Is it true that all convex polyhedra can be represented as Wulff shapes of simple idealized materials? To make progress towards this conjecture, we first prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours), to a crystalline perimeter, and discuss the possible Wulff shapes obtainable in this way. Next, exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling. This is joint work with Giacomo Del Nin from University of Warwick.
Paul Pegon
