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Page 28

		1.3.1— Self-Similarity Implies a Scaling Relationship

		Smaller pieces of a fractal will be seen at finer resolution. A measurement made at finer resolution will include more of these smaller pieces. Thus the value measured of a property, such as length, surface, or volume, will depend on the resolution used to make the measurement. How a measured property depends on the resolution used to make the measurement is called the scaling relationship.

		Self-similarity specifies how the small pieces are related to the large pieces. Thus self-similarity determines the scaling relationship.

		The mathematical form of self-similarity determines the mathematical form of the scaling relationship. The mathematical form of self-similarity is that the value of a property measured at resolution ar is proportional to the value measured at resolution r. That is, , where k is a constant. From this form, it can be shown that the scaling relationship has one of two possible forms.

		1— Power Law

		The simplest form of the scaling relationship is that the measured value of a property depends on the resolution used to make the measurement with the equation: . In this equation B and b are constants. This form is called a power law.

		2— Full Form

		The full form of the scaling relationship is the equation:

		where B, b, and a are constants and f(x) is a periodic function such that f(1+x)=f(x).

 

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