The <i>N</i> -vortex problem on a rotating sphere. II. Heterogeneous Platonic solid equilibria

Abstract

We describe a new method of constructing point vortex equilibria on a sphere made-up of N vortices with different strengths. Such equilibria, called heterogeneous equilibria , are obtained for the five Platonic solid configurations, hence for . The method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant. By symmetries inherent in the Platonic solid configurations, this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure. For the tetrahedron ( N =4) and octahedron ( N =6), the degree of heterogeneity is 4 and 6, respectively, hence we are free to choose each of the vortex strengths independently. For the cube ( N =8), the degree of heterogeneity is 5, for the icosahedron ( N =12) it is 7, while for the dodecahedron ( N =20) it is 4. Thus, the entire set of equilibria based on the Platonic solid configurations is obtained, including substructures associated with each configuration constructed by taking different linear combinations of the basis elements.