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Article Dans Une Revue Journal de Physique I Année : 1995

Topologiacl Models of 2D Fractal Cellular Structures

Gérard Le Caër
Renaud Delannay

Résumé

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 ≥3 and P(3) ≠0 evolve regularly when topological disorder, conveniently measured by the variance μ2 of P(n), increases. The strong correlations which exist among cells, in particular in natural structures $(\mu_{2}\lesssim 5)$, decrease progressively when μ2 increases, a cell repartition close to a random one being reached for μ2∼12. We argue that the structures finally evolve to fractal structures (for which μ2 is infinite) but we have not characterized the latter transition.

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jpa-00247147 , version 1 (04-02-2008)

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Gérard Le Caër, Renaud Delannay. Topologiacl Models of 2D Fractal Cellular Structures. Journal de Physique I, 1995, 5 (11), pp.1417-1429. ⟨10.1051/jp1:1995207⟩. ⟨jpa-00247147⟩

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