Abstract:

         We aim to clarify several fundamental terms used in fractal  analysis and examine how the estimates of the fractal dimension  can be made clearer to best serve as descriptive indices. The problem is essentially one of clarifying the semantics of the term `fractal', since the syntax of calling something `fractal' is often used with little regard to the principles underlying scaling theory. Towards this aim we discuss the use of language and the necessity to establish a linguistic base that serves as the template for communication across different disciplines.

1 Introduction

            Our position deals with the use and understanding of language  from a pragmatic standpoint. That is, we discuss the misconception of some vital definitions and terms in the field of fractal analysis. The paper is not about how we come to know but rather how we should communicate in order to gain knowledge and understanding. We offer practical solutions by identifying several terms associated with fractal analysis that need to be clarified. These terms, such as `fractal' have been socialised and come to mean something in the literature that may be misleading and from an applied perspective not useful. Further, fractal analysis procedures such as the box-counting  method and variations in sampling and preparing images for analysis  can have non-trivial effects on the estimation and interpretation of D[3, 8, 12, 16, 19].

          Applications of fractal analyses have been extensively used in diverse scientific, sociological and philosophical areas of research. Despite this large volume of literature, there is still a lack of clarity regarding the meaning of the terms used this field and therefore inferences drawn from the results are questionable[3, 18, 12, 19].

            The concept of `fractal scaling' is revolutionary but not new in that it offers a means to describe how things `are' in terms of the object's scaling characteristics[11, 21]. What we mean here is, that D quantifies `something' in terms of shape, texture, size, number, colour, repetitiveness, similarity, randomness, regularity, heterogeneity, or any other adjectival descriptions that are commonly used to define the properties of some object or event. These descriptions can not be quantified using Euclidean geometry that idealises form. The properties mentioned above reflect the complexity of the object or event. Fractal analysis has provided a means of quantifying these properties as a measure of complexity  or scale-dependency of the pattern. This is not to say that fractal analysis is the only means of quantifying complexity. Other analytical methods include Fourier analysis,  fractal harmonics, polygonal harmonics and wavelets [13, 26]. In principal, fractal analysis should improve on the description of morphological features compared to conventional shape parameters. Common examples from the literature include heart rate irregularities, grazing effects on pastoral lands or stock market fluctuations[2, 16, 17].

           It is our observation however, that conclusions drawn from fractal research remain at best tentative -- with some research areas offering more conclusive results[1, 4]. The lack of conclusive results can, in many cases, be explained by the apparent lack of a linguistic base, that is a sound description of fractal theory and its relationship to the associated analysis procedures. Fractal research and discussion is characterised by the repetition of definitions and procedures that were intended to be vague[17]. Fractal analysis measures length as a process and therefore is defined as a limit which allows the image (if fractal) to be analysed in arbitrarily high resolution[22]. In practice any object is represented by a finite data set and the measurement is restricted to a finite magnification range. The image can be interpreted by D if it is assumed that the finite data set of the image reflects a fractal set and is self-similar[20]. If it is assumed that the image does not reflect an ideal fractal in a statistical sense (this is the case for biological images), than interpreting the image using D is meaningless. The fractal dimension may still be useful though by using it as a quantitative parameter that indicates complexity or the scale dependence of a pattern[14]. This fundamental concept is not made clear in the literature where D is taken to indicate fractality[19]. Communication becomes meaningful if all involved understand the terms. That is a transparent linguistic base exists.

2   Communication

               A fractal set is a set in metric space for which the Hausdorff-Besicovitch dimension  D > topological dimension

.[17, p361,] The above quote is the most often quoted sentence found in journal articles to describe fractals, even though Mandelbrot states, on the next page, that this definition is rigorous, but also tentative. The definition is an example of communication that requires linguistic literacy as it requires an understanding of what is meant by metric space, Hausdorff dimension and topological dimension. It does not help in the understanding of fractal theory nor how this relates to fractal analysis. It would be more appropriate to point out that the definition applies to theoretical fractals and may be totally useless in practice if the image to be analysed does not reflect, albeit even statistically, a fractal set. In practice, the estimate of D quantifies scale invariance over a limited scaling range and does not indicate whether the image is fractal or not. We suggest that communication occurs at different levels and that the literature can be divided into three main categories.

1. Theoretical mathematical research (those in the know who already posses a solid knowledge base in fractal and scaling theory). That is, those to whom the above definition by Mandelbrot means something quite concrete.

2. Applied science research such as the biological and social sciences (those with a solid knowledge base in other disciplines and using fractals).

3. Methodological analysts (those that develop tools for fractal analyses and aim at describing these clearly).

Once this division of the scientific/professional community is established it becomes obvious that a type of professional socialisation has taken place. Specific language used by each group establishes an identity within this group and marks it off as a specialist domain of knowledge and expertise [5]. To apply fractal analysis successfully it is important to obtain fractal literacy. For language to be a resource[10], with a potential to create meaning, it is important that novices are able to obtain the appropriate linguistic skills. Procedures used to determine the fractal dimension of images need to be made explicit. Some aspects of fractal analyses are like a black box in that one obtains a program, loads the image and obtains a fractal dimension. For the fractal dimension to be meaningful, the black box needs to be opened.

3   Conclusion

                 Language can be viewed as a resource for creating meaning[9, 10]. 

What a novice needs to do is construct a linguistic system that enables them to participate in the scientific debate. This is achieved, in our case, by helping novices to appropriate the fractal linguistic system. Specifically, knowledge needs to be encoded in a language that is understandable when transmitted from a specialist source. It should not be necessary for the learner to first decode language in order to learn anything from it. Our paper represents a start in this direction. Information must be transmitted in many different ways to incorporate different learners. Learners on the other hand must be able to freely communicate their thoughts as they attain the specific linguistic literacy.

        We explored the meaning of the descriptor `fractal' and one of its characteristics -- `similarity'. Our aim was to demonstrate that an understanding of how language is used in this specialised field reflects on to the application and conclusions drawn from the research. In this respect, we discussed some of the considerations necessary when choosing an algorithm that determines the complexity of an image.

4   Acknowledgments

       MDW would like to acknowledge the support of the National Science Foundation, Environmental and Ocean Systems Program (Grant No. BES-9312825) and of the principle investigator, Dr R.I. Dick. The work cited here was part of MDW's Masters research. 

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