  The fast multipole method (FMM) performs fast approximate kernel summation to
a specified tolerance $\epsilon$ by using a hierarchical division of the
domain, which groups source and receiver points into regions that satisfy local
separation and the well-separated pair decomposition properties. While square
tilings and quadtrees are commonly used in 2D, we investigate alternative
tilings and associated spatial data structures: regular hexagons (septree) and
triangles (triangle-quadtree). We show that both structures satisfy separation
properties for the FMM and prove their theoretical error bounds and
computational costs. Empirical runtime and error analysis of our
implementations are provided.
