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 Ai (i=1,..., N), P can be written in the form P=R(SigmaN(i=1) square root Ai)/2, where R is topology dependent. We seek the set of areas Ai 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 2pi/3 angles at vertices (Plateau cells), and ii) infinite periodic bubble clusters.