2019/01/02 by Headley, Francis, Cox, Simon
#FOS: Physical sciences #Soft Condensed Matter (cond-mat.soft)
paper · doi:10.48550/arxiv.1901.00319
We present conjectured candidates for the least perimeter partition of a disc into N ≤ 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.