  There is a very natural map from the configuration space of n distinct points
in Euclidean 3-space into the flag manifold U(n)/U(1)^n, which is compatible
with the action of the symmetric group. The map is well-defined for all
configurations of points provided a certain conjecture holds, for which we
provide numerical evidence. We propose some additional conjectures, which imply
the first, and test these numerically. Motivated by the above map, we define a
geometrical multi-particle energy function and compute the energy minimizing
configurations for up to 32 particles. These configurations comprise the
vertices of polyhedral structures which are dual to those found in a number of
complicated physical theories, such as Skyrmions and fullerenes. Comparisons
with 2-particle and 3-particle energy functions are made. The planar
restriction and the generalization to hyperbolic 3-space are also investigated.
