Platonic and Archimedean solids have beguiled scientists, artists and mathematicians for millennia. In recent times polyhedral shapes have been identified at the microscopic level in crystalline shells such as fullerenes, viral capsids and protein-based bacterial organelles. The most frequently found polyhedron in homogeneous crystalline shells is the icosahedron. We demonstrate that other geometries arise spontaneously in shells formed by more than one component. Our work explains the existence of various regular and irregular polyhedral shells found in nature, and provides the principles for designing nanocontainers with specific shapes and symmetries for numerous applications in material and life sciences.