This chapter discusses the paradox again after exploring the use of fractals to describe rugged fineparticle boundaries. To describe the boundaries similar to those of J. Koch-Weser's triadic island, B. P. Mandelbrot suggested allocating a number between 1 and 2 to the boundary to describe its space-filling ability. Mandelbrot's major contribution to the art of describing a rugged boundary was to demonstrate a technique for allocating a numerical fractal dimension to a natural boundary. Investigations at Laurentian University are seeking to link the fracture ruggedness of a pigmented plastic with the dispersion characteristics of the fineparticles loaded into the plastic. When investigating the structure of a small number of fineparticles the method is straightforward and adequate. As more automated procedures for characterizing fractal boundaries are developed the technique should prove useful in the study of the properties of a major class of fineparticles formed by this type of random process.