  It is well known that the toroidal dimensional reduction of supergravities
gives rise in three dimensions to theories whose bosonic sectors are described
purely in terms of scalar degrees of freedom, which parameterise sigma-model
coset spaces. For example, the reduction of eleven-dimensional supergravity
gives rise to an E_8/SO(16) coset Lagrangian. In this paper, we dispense with
the restrictions of supersymmetry, and study all the three-dimensional scalar
sigma models G/H where G is a maximally-non-compact simple group, with H its
maximal compact subgroup, and find the highest dimensions from which they can
be obtained by Kaluza-Klein reduction. A magic triangle emerges with a duality
between rank and dimension. Interesting also are the cases of Hermitean
symmetric spaces and quaternionic spaces.
