| | | | | | 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. | | | | | | | | | 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. | | | | | | | | | 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). | | | | |