The structure of froths through a dynamical map

The shell map is a very simple representation of the structure of foams, combining the geometrical (random tiling) and dynamical (loss of information from an arbitrary cell out) aspects of disorder. We will illustrate it and give several examples, including a few arising from discussions in Cargese. This chapter is written by following the main lines of two previously published papers [1, 2]. In Nature, space-filling disordered patterns and cellular structures are widespread [3, 4]. These structures (froths) are partitions of Ddimensional space by convex cells. Disorder imposes that each vertex has minimal number of incident edges, faces and cells (D + 1 edges incident on a vertex, D faces incident on a edge, D − 1 cells incident on a face, in D-dimensions, Fig.1). In this respect, a froth is a regular graph, but the number of edges bounding each face, the number of faces bounding a polyhedral cell, etc., are random variables [5]. Minimal incidences implies also that the topological dual of a froth is a triangulation (Fig.1), a useful representation of packings. Indeed, for any given packing, or point set, one can construct the Voronöı tessellation [6], which is a space-filling assembly of polyhedral cells. When the starting points are disordered (no special symmetries) the Voronöı tessellation is a froth. The space filled by the froth can be curved (Fig.1). This is for instance the case in amphiphilic membranes [7], fullerenes, the basal layer of the epidermis of mammals [8] or the ideal structure of amorphous materials [9, 10]. Disorder does not necessarily imply inhomogeneity. On the contrary, in many cases, disordered froths are very homogeneous (cells with very similar sizes and regular shapes). This is -for instancethe case in the epidermis, where the biological ∗Published in in Foams and Emulsions, eds. J. F. Sadoc and N. Rivier, (Kluwer Academic Publisher, Netherlands 1999) pag.497510.