Saturday, December 25, 2010
Marcel Minnaert does not write about fractal shapes. In his three-volume encyclopedia of observations in nature he never mentions anything that comes even near.
He wrote his books in the 1940's. Could it be that the concept of "fractal" was still so remote that even this master observer didn't notice it?
My personal theory of this phenomenon:
- swamp gas bubbles rise to the surface, keeping the water in motion and preventing freezing
- the water current created by the bubbles tries to raiate outward from the hole
- it searches for the easiest path, and thus also creating and stabilizing the path.
The same process is seen in lightning and the creation af a river delta (I think).
Looking this up in The Self-Made Tapestry: Pattern Formation in Nature by Philip Ball I'm led to "dendritic patterns" and "lichtenberg figures" - a force forms a path through a medium that resists the force.
From a review of this book (Reuben Rudman):
In the chapter on Branches we find detailed descriptions of dendritic growth of crystals, formation of snowflakes, tree-branch growth, and bacterial colony formation presented in terms of diffusion-limited aggregation (DLA) and fractal geometry. DLA describes systems in which the “rate of growth is governed by the rate of diffusion of particles. It differs from the way regular, faceted crystals grow, in that there is no opportunity for the impinging particles to rearrange themselves so that they pack together most efficiently. Since this takes place at the surface of the growing crystal, it soon becomes jagged and disorderly.”
The details of the final shape are a function of the kinetics of crystal growth, where the branched clusters are nonequilibrium structures. The fractal dimension is a measure of how densely packed the branches are and is a property that is precise, reproducible, and characteristic of the apparently irregular, branched objects. We also learn that the algorithms used to describe the shape of a tree are much more complex than one would imagine; several examples are described.
In Breakdowns, which in some ways is a continuation of the previous chapter, glass fracture, electrical discharges, earthquake-induced crustal fractures, and landscape evolution are analyzed in terms of fractal geometry. The pattern developed as a river forms from its converging streams is described best by a fractal scaling law that is characteristic of most branched networks including cracks and DLA clusters.