi }{3}\frac{\pi }{2}$	File:Octahedron.png 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	File:Hexahedron.png 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	File:Polyhedron 6-8 max.png 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	File:Polyhedron truncated 6 max.png 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	File:Polyhedron truncated 8 max.png 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	File:Polyhedron small rhombi 6-8 max.png 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	File:Polyhedron great rhombi 6-8 max.png 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	File:Polyhedron snub 6-8 left max.png Snub cube Snic V 24,E 60,F 38=(8+24){3}+6{4} χ=2, group=O, Template:SfracB3, [4,3]+, (432), order 24 | 2 3 4 - 3.3.3.3.4 W17, U12, K17, C24
$\frac{\pi }{5}\frac{\pi }{3}\frac{\pi }{2}$	File:Icosahedron.png 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	File:Dodecahedron.png 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	File:Polyhedron 12-20 max.png 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	File:Polyhedron truncated 12 max.png 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	File:Polyhedron truncated 20 max.png 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	File:Polyhedron small rhombi 12-20 max.png 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	File:Polyhedron great rhombi 12-20 max.png 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	File:Polyhedron snub 12-20 left max.png Snub dodecahedron Snid V 60,E 150,F 92=(20+60){3}+12{5} χ=2, group=I, Template:SfracH3, [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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r
$\frac{\pi }{3}\frac{\pi }{2}\frac{2\pi }{3}$					File:Tetrahemihexahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r
$\frac{\pi }{4}\frac{2\pi }{3}\frac{\pi }{2}$	octahedron	cube		File:Stellated truncated hexahedron.png 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		File:Uniform great rhombicuboctahedron.png 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	File:Small rhombihexahedron.png 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
$\frac{3\pi }{4}\frac{\pi }{3}\frac{\pi }{2}$							File:Great truncated cuboctahedron.png 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
$\frac{3\pi }{4}\frac{2\pi }{3}\frac{\pi }{2}$							File:Great rhombihexahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r
$\frac{2\pi }{5}\frac{\pi }{3}\frac{\pi }{2}$	File:Great icosahedron.png 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	File:Great stellated dodecahedron.png 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	File:Great icosidodecahedron.png 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	File:Great stellated truncated dodecahedron.png 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	File:Great truncated icosahedron.png 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	File:Uniform great rhombicosidodecahedron.png 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
$\frac{3\pi }{5}\frac{\pi }{3}\frac{\pi }{2}$	File:Rhombicosahedron.png 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	File:Great truncated icosidodecahedron.png 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	File:Great rhombidodecahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r
$\frac{\pi }{5}\frac{2\pi }{5}\frac{\pi }{2}$	File:Small stellated dodecahedron.png 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	File:Great dodecahedron.png 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	File:Dodecadodecahedron.png 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	File:Small stellated truncated dodecahedron.png 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	File:Great truncated dodecahedron.png 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	File:Rhombidodecadodecahedron.png 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
$\frac{\pi }{5}\frac{3\pi }{5}\frac{\pi }{2}$	File:Small rhombidodecahedron.png 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	File:Truncated dodecadodecahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r
$\frac{\pi }{3}\frac{\pi }{3}\frac{2\pi }{3}$				File:Octahemioctahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r

5 3 3

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r
$\frac{3\pi }{5}\frac{\pi }{3}\frac{\pi }{3}$	File:Great ditrigonal icosidodecahedron.png 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	File:Small ditrigonal icosidodecahedron.png 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		File:Great icosihemidodecahedron.png 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	File:Small icosihemidodecahedron.png 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	File:Great icosicosidodecahedron.png 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
$\frac{\pi }{5}\frac{2\pi }{3}\frac{\pi }{3}$	File:Small icosicosidodecahedron.png 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	File:Small dodecicosahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r
$\frac{\pi }{4}\frac{\pi }{3}\frac{3\pi }{4}$					File:Cubohemioctahedron.png 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	File:Great cubicuboctahedron.png 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	File:Cubitruncated cuboctahedron.png 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
$\frac{\pi }{4}\frac{\pi }{4}\frac{2\pi }{3}$				File:Small cubicuboctahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r
$\frac{\pi }{3}\frac{2\pi }{5}\frac{3\pi }{5}$				File:Small dodecahemicosahedron.png 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	File:Great dodecicosahedron.png 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	File:Small dodecicosidodecahedron.png 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
$\frac{\pi }{3}\frac{\pi }{5}\frac{4\pi }{5}$				File:Great dodecahemicosahedron.png 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	File:Small ditrigonal dodecicosidodecahedron.png 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	File:Great ditrigonal dodecicosidodecahedron.png 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
$\frac{\pi }{5}\frac{\pi }{5}\frac{2\pi }{3}$				File:Small dodecicosidodecahedron.png 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	File:Great dodecicosidodecahedron.png 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
$\frac{\pi }{5}\frac{\pi }{3}\frac{3\pi }{5}$		File:Ditrigonal dodecadodecahedron.png 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		File:Icosidodecadodecahedron.png 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	File:Small ditrigonal dodecicosidodecahedron.png 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	File:Icositruncated dodecadodecahedron.png 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

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

Spherical triangle $\frac{\pi }{p}\frac{\pi }{q}\frac{\pi }{r}$	q r	p r	p q	p	q	r		p q r
$\frac{2\pi }{5}\frac{3\pi }{5}\frac{3\pi }{5}$					File:Great dodecahemidodecahedron.png 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