TOPOLOGICAL MODELS OF 2D FRACTAL CELLULAR STRUCTURES

In space-filling 2D cellular structures with trivalent vertices and in which each cell is constrained to share at most one side with any cell and no side with itself, the maximum fraction of three-sided cells is produced by a decoration of vertices of any initial structure by three-sided cells. Fractal cellular structures are obtained if the latter decoration process is iterated indefinitely. Other methods of constructions of fractal structures are also described. The probability distribution P(n) of the number n of cell sides and some two-cell topological properties of a 2D fractal cellular structure constructed from the triangular Sierpinski gasket are investigated. On the whole, the repartition of cells in 2D structures with n greater than or equal to 3 and P(3) not equal 0 evolve regularly when topological disorder, conveniently measured by the variance mu(2) of P(n), increases. The strong correlations which exist among cells, in particular in natural structures (mu(2) less than or similar to 5), decrease progressively when mu(2) increases, a cell repartition close to a random one being reached for mu(2) similar to 12. We argue that the structures finally evolve to fractal structures (for which mu(2) is infinite) but we have not characterized the latter transition.