Pattern Growth: From Smooth Interfaces to Fractal Structures

  • A. Arnéodo
  • Y. Couder
  • G. Grasseau
  • V. Hakim
  • M. Rabaud
Part of the NATO ASI Series book series (NSSB, volume 225)


Fractal structures can be obtained in several growth experiments, the resulting patterns being fractal only in a limited range between a smallest and a largest length scale. In this regard they differ from mathematically defined fractals which cover an infinite range of length scales. In the systems that we will discuss the smallest scale lmin can be linked with the instability process or with the computational technique, the largest scale lmax can be imposed by either geometrical boundary conditions or, in free growth, the overall size reached by the pattern. We can therefore tune the extent of the fractal range and go from compact objects (when the two length scales are close to each other) to structures with a large fractal range (when the two scales are far from each other). Limiting ourselves here to patterns grown in Laplacian fields, we investigate their statistical properties and show that, as the fractal range opens up, some properties of the compact forms are retained by the fractal structure. We use two isotropic Laplacian pattern forming systems: The Saffman Taylor instability1,2 and the formation of clusters in the numerical model of Diffusion Limited Aggregation3 (D.L.A.). These two systems are very different in nature but their ruling laws have strong similarities.


Fractal Structure Fractal Range Large Length Scale Linear Channel Geometrical Boundary Condition 
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Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • A. Arnéodo
    • 1
  • Y. Couder
    • 2
  • G. Grasseau
    • 1
  • V. Hakim
    • 2
  • M. Rabaud
    • 2
  1. 1.Centre de Recherche Paul PascalPessacFrance
  2. 2.Laboratoire de Physique StatistiqueParis Cedex 05France

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