Abstract

The minimal perimeter enclosing N planar regions, each being simply connected and of the same area, is an open problem, solved only for a few values of N . The problems of how to construct the configuration with the smallest possible perimeter E and how to estimate the value of E are considered. Defect-free configurations are classified and we start with the naïve approximation that the configuration is close to a circular portion of a honeycomb lattice. Numerical simulations and analysis that show excellent agreement to within one free parameter are presented; this significantly extends the range of values of N for which good candidates for the minimal perimeter have been found. We provide some intuitive insight into this problem in the hope that it will help the improvement in future numerical simulations and the derivation of exact results