A diffusion generated method for computing Dirichlet partitions

A Dirichlet k-partition of a closed d-dimensional surface is a collection of k pairwise disjoint open subsets such that the sum of their first Laplace-Beltrami-Dirichlet eigenvalues is minimal. In this paper, we develop a simple and efficient diffusion generated method to compute Dirichlet k-partitions for d-dimensional flat tori and spheres. For the two-dimensional flat torus, for most values of k = 3-9, 11, 12, 15, 16, and 20, we obtain hexagonal honeycombs. For the three-dimensional flat torus and k = 2, 4, 8, 16, we obtain the rhombic dodecahedral honeycomb, the Weaire-Phelan honeycomb, and Kelvin's tessellation by truncated octahedra. For the four-dimensional flat torus, for k = 4, we obtain a constant extension of the rhombic dodecahedral honeycomb along the fourth direction and for k = 8, we obtain a 24-cell honeycomb. For the two-dimensional sphere, we also compute Dirichlet partitions for k = 3-7, 9, 10, 12, 14, 20. Our computational results agree with previous studies when a comparison is available. As far as we are aware, these are the first results for Dirichlet partitions of the four-dimensional flat torus. (C) 2018 Elsevier B.V. All rights reserved.