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A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail with change in scale.[2]: 1 There are several types of fractal dimension that can be determined theoretically and empirically (see Figure 1).[3][4] The sets that fractal dimensions are used for characterizing come from a broad spectrum ranging from the abstract[5][4] to a host of practical phenomena, including turbulence[2]: 97–104 , river networks: 247–246 , urban growth[6][7], human physiology[8][9], medicine[3], and market trends[10]. The essential idea of "fractional" or "fractal" dimensions has a long history in mathematics that can be traced as far back as the 1600s[2]: 19 [11], but the term itself was brought to the fore by mathematician Benoît Mandelbrot who, in 1975, coined the terms fractal and fractal dimension.[12] [3] [2] [5][13][4][10]
Fractal dimensions were first applied as an index characterizing certain complex geometric forms for which the details seemed more important than the gross picture[12]. To elaborate, for sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points (0-dimensional sets); 1 for sets describing lines (1-dimensional sets having length only); 2 for sets describing surfaces (2-dimensional sets having length and width); and 3 for sets describing volumes (3-dimensional sets having length, width, and height). But this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry[14]. Moreover, unlike topological dimensions, the fractal index can fall between integer values, attesting that a set fills its space qualitatively and quantitatively differently than an ordinary geometrical set does.[4][5][13] For instance, a curve with fractal dimension very near to 1, say 1.10, behaves quite like ordinary lines, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface; in turn, a surface with fractal dimension of 2.1 fills space very much like ordinary surfaces, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume.[14]: 48 [15]
The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so; the two are not strictly correlated[1]. Rather, what a fractal dimension measures is complexity, a concept tied up in certain key features of fractals: self-similarity and detail or irregularity[16]. These features are evident in the exemplary fractal Koch curve illustrated in Figure 2. It is a curve with a topological dimension of 1, so one might hope to be able to measure its length or slope, as with ordinary lines. But we cannot do either of these things, because the fractal curve has complexity in the form of self-similarity and detail that ordinary lines lack but necessarily define fractals.[2] The self-similarity lies in the infinite scaling, and the detail in the defining element of the Koch set. The length between any two points on a Koch curve is infinitely unmeasurable because the curve is a theoretical construct that never stops repeating itself. Every smaller piece of it is composed of an infinite number of scaled segments that look exactly like the first iteration. It is by no means a rectifiable curve, meaning it cannot be measured by being broken down into many segments approximating its length. Thus, we cannot characterize it by finding its length or slope, but we can determine its fractal dimension, which turns out to be 1.2619 (see calculations), and tells us that the Koch curve fills space somewhat more than ordinary lines, but notably less than if it were a surface.