206.

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		1.4.7— Example of Determining the Fractal Dimension: Using the Scaling Relationship

		The fractal dimension can be found from the scaling relationship. This is the most often used method to determine the fractal dimension from experimental data.

		The fractal dimension d means that the number of pieces N(r) measured at resolution r is proportional to r-d. The scaling relationship evaluated from the experimental data tells us that the value of a property measured at resolution r is proportional to rb. If we know how the property depends on the number of pieces and the size of each piece, we can determine the fractal dimension d from the exponent b in the scaling relationship.

		For example, Richardson measured the length of the coastline of Britain by laying line segments of length r end to end along the coastline. The length r of the line segments set the resolution used to make the measurement. The total length of the coastline is equal to the number N(r) of these line segments multiplied by the length r of each one. That is, . The definition of the fractal dimension is that the number of line segments N(r) is proportional to r-d. Thus the total length of the coastline is proportional to r1-d. Richardson repeated the measurement using line segments of different size r. In this way he determined the scaling relationship that the total length of the coastline was proportional to r-.25. Equating the exponent of the scaling relationship to the exponent determined from the properties of the dimension, we find -.25 = 1 - d. Thus the fractal dimension d of the length of the coastline is equal to 1.25.

 

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