The Snub Dodecahedron is the most spherical of the Archimedean solids (Aravind, P. K., "How Spherical Are the Archimedean Solids and Their Duals?" College Mathematics Journal, 42:2, 2011, pp. 98-107.). To me, as a chemist, it is among the most beautiful, because, like the Snub Cube, it exists in two chiral forms. It requires great care in assembly. This polyhedron is also among the most rigid because of the large number of triangles.
(3,3,3,3,5) (2 chiral forms)
80 triangle and 12 pentagon faces, 60 vertices, 150 edges
Circumsphere diameter: 32.8 cm/12.9 in