Molecules and crystals with both icosahedral and cubic symmetry

Abstract

Notwithstanding the apparent incompatibility between octahedral and icosahedral symmetries, fragments with the two types of symmetry coexist in many molecules and crystals, as evidenced by continuous shape and symmetry measures. A geometric analysis of Platonic and Archimedean polyhedra and of a variety of molecular and crystal structures strongly suggests that octahedral symmetry is latent in icosahedral polyhedra and vice versa. In this Feature Article, new concepts and structural data from the literature combine to offer a perspective view of complex molecular and extended structures. Its influence on the common cubic packing of icosahedral molecules is discussed for a variety of examples, including water clathrates, dodecahedrane, Buckminsterfullerene, the Pd₁₄₅ and Mo₁₃₂ clusters and several intermetallic phases.

 Kepler stared, weeping, at the five polyhedrafit into each other, systematic, perfect,with musical order up to the great sphere.He loved the dodecahedron, wept over the icosahedronfor its inconsistencies and intricacieslovely and odd but, oh!, so necessary,since one cannot conceive more perfect solidsbeyond the five known, in three dimensions. Gabriel Celaya