In an answer to this question, I showed that a general electric quadrupole can be modeled by a combination of seven point charges (or possibly less if one or two of the principal quadrupole moments is zero, or if they add to zero). This is likely the minimal number, or at least it is likely to be the minimal systematic expansion (or, if it isn't, I'll be happy to be corrected).
However, that construction strongly relies on the diagonalization of the quadrupole moment matrix, so it does not easily generalize to higher orders of the multipole tensor hierarchy.
Thus, I would like to ask: given an arbitrary electric octupole moment tensor $Q_{ijk}$, what is the minimal number of point charges for which the leading term in the multipole expansion will be precisely that octupole term? (More technically, I'm looking for a systematic procedure that will take an octupole moment tensor $Q_{ijk}$ and produce a collection of point charges $\{q_i,\mathbf r_i\}_{i=1}^N$, where $N$ is minimal across all $Q_{ijk}$.)
I would expect the answer to this question to hinge on the ways one can analyze and simplify tensors of rank 3 and higher. (As an example of what I mean, the quadrupole moment tensor for two point charges $+q$ at $\pm d\,\hat{\mathbf e}$ and a point charge $-2q$ at the origin has all-nonzero components unless $\hat{\mathbf e}$ falls on a coordinate plane, even though the minimal model is much simpler than that. Moving to the principal-moments frame strips away this unnecessary complexity.) I am mostly interested in those tools to analyse and understand the general case, rather than some abstruse mathsy way to pare down the number of charges in some weird edge case.
