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 that are dual to those found in a number of complicated physical theories, such as Skyrmions and fullerenes. Comparisons with 2– and 3–particle energy functions are made. The planar restriction and the generalization to hyperbolic 3–space are also investigated.