We investigate constructions made from magnetic spheres. We give heuristic rules for making stable constructions of polyhedra and planar tilings from loops and saddles of magnetic spheres, and give a theoretical restriction on possible configurations, derived from the Poincaré-Hopf theorem. Based on our heuristic rules, we build relatively stable new planar tilings, and, with the aid of a 3D printed scaffold, a construction of the buckyball. From our restriction, we argue that the dodecahedron is probably impossible to construct. We finish with a simplified physical model, within which we show that a hexagonal loop is in static equilibrium.