The unusual magnetism exhibited by dysprosium triangles [Dy-3(mu(3)-OH)(2)L3Cl2(H2O)(4)]-[Dy-3(mu(3)-OH)(2)L3Cl(H2O)(5)]Cl-5 center dot 19H(2)O is explained using the recently developed ab initio methodology for the simulation of magnetic properties of complexes. The local anisotropy axes on the dysprosium sites are found to lie in the plane of the Dy-3 triangle and to make angles of ca. 120 degrees with each other. The small antiferromagnetic exchange interaction between sites leads to a non-magnetic Kramers doublet in the ground state of the complex. The arrangement of the local magnetization vectors in this state is close to toroidal. By contrast, the lowest excited states are characterized by a huge magnetic moments of ca. 20 mu(B) and show very different behavior of magnetization for. fields applied along and perpendicular to the plane of the Dy-3 triangle.