Existence of Tripoles? With multipole expansions, we speak only of monopoles, dipoles, and $2^n$-poles. Why is there nothing like a tripole? So how would something like $rsin(3 \theta)$ be expressed with a multipole expansion? Is the reason for not having terms like this in a multipole expansion that it is redundant, or is there some more fundamental reason? 
 A: The function $\sin 3\theta$ on the unit sphere is not an eigenfunction of the Laplacian on the sphere, i.e. the angular part of the Laplacian, i.e. of $L^2$, so it is not convenient a basis vector in problems whose Hamiltonian involves the Laplacian.
The function $\sin 3\theta$ may be written as a combination of spherical harmonics $Y_{lm}$ with many different values of $(l,m)$ so it is a "mixture" of multipoles of different "rank". For the more natural basis of functions on the sphere that may see as basis vectors, see

The table of spherical harmonics
https://en.wikipedia.org/wiki/Table_of_spherical_harmonics

For example, the spherical harmonics $Y_{3,\pm 3}$ are proportional to
$$ Y_{3,\pm 3} \sim \exp(\pm 3i\phi) \sin^3 \theta $$
which is very similar to $\sin 3\theta$ but has the extra $\phi$-dependence. Similarly, one may look at the function $Y_{30}$ which is similar to $\sin 3\theta$ but prefers cosines and so on. Either $\sin^3 \theta$ or $\cos^3\theta$ (check it!) without any $\phi$-dependence is a combination of $Y_{30}$ and $Y_{10}$.
Once one realizes why the spherical harmonics are the preferred, more natural basis, we may carefully discuss the spherical harmonics' association with the multipole expansion. For example, we learn that $Y_{3,m}$ for any $m$, including the functions similar to yours above, are associated with octupoles, not "tripoles"!
More generally, $Y_{\ell m}$ is the angular part of the $2^\ell$-pole.
The powers of two are a natural way to describe the terms in the multipole expansions for reasons explained elsewhere, e.g. here:

https://physics.stackexchange.com/a/127496/1236

In the multipole terminology, a "tripole" would correspond to a triplet (e.g. vertices of a triangle) of charges. If their total charge would be nonzero, there would be a leading "monopole" term. If the total charge cancelled, the system of 3 charges would still have a dipole moment. Unless the three (nonzero) charges would lie on the same line, the dipole moment couldn't be canceled.
