When a soft magnetic particle suspension is subjected to a vertical uniaxial magnetic field the particles polarize, the positive dipolar interactions causing particle chain formation. If instead an audio-frequency rotating magnetic field is applied in a horizontal plane, the particles experience an average interaction that to first order is a negative dipolar interaction, causing particle sheet formation in the biaxial field plane. When a vertical uniaxial field and a horizontal biaxial field are simultaneously applied with the field amplitudes balanced, the positive and negative dipolar interactions cancel to first order, and one might expect no dipolar interactions at all. But in this balanced triaxial field an isotropic second-order dipolar interaction of surprising magnitude remains. This triaxial interaction can be attractive or repulsive, and exhibits strong many-body interactions that lead to a variety of unexpected effects, including stable clusters with molecular geometries, the emergence of a particle foam, and the production of vortices in the fluid. By manipulating the triaxial field, a variety of particle structures can be made that cannot be produced by any other known means. (C) 2003 American Institute of Physics.