11/21/2023 0 Comments Fractal dimensionIn light of this apparent fundamental indeterminacy, Mandelbrot posits that familiar geometrical metrics such as length are inadequate for describing the complexity found in nature. This quality reflects the fact that the outline of the British coastline is an example of a “self-similar” structure-that is, a structure that exhibits the same statistical qualities, or even the exact details, across a wide range of length scales. Indeed, as the precision of such measurements increases-that is, by decreasing the length of the “ruler” used to trace the profile-the measured total length appears to increase as well. As Richardson and Mandelbrot note, such a hypothesis is difficult to evaluate, since individual records of the length of Britain’s west coast varied by up to a factor of three. Richardson, a pacifist and mathematician, sought to investigate the hypothesis that the likelihood that war would erupt between a pair of neighboring nations is related to the length of the nations’ shared border. In fact, however, the curious nature of coastline measurements had been discussed by Lewis Fry Richardson 6 years prior in the General Systems Yearbook. Mandelbrot often is credited with introducing the notion of a fractional, or fractal, dimension in his 1967 paper, “How long is the coast of Britain?”. Additionally, we investigate the inherent challenges in quantifying fractal characteristics (and indeed of verifying the presence of such fractal characteristics) in time-series traces modeled to resemble physical data sets. We investigate the fidelity of such techniques by applying each technique to sets of computer-generated time-series data sets with well-defined fractal characteristics. In this chapter, we introduce and investigate a variety of fractal analysis techniques directed to time-series structures. However, such spatial analyses generally are not well-suited for the analysis of so-called “time-series” fractals, which may exhibit exact or statistical self-affinity but which inherently lack well-defined spatial characteristics. As a traditional example, a fractal dimension of a spatial fractal structure may be quantified via a box-counting fractal analysis that probes a manner in which the structure fills space. Next, we'll determine the dimension of the Sierpinski Triangle.Many methods exist for quantifying the fractal characteristics of a structure via a fractal dimension. And this describes the Koch Curve - it's wigglier than a straight line, but it doesn't fill up a whole 2-Dimensional plane either.Īs we'll see soon, the more of a plane that a fractal covers the closer its dimensions is to 2. If a line is 1-Dimensional, and a plane is 2-Dimensional, thenĪ fractional dimension of 1.26 falls somewhere in between a line and a plane. We can make some sense out of the dimension, by comparing it to the simple, whole number dimensions. In fact, all fractals have dimensions that are fractions, not whole numbers. What could a fractional dimension mean?įractional dimensions are very useful for describing fractal shapes. We're used to dimensions that are whole numbers, 1,2 or 3. Use a calculator (or Google) to find the value for log(4): So according to the formula D = log(N) / log(r), we can say that D = log(4) / log(3) = 1.26 Order 4 has four times as many pieces as order 3, and each piece is 1/3 the scale. In this case, we can see that the number of pieces in the generator, N, is 4, and the magnification factor is 3, because each section of the generator is 1/3 of the unit length.This same relationship holdsīetween each of the orders of the curve. Remembering that D = log(N) / log(r), we can calculate the dimension D by seeing how the number of units, N, changes with the magnification factor, r. The third order curve follows the same pattern, and it has 64 tiny segments, each of which is 1/27 of the unit length, making a total length of 64/27.Īs the progression continues, the curve gets longer and longer, and eventually becomes infinitely long! Now, it is not very useful to know that a curve is infinitely long,Īnd this is where the concept of Fractal Dimension becomes very useful. Of the unit length, That means the total length of the second order curve is 16/9. The second order of the Koch Curve has had each of the 4 sections of the generator replaced with the same shape, so it has 16 small segments, and each segment is 1/9 The generator (order 1) is made of 4 sections, and each section is 1/3 of the length of the initiator (order 0), which has a unit length of 1. Let's look at the way the length of the curve changes as we iterate the fractal. As we learned in Chapter 2, geometric fractals can be made by starting with a simple generator pattern and replacing every section of the pattern with a smaller copy
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