FROM ONE-CELL TO THE WHOLE FROTH - A DYNAMICAL MAP

We investigate two- and three-dimensional shell-structured-inflatable froths, which can be constructed by a recursion procedure adding succe ssive 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 restric ted set of local configurations for which the recursive inflation tran sformation is not applicable. These configurations are inclusions betw een successive layers and can be treated as vertices and edges decorat ions of a shell-structured-inflatable skeleton. The recursion procedur e is described by a logistic map, which provides a natural classificat ion into Euclidean, hyperbolic, and elliptic froths. Froths tiling man ifolds with different curvatures can be classified simply by distingui shing between those with a bounded or unbounded number of elements per shell, without any a priori knowledge on their curvature. A result, a ssociated with maximal orientational entropy, is obtained on topologic al properties of natural cellular systems. The topological characteris tics of all experimentally known tetrahedrally close-packed structures are retrieved.