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arXiv · 1901.00319

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

Francis Headley·Simon Cox
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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.

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Francis Headley, Simon Cox. 2019-01-02. Least-perimeter partition of the disc into $N$ regions of two different areas. https://doi.org/10.1140/epje%2Fi2019-11857-0

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