A diffusion generated method for computing Dirichlet partitions

Abstract

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.

24 pages, 11 figures