$n = 1, \ldots, k$ requires ${\mathcal{O}}(k^2)$ evaluations which is too expensive for the hypergraphs we envision, with a huge number of vertices $k$. Luckily, we can circumvent this problem using techniques from [@Hu2005] and [@Nakasato2009]: we will build the $d$-dimensional equivalent of an octree to group the repelling charges into clusters and treat far-away clusters of charges as a single, but heavier charge. To ensure that this works properly the location of this larger charge will be the weighted average location of the cluster and the weight of the charge will be set equal to the sum of the charge weights in the cluster. Note that the repelling-charges part of $f$ in consists of applications of the map $x \mapsto \|x\|^{-\delta}$ for $x \in {\mathbf{R}}^d$, the Laplacian of which is given by $x \mapsto \delta \, (\delta - (d - 2)) \, \|x\|^{-\delta - 2}$. Hence, we can ensure that this part of $f$ is harmonic (i.e. with vanishing Laplacian) by choosing $\delta = d - 2$. This will in turn make our energy function behave well when treating clusters of far-away charges as a single, heavier charge, because of the *mean-value property* of harmonic functions (see [@Axler1992 Theorem 