T. Aste et al., FROM ONE-CELL TO THE WHOLE FROTH - A DYNAMICAL MAP, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 53(6), 1996, pp. 6181-6191
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.