P. Baveye's Biophysical Chemistry of Fractal Structures and Processes in PDF

By P. Baveye

ISBN-10: 0470511206

ISBN-13: 9780470511206

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Example text

Following in Cantor’s steps, another assault on the concept of dimension was made simultaneously by Peano [12] and Hilbert [13], in two short but influential articles. Both describe polygons that appear at first glance to be perfectly innocent, but nevertheless happen to fill a square more and more completely, so that, in the limit, they pass through every single point in the square. 3). 3. 3 First stages of the construction of Peano’s space-filling curve. At each step, each line segment is replaced by nine line segments scaled down by a factor of 1/3.

For these, other easily measurable dimensions are necessary, like the box-counting dimension. 14), on the route to defining the Hausdorff dimension, the covering sets Ui were allowed to have various diameters, within the constraint |Ui | ≤ δ. 14). 19) i In other words, Nδ (F ) is the smallest number of sets of diameter at most δ that can cover F . 20) The motivation for this (apparently arbitrary) definition is that, for many sets of points, there is a nearly linear relationship between ln Nδ (F ) and ln δ for small δ, and, as δ → 0, the slope of this relationship approaches DBC (F ).

In the next step, one divides the initiator in nine smaller squares of side length l0 /3 and removes the central square. g. 10 (p. 25)], which is Cantor-like in many respects, even though its initiator is two-dimensional. Like the Cantor set, the Sierpinski carpet is a self-similar, uncountable set with a topological dimension equal to zero. e. practically, its area in the plane, is equal to zero, so the traditional concept of area does not provide a very useful description of the spatial coverage of the Sierpinski carpet.

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Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems by P. Baveye

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