Fractal Dimension

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Contributed by Daniel Daniel

Introduction

A fractal dimension is a statistical quantity that describes how a fractal appears to fill space. A fractal is an object that displays a property known as self-similarity, i.e. a geometric shape that can be reduced to smaller parts, with each smaller part being a reduced copy of the whole. There are several specific definitions of fractal dimensions, but the most important ones include Renyi dimensions and Haussdorf dimensions.

One possible working definition of the fractal dimension D is

<math>D = \lim_{l \rightarrow 0} \frac{\log N(l)}{\log\frac{1}{l}}</math>

Eq(1)

where N(l) is the number of self-similar structures of linear size l required to cover the original object.

Examples of fractals

Fractaldimensions1.png Figure 1. Koch snowflake

There are many examples of fractals in nature. A koch snowflake is an idealization (and a good one) of an actual snowflake. Figure 1 shows the first four iterations of a Koch snowflake. If we continue with the iterations infinitely, we will have a Koch snowflake and the length of curve between any two points is infinite. The fractal dimension of a fractal line can be understood intuitively to describe an object that is too big to be a one-dimensional object, but too thin to be a two-dimensional object.

Fractaldimension2.jpg

Figure 2. The coastline of the United Kingdom as measured with measuring rods of 200 km, 100 km and 50 km in length. The resulting coastline is about 2350 km, 2775 km and 3425 km; the shorter the scale, the longer the measured length of the coast.

Another example of a fractal is the coastline of a country. The length of a coastline can be measured more and more accurately by using a series of shorter and shorter measuring rods. For a rectifiable curve, such as a circle, this procedure will converge to an actual perimeter as we get to shorter and shorter measuring rods, but in the case of a fractal structure like the coast line, there is no convergence. This is illustrated in figure 2. It was found empirically if L is the measured length of the coast-line and l is the length of the measuring rod, the relationship between L and l is given by <math>L =M l^{(1-D)} </math>, where M is some positive constant and D is the fractal dimension. It is trivial to show that this definition of D is consistent with equation (1).

A non-exhaustive list of fractals and their fractal dimension can be found at http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Fractals and Polymers

A polymer is an example of a fractal structure. A Gaussian chain has an end-to-end distance given by <math>R^2 = N l^2</math>, giving a fractal dimension of 2 using the definition in eq. (1) whatever dimension of space it is occupying. Flory has shown that a polymer can have D < 2 in particular when self-avoiding walk is accounted for, in which case D = 1.66. A collapsed polymer has D=3 and fills space completely.

Keyword in references:

G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907