Rationalizing Euclidean Assemblies of Hard Polyhedra from Tessellations in Curved Space

Entropic self-assembly is governed by 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. At the same time, most polyhedra are not space filling
due to geometric constraints, but these constraints can be relaxed or even eliminated by
sufficiently curving space. We show using Monte Carlo simulations that the majority of
hard Platonic shapes self-assemble entropically into space-filling crystals when constrained
to the surface volume of a 3-sphere. As we gradually decrease curvature to “flatten” space
and compare the local morphologies of crystals assembling in curved and flat space, we
show that the Euclidean assemblies can be categorized either as remnants of tessellations in
curved space (tetrahedra and dodecahedra) or non-tessellation-based assemblies caused by
large-scale geometric frustration (octahedra and icosahedra)



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