Various kinds of materials classified as fibers are thin, thread-like, and flexible. A general definition of fibers pays attention to their shape, that is, the least aspect ratio of practical fibers is 100. Another important characteristic of fibers is their flexibility. Some are naturally crimped and others have artificial waves added to them. The randomness of the twisted fiber shapes is difficult to describe, but it is desirable to be able to express the complexity of fiber shapes numerically, like the aspect ratio.
Mandelbrot's concept of fractal geometry is one of the most mathematically convenient when estimating irregular sets. Fractal geometry is characterized by self-similarity (both exact and statistical) and non-integral dimensions. The images of single fibers, multi-pored films, and so on, are regarded as a random fractal set, a kind of statistically self-similar object. Roughly speaking, the fractal dimension is a criterion of the prominence of irregularities in a set.
When Mandelbrot defined fractal, he singled out Hausdorff's dimension, which is defined by covering and can be applied to any set. The advantages of Hausdorff's dimension are mathematical strictness and clarity. On the other hand, the major disadvantages are that it is hard to estimate by computational methods, it is ambiguous in finding the most efficient covering condition, and it is troublesome to take a limit. In many nonintegral covering practical dimensions, the box-counting dimension has been chosen to describe the complexity of the textile products.
The box-counting method for determining the fractal dimensions of crimped fibers, porous polyurethane-blend films, and modified random Koch curves. The box-counting dimension of nylon 6 crimped filament has a distribution of 1.00-1.65, polyurethane-blend films has 1.8660-1.8793,and modified random Koch curves has 1.033-1.310.