Icosahedral shapes have been identified in molecular crystalline shells such as large viral shells or fullerenes. We have demonstrated that other geometries, including some Archimedean geometries, arise spontaneously in shells formed by more than one component (1). We describe the buckling of a crystalline shell with two coexisting elastic components, at different relative concentrations. By using theoretical arguments and numerical simulations we find various irregular and regular polyhedra such as Platonic and Johnson solids and n-gonal hosohedra. Our work explains the principles to design various hallow polyhedra and the existence of regular and irregular polyhedral shells observed in organelles and in halophilic organisms wall envelopes, as well as viral capsids made of various proteins. We provide experimental evidence of the spontaneous buckling phenomena in shells made of mixtures of cationic and anionic amphiphiles (2), where electrostatics drives their co-assembly, and orders the assembly into faceted ionic structures with various crystalline domains. Though these shells are stable at high monovalent salt concentrations, their crystalline structure and shape are modified by the pH value of the solution. Our work provides guidelines for the fabrication of nanocontainers with specific shapes and may aid to elucidate paradigms that relate shape and composition of cellular shells.

(1) G. Vernizzi, R. Sknepnek, and M. Olvera de la Cruz "Platonic and Archimedean geometries in multi-component elastic membranes" Proc. Natl. Acad. Sci. USA, 118, 4292–4296 (2011). (2) M. A. Greenfield, L. C. Palmer, G. Vernizzi, M. Olvera de la Cruz, and S. I. Stupp “Buckled Membranes in Mixed-Valence Ionic Amphiphile Vesicles” J. Am. Chem. Soc., 131, 12030–12031 (2009).