 
Summary: THE SHELL MAP *
The structure of froths through a dynamical map
TOMASO ASTE
Laboratoire de Dynamique des Fluides Complexes,
Universit´e Louis Pasteur Strasbourg, 67084 France.
tomaso@ldfc.ustrasbg.fr
1. Introduction
The shell map is a very simple representation of the structure of foams,
combining the geometrical (random tiling) and dynamical (loss of infor
mation 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, spacefilling disordered patterns and cellular structures
are widespread [3, 4]. These structures (froths) are partitions of D
dimensional 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 Ddimensions, Fig.1). In this respect, a froth is a regular
graph, but the number of edges bounding each face, the number of
