Packing of twinned circles on a sphere

Abstract

The twinned-circle problem is to pack 2 N non-overlapping equal circles forming N pairs of twins (rigidly connected neighbours) on a sphere so that the angular radius of the circles will be as large as possible. In the case that the contact graph(s) of the unconstrained circle packing support(s) at least one perfect matching, a complete solution to the twinned circles problem is found, with the same angular radius as the unconstrained problem. Solutions for N =2–12 pairs of twins are counted and classified by symmetry. For N =2–6 and 12, these are mathematically proven to be the best solutions; for N =7–11, they are based on the best known conjectured solutions of the unconstrained problem. Where the contact graph of the unconstrained problem has one or more rattling circles, the twinned problem is most easily solved by finding perfect matchings of an augmented graph in which each rattling circle is supposed to be simultaneously in contact with all its contactable neighbours. The underlying contact graphs for the unconstrained packings for N =2–12 are all Hamiltonian, guaranteeing the existence of perfect matchings, but Hamiltonicity is not a necessary condition: the first solution to the twins problem based on an example of a non-Hamiltonian contact graph occurs at N =16.