For bidisperse packings a stronger question whether area fraction increases as the area ratio increases from 0 to 1. There is an analogous question for bidisperse partitions, namely whether the perimeter increases as the area ratio increases from 0 to 1 (fixing the sums of the areas). Work of Fortes and Teixeira [FT] indicates that it does, although the perimeter function graphed in their paper should be divided by sqrt(1+lambda) for normalization to fixed sum of areas. If the 6_1 6_1 structure is replaced by disjoint phases of hexagons then the perimeter looks strictly increasing.

[FT] M. A. Fortes and P. I. C. Teixeira, Minimum perimeter partitions of the plane into equal numbers of regions of two different areas, Eur. Phys. J. E 6 (2001), 133–137.

For bidisperse packings a stronger question whether area fraction increases as the area ratio increases from 0 to 1. There is an analogous question for bidisperse partitions, namely whether the perimeter increases as the area ratio increases from 0 to 1 (fixing the sums of the areas). Work of Fortes and Teixeira [FT] indicates that it does, although the perimeter function graphed in their paper should be divided by sqrt(1+lambda) for normalization to fixed sum of areas. If the 6_1 6_1 structure is replaced by disjoint phases of hexagons then the perimeter looks strictly increasing.

ReplyDelete[FT] M. A. Fortes and P. I. C. Teixeira, Minimum perimeter partitions of the plane into equal numbers of regions of two different areas, Eur. Phys. J. E 6 (2001), 133–137.