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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

paulo@ist.utl.pt

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

Abstract
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.

PACS
83.80.Iz - Emulsions and foams.


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