By P. Baveye
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Additional resources for Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems
Following in Cantor’s steps, another assault on the concept of dimension was made simultaneously by Peano  and Hilbert , 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.
Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems by P. Baveye