Sommaire:

The densest way to pack objects in space, also known as the packing problem, has intrigued scientists and philosophers for millenia. Today, packing comes up in various systems over many length scales from batteries and catalysts to the self-assembly of nanoparticles, colloids and biomolecules. Despite the fact that so many systems' properties depend on the packing of differently-shaped components, we still have no general understanding of how packing varies as a function of particle shape. Here we carry out an exhaustive study of how packing depends on shape by investigating the packings of over 55,000 polyhedra. By combining simulations and analytic calculations, we study families of polyhedra interpolating between Platonic and Archimedean solids such as the tetrahedron, the cube, and the octahedron, via continuous vertex and edge truncations. We find maximum packing-density surfaces that reveal surprising richness and complexity. We expect our density surface plots to guide experiments that use shape and packing in a similar way that phase diagrams are used to do chemistry. We suggest to think about a shape no longer as a static property but rather as but one point in a higher dimensional “shape space” where the neighborhood around the given shape, as achieved by small deformations, needs to be taken into account as it may reveal why we can assemble certain crystals, transition between them, or get stuck in kinetic traps.

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