Truncated icosidodecahedron

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron 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. The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection). There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron. The surface area A and the volume V of the truncated icosidodecahedron of edge length a are: If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest. Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of: (±1/φ, ±1/φ, ±(3 + φ)), (±2/φ, ±φ, ±(1 + 2φ)), (±1/φ, ±φ2, ±(−1 + 3φ)), (±(2φ − 1), ±2, ±(2 + φ)) and (±φ, ±3, ±2φ), where φ = 1 + /2 is the golden ratio.