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 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 shell-structured-inflatable 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 topological properties of natural cellular systems. The topological characteristics of all experimentally known tetrahedrally close-packed structures are retrieved.