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 2d flat torus, for most values of k=3-9,11,12,15,16, and 20, we obtain hexagonal honeycombs. For the 3d 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 4d 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 2d 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 published results for Dirichlet partitions of the 4d flat torus.