By P. Baveye
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Extra info for Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems
The scaling factor is equal to 1/3. The curve I2 consists of 42 = 16 segments each having a length of 1/32 = 1/9, so that the total length of I2 is equal to (4/3)2 = 16/9. At the nth stage, there are 4n segments of length δn = 3−n , with a total length of L(In ) = (4/3)n . This total length L(In ) may be expressed differently, in a way that makes explicit the dependence of L(In ) on δn . g. 5 (p. 2) corresponds to a power-law (or ‘Paretian’) relationship between L and δn . Similarly, the number Nn of segments of length δn is also a power-law function of δn .
Nevertheless, it has astonishing properties, which challenge the traditional concepts of dimension. Indeed, since the von Koch curve is inﬁnite in length and contains exact replicates of itself, any scaled-down sub-image is also of inﬁnite length . This leads to the conclusion that for any two points on the curve, no matter how close they are, the curve between them is of inﬁnite length! 4, the resulting geometrical construct, commonly known as the von Koch ‘island’ or ‘snowﬂake’, has inﬁnite length, even though it ﬁts within a ﬁnite area!
This produces mathematical relationships that will be useful later on. If the initiator is a cube of side length l0 , made of some material of uniform mass density ρ0 , the ﬁrstorder structure (obtained after one iteration) would have a porosity φ1 (void volume divided by total volume) equal to 7/27 and a mass density ρ1 equal to 20ρ0 /27. 55ρ0 ). 4) The second of these two equations again gives rise to a power-law relationship (as in the case of the von Koch curve). 4) will be mentioned again in a later section.
Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems by P. Baveye