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Soft Matter and Biological Physics

Eur. Phys. J. E 6, 133-137 (2001)

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

M.A. Fortes and P.I.C. Teixeira

Departamento de Engenharia de Materiais e Instituto de Ciência e Engenharia de Materiais e Superfícies, Instituto Superior Técnico, Avenida Rovisco Pais, P-1049-001 Lisboa, Portugal


(Received 27 June 2001 and Received in final form 29 August 2001)

We identify the minimum-perimeter periodic tilings of the plane by equal numbers of regions (cells) of areas 1 and $\lambda $ (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 $\lambda $ close to 1 the minimal tiling is hexagonal. For smaller values of $\lambda $ 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.

83.80.Iz - Emulsions and foams.

© EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2001