Explore periodic tessellations of Euclidean space through the lens of energy minimization in this 31-minute mathematical lecture. Delve into the minimization of local and non-local perimeter functionals to understand how periodic tilings emerge naturally from variational principles. Examine the existence, regularity, and qualitative properties of minimizers, with particular emphasis on non-local perimeter functionals and their unique characteristics. Discover the mathematical framework underlying these geometric structures and learn about current research challenges in the field. Gain insights into collaborative research connecting the Politecnico di Milano and Università di Pisa, focusing on the intersection of geometric measure theory, calculus of variations, and periodic structures. Understand how these mathematical concepts relate to broader questions in free boundary problems and their applications in mathematical physics and geometry.