  It is argued that spectral features of quantal systems with random
interactions can be given a geometric interpretation. This conjecture is
investigated in the context of two simple models: a system of randomly
interacting d bosons and one of randomly interacting fermions in a j=7/2 shell.
In both examples the probability for a given state to become the ground state
is shown to be related to a geometric property of a polygon or polyhedron which
is entirely determined by particle number, shell size, and symmetry character
of the states. Extensions to more general situations are discussed.
