Por Paulo Teixeira (Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa; e Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Portugal).

The least-perimeter partition of the Euclidean plane into regions all of the same area is the regular honeycomb [1], i.e., a tiling by identical regular hexagons. Much less attention has been devoted to cases where the regions are not all of the same area, or do not cover the entire plane, as they inevitably will in physical realisations of these mathematical objects.
In this spirit I review our work on four different problems: (i) What is the least-perimeter periodic partition of the plane into regions of two different areas [2]? (ii) What is the least-perimeter partition of a finite two-dimensional domain into regions of two different areas? Is it mixed or size-sorted [3]? (iii) What are the least-perimeter periodic two-dimensional chains of (possibly curved) polygons [4]? (iv) How do (i)-(iii)  relate to results for other systems, namely hard discs/spheres and soft colloids [5]?  For all these we drew candidate structures, calculated their perimeters, and minimised them. In some cases this could be done analytically; in others, only numerically. There is absolutely no guarantee that these are all the possible structures, nor do we offer any mathematical proofs whatsoever. These results should therefore be treated as conjectures.

References:
[1] T. C. Hales, The honeycomb conjecture. Discrete Comput. Geom. 25, 1 (2001).
[2] 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, 133 (2001).
[3] P. I. C. Teixeira, F. Graner, and M. A. Fortes, Mixing and sorting of bidisperse two-dimensional bubbles. Eur. Phys. J. E 9, 161 (2002).
[4] P. I. C. Teixeira and M. A. Fortes, Periodic chain clusters of two-dimensional bubbles. J. Phys.: Condens. Matter 14, 5719 (2002).
[5] C. N. Likos and C. L. Henley, Complex alloy phases for binary hard-disc mixtures. Phil. Mag. B 68, 85 (1993).