Self-assembled clusters of magnetically tilted dipoles

Motivated by the idea of using simple macroscopic examples to illustrate the physics of complex systems, we modify a historic experimental setup in which interacting floating magnets spontaneously self-assemble into ordered clusters. By making the cluster components mechanically stable against their natural tendency to flip and coalesce, we can monitor the torque experienced by individual magnets through the macroscopic tilt angles they acquire when exposed to noncollinear external fields. A mathematical model that reproduces the empirical observations is introduced, enabling us to go beyond the experimental cases considered. The model confirms the existence of alternative orderings as the number of objects increases. Furthermore, a simpler and more mathematically transparent version of our model enables us to establish the conditions under which the cluster structure is stable and when it will collapse. We argue that our simple experimental setup combined with the accompanying models may be useful to describe general features seen in systems composed of mutually repulsive particles in the presence of modulating confining fields.