 
Summary: From one cell to the whole froth: A dynamical map
T. Aste,* D. Boose´, and N. Rivier
Laboratoire de Physique The´orique, Universite´ Louis Pasteur, F 67084 Strasbourg Cedex, France
Received 24 July 1995; revised manuscript received 14 December 1995
We investigate two and threedimensional shellstructuredinflatable froths, which can be constructed by a
recursion procedure adding successive layers of cells around a germ cell. We prove that any froth can be
reduced into a system of concentric shells. There is only a restricted set of local configurations for which the
recursive inflation transformation is not applicable. These configurations are inclusions between successive
layers and can be treated as vertices and edges decorations of a shellstructuredinflatable skeleton. The
recursion procedure is described by a logistic map, which provides a natural classification into Euclidean,
hyperbolic, and elliptic froths. Froths tiling manifolds with different curvatures can be classified simply by
distinguishing between those with a bounded or unbounded number of elements per shell, without any a priori
knowledge on their curvature. A result, associated with maximal orientational entropy, is obtained on topo
logical properties of natural cellular systems. The topological characteristics of all experimentally known
tetrahedrally closepacked structures are retrieved.
PACS number s : 82.40.Ck, 82.70.Rr
I. INTRODUCTION
A froth is a topologically stable division of space by
cells, which are convex polytopes polygons in two
dimensions 2D , polyhedra in three dimensions 3D of
