Abstract
Branching patterns are ubiquitous in nature; consequently, over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties; however, their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log–log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions.
Highlights
We propose that results such as these should be treated with skepticism because the methods used to estimate dimension rely on an assumption of self-similarity which branching structures do not satisfy
Some more recent criticisms have emerged, including [12], in which the authors discuss some of the difficulties in applying fractal methods to ecological data, concluding that evidence of a scaling relationship which spans only a few orders of magnitude is not sufficient evidence for true fractality, and [7], which cautions against computing fractal dimensions of root systems without rigorously testing for self-similarity, or statistical self-similarity, first
As we have discussed in this paper, estimating the fractal dimension of naturally occurring branching structures is a much more complex topic than previous works have suggested
Summary
Ever since Benoît Mandelbrot first coined the term fractal in 1975 [1], mathematicians have struggled to reach a consensus as to the formal definition of a fractal. Demonstrated that random distributions exhibit apparent fractal behaviour over a range of scales consistent with the typical range observed in experimental measurements of fractal objects These early words of warning were not generally heeded, and studies on the estimation of the fractal dimensions of natural objects have continued to appear. Amidst these studies, some more recent criticisms have emerged, including [12], in which the authors discuss some of the difficulties in applying fractal methods to ecological data, concluding that evidence of a scaling relationship which spans only a few orders of magnitude is not sufficient evidence for true fractality, and [7], which cautions against computing fractal dimensions of root systems without rigorously testing for self-similarity, or statistical self-similarity, first. It should be noted here that these comments extend to “multifractal” methods as well, since the determination of the famous f (α) curve is dependent on least-squares fitting of box-counting data and thereby suffers from the same problems we will discuss in this paper
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