All the faces are identical, each edge is identical and each vertex is identical.
The all have a Wythoff symbol of the form p|q r, and at one of q or r is 2.
Icosahedron
Ike
V 12,E 30,F 20=20{3}
χ=2, group=Ih 5 | 3 2 - 3.3.3.3.3
W4, U22, K27, C25
Dodecahedron
Doe
V 20,E 30,F 12=12{5}
χ=2, group=Ih 3 | 5 2 - 5.5.5
W5, U23, K28, C26
Non-convex
The Kepler Posisot solids.
Great icosahedron
Gike
V 12,E 30,F 20=20{3}
χ=2, group=Ih 5/2 | 2 3 - 3.3.3.3.3
W41, U53, K58, C69
Great dodecahedron
Gad
V 12,E 30,F 12=12{5}
χ=-6, group=Ih 5/2 | 2 5 - 5.5.5.5.5
W21, U35, K40, C44
Small stellated dodecahedron
Sissid
V 12,E 30,F 12=12{5/2}
χ=-6, group=Ih 5 | 25/2 - 5/2.5/2.5/2.5/2.5/2
W20, U34, K39, C43
Great stellated dodecahedron
Gissid
V 20,E 30,F 12=12{5/2}
χ=2, group=Ih 3 | 25/2 - 5/2.5/2.5/2
W22, U52, K57, C68
Quasi-regular
Each edge is identical and each vertex is identical. There are two types of faces
which apear in an alternating fashon around each vertex.
The first row are semi-regular with 4 faces around each vertex. They fave Wythoff symbol 2|p q.
The second row are ditriogonal with 6 faces around each vertex. They have Wythoff symbol 3|p q or 3/2|p q.
Cuboctahedron
Co
V 12,E 24,F 14=8{3}+6{4}
χ=2, group=Oh 2 | 3 4 - 3.4.3.4
W11, U07, K12, C19
Icosidodecahedron
Id
V 30,E 60,F 32=20{3}+12{5}
χ=2, group=Ih 2 | 3 5 - 3.5.3.5
W12, U24, K29, C28
Great icosidodecahedron
Gid
V 30,E 60,F 32=20{3}+12{5/2}
χ=2, group=Ih 2 | 3 5/2 - 3.5/2.3.5/2
W94, U54, K59, C70
Dodecadodecahedron
Did
V 30,E 60,F 24=12{5}+12{5/2}
χ=-6, group=Ih 2 | 5 5/2 - 5.5/2.5.5/2
W73, U36, K41, C45
Each vertex has three faces surronding it, two of which are identical. These all have Wythoff symbols 2 p|q, some are constructed by truncating the regular soilds.
Small stellated truncated dodecahedron
Quitsissid
V 60,E 90,F 24=12{5}+12{10/3}
χ=-6, group=Ih 2 5 | 5/3 - 5.10/3.10/3
W97, U58, K63, C74
Quasitruncated small stellated dodecahedron
Small stellatruncated dodecahedron
Great stellated truncated dodecahedron
Quitgissid
V 60,E 90,F 32=20{3}+12{10/3}
χ=2, group=Ih 2 3 | 5/3 - 3.10/3.10/3
W104, U66, K71, C83
Quasitruncated great stellated dodecahedron
Great stellatruncated dodecahedron
Hemi-hedra
The hemi-hedra all have faces which pass through the origin. Their Wythoff symbols are of the form p p/m|q or p/m p/n|q. With the exception of the tetrahemihexahedron they occur in pairs, and are closely related to the semi-regular polyhedra, like the cuboctohedron.
Cubohemioctahedron
Cho
V 12,E 24,F 10=6{4}+4{6}
χ=-2, group=Oh 4/34 | 3 - 4.6.4.6
W78, U15, K20, C51
Small icosihemidodecahedron
Seihid
V 30,E 60,F 26=20{3}+6{10}
χ=-4, group=Ih 3/23 | 5 - 3.10.3.10
W89, U49, K54, C63
Small dodecahemidodecahedron
Sidhid
V 30,E 60,F 18=12{5}+6{10}
χ=-12, group=Ih 5/45 | 5 - 5.10.5.10
W91, U51, K56, C65
Great icosihemidodecahedron
Geihid
V 30,E 60,F 26=20{3}+6{10/3}
χ=-4, group=Ih 3 3 | 5/3 - 3.10/3.3.10/3
W106, U71, K76, C85
Great dodecahemidodecahedron
Gidhid
V 30,E 60,F 18=12{5/2}+6{10/3}
χ=-12, group=Ih 5/35/2 | 5/3 - 5/2.10/3.5/2.10/3
W107, U70, K75, C86
Great dodecahemicosahedron
Gidhei
V 30,E 60,F 22=12{5}+10{6}
χ=-8, group=Ih 5/45 | 3 - 5.6.5.6
W102, U65, K70, C81
Small dodecahemicosahedron
Sidhei
V 30,E 60,F 22=12{5/2}+10{6}
χ=-8, group=Ih 5/35/2 | 3 - 6.5/2.6.5/2
W100, U62, K67, C78
Rhombic quais-regular
Four faces around the vertex in the pattern p.q.r.q. The name rhombic stems from inserting
a square in the cubeoctohedron and icodocehedron. The Wythoff symbol is of the form p q|r.
Small rhombicuboctahedron
Sirco
V 24,E 48,F 26=8{3}+(6+12){4}
χ=2, group=Oh 3 4 | 2 - 3.4.4.4
W13, U10, K15, C22
Rhombicuboctahedron
Small cubicuboctahedron
Socco
V 24,E 48,F 20=8{3}+6{4}+6{8}
χ=-4, group=Oh 3/24 | 4 - 3.8.4.8
W69, U13, K18, C38
Uniform great rhombicuboctahedron
Querco
V 24,E 48,F 26=8{3}+(6+12){4}
χ=2, group=Oh 3/24 | 2 - 3.4.4.4
W85, U17, K22, C59
Quasirhombicuboctahedron
Small rhombicosidodecahedron
Srid
V 60,E 120,F 62=20{3}+30{4}+12{5}
χ=2, group=Ih 3 5 | 2 - 3.4.5.4
W14, U27, K32, C30
Rhombicosidodecahedron
Small dodecicosidodecahedron
Saddid
V 60,E 120,F 44=20{3}+12{5}+12{10}
χ=-16, group=Ih 3/25 | 5 - 3.10.5.10
W72, U33, K38, C42
Uniform great rhombicosidodecahedron
Qrid
V 60,E 120,F 62=20{3}+30{4}+12{5/2}
χ=2, group=Ih 5/33 | 2 - 3.4.5/2.4
W105, U67, K72, C84
Quasirhombicosidodecahedron
Great icosicosidodecahedron
Giid
V 60,E 120,F 52=20{3}+12{5}+20{6}
χ=-8, group=Ih 3/25 | 3 - 3.6.5.6
W88, U48, K53, C62