Minimum perimeter partitions of the plane into equal numbers of regions of two different areas

Abstract:

We identify the minimum-perimeter periodic tilings of the plane by equal numbers of regions (cells) of areas 1 and λ (minimal tilings), with at most two cells of each area per period and for which all cells of the same area are topologically equivalent. For λ close to 1 the minimal tiling is hexagonal. For smaller values of λ the minimal tilings contain pairs of 5/7, 4/8 and 3/9 cells, the cells with fewer sides having smaller area. The correlation between area fraction and number of sides in the minimal tilings is approximately linear and consistent with Lewis' law.