Truncated icosidodecahedron

Truncated icosidodecahedron
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 62, E = 180, V = 120 (χ = 2)
Faces by sides	30{4}+20{6}+12{10}
Conway notation	bD or taD
Schläfli symbols	tr{5,3} or ${\displaystyle t{\begin{Bmatrix}5\\3\end{Bmatrix}}}$
	t0,1,2{5,3}
Wythoff symbol	2 3 5 |
Coxeter diagram
Symmetry group	Ih, H3, [5,3], (*532), order 120
Rotation group	I, [5,3]+, (532), order 60
Dihedral angle	6-10: 142.62° 4-10: 148.28° 4-6: 159.095°
References	U28, C31, W16
Properties	Semiregular convex zonohedron
Colored faces	4.6.10 (Vertex figure)
Disdyakis triacontahedron (dual polyhedron)	Net

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,^[1] great rhombicosidodecahedron,^[2][3] omnitruncated dodecahedron or omnitruncated icosahedron^[4] is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all vertex-transitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a 15-zonohedron.