(362h) Rationalizing Euclidean Assemblies of Hard Polyhedra from Tessellations in Curved Space

The most important system attribute that governs entropic self-assembly is the shape of the constituent particles. Yet, a priori prediction of crystal structures from particle shape alone is non-trivial for anything but the simplest of space-filling shapes, especially when the thermodynamically preferred structure differs from its densest packing. While geometric constraints prevent these polyhedra from tessellating 3D space, we can eliminate them by sufficiently curving space. When constrained to the surface volume of a 3-sphere, we show using Monte Carlo simulations that most hard polyhedra in the family of Platonic solids self-assemble entropically into space-filling crystals. As we gradually increase the 3-sphere radius to ``flatten'' space, defects arise due to geometric frustration. By comparing the local particle environments in crystals assembling in curved and flat space, we show that the Euclidean assemblies can be categorized either as remnants of tessellations on the 3-sphere or non-tessellation-based assemblies caused by large-scale geometric frustration.