Least-perimeter partition of the disc into $N$ regions of two different areas

Abstract

We present conjectured candidates for the least perimeter partition of a disc into $N \le 10$ regions which take one of two possible areas. We assume that the optimal partition is connected, and therefore enumerate all three-connected simple cubic graphs for each $N$. Candidate structures are obtained by assigning different areas to the regions: for even $N$ there are $N/2$ regions of one area and $N/2$ regions of the other, and for odd $N$ we consider both cases, i.e. where the extra region takes either the larger or the smaller area. The perimeter of each candidate is found numerically for a few representative area ratios, and then the data is interpolated to give the conjectured least perimeter candidate for all possible area ratios. At larger $N$ we find that these candidates are best for a more limited range of the area ratio.