Lower bounds for the surface energy of two-dimensional foams

Amongst the two-dimensional cellular patterns that fill a plane, dry foams at stable equilibrium typify a particular subset for which the total perimeter $P$ of cell boundaries (i.e., films between bubbles) has a local minimum. For a given set of bubble areas $A_i$ ($i=1,\ldots,N$), $P$ can be written in the form $P=R(\sum_{i=1}^N\sqrt{A_i})/2$, where $R$ is topology dependent. We seek the set of areas $A_i$ and the cluster topology that minimise $R$, and propose lower bounds for $R$ that set lower bounds for the surface energy of i) individual bubbles, with circular edges meeting at $2\pi/3$ angles at vertices (Plateau cells), and ii) infinite periodic bubble clusters