  Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a
unit sphere $S^2$, admits essentially a unique representation of the form
$\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are
linear forms in $x, y$ and $z$ and $\lambda$ is a real number. The coefficients
of these linear forms, viewed as 3D vectors, are called \emph{multipole}
vectors of $P$. In this paper we consider similar multipole representations of
polynomial and analytic functions on other quadratic surfaces $Q(x, y, z) = c$,
real and complex. Over the complex numbers, the above representation is not
unique, although the ambiguity is essentially finite. We investigate the
combinatorics that depicts this ambiguity. We link these results with some
classical theorems of harmonic analysis, theorems that describe decompositions
of functions into sums of spherical harmonics. We extend these classical
theorems (which rely on our understanding of the Laplace operator
$\Delta_{S^2}$) to more general differential operators $\Delta_Q$ that are
constructed with the help of the quadratic form $Q(x, y, z)$. Then we introduce
modular spaces of multipoles. We study their intricate geometry and topology
using methods of algebraic geometry and singularity theory. The multipole
spaces are ramified over vector or projective spaces, and the compliments to
the ramification sets give rise to a rich family of $K(\pi, 1)$-spaces, where
$\pi$ runs over a variety of modified braid groups.
