The use of the perimeter-area method to calculate the fractal dimension of aggregates

Abstract

Complicated geometrical objects like aggregate clusters can be characterised by a simple parameter: the fractal dimension. There are many ways to measure this fractal dimension. For most methods, it is found by the exponent of a relationship between an intrinsic value, such as mass or area, to a characteristic value, such as bulk aggregate length. The perimeter-area method was derived by Mandelbrot to measure the fractal dimension of chips of ore. In this method, there is no distinction of whether the perimeter and area values are characteristic or intrinsic.Various methods of measuring the fractal dimension are used on different known fractal objects to demonstrate these issues in an idealised setting.We show that while the distinction is not important for Mandelbrot's ore chips, which are island-type objects, it is very important in the analysis of cluster-type objects. The perimeter-area method is a valid tool in the fractal characterisation of aggregates and clusters, however, researchers must be careful to take the appropriate intrinsic and characteristic measurements.