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Fractal Dimension of Coastline of Australia Akhlaq Husain1*, Jaideep Reddy2, Deepika Bisht2 & Mohammad Sajid3

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Fractal Dimension of Coastline of Australia Akhlaq Husain1*, Jaideep Reddy2, Deepika Bisht2 & Mohammad Sajid3 www.nature.com/scientificreports OPEN Fractal dimension of coastline of Australia Akhlaq Husain1*, Jaideep Reddy2, Deepika Bisht2 & Mohammad Sajid3 Coastlines are irregular in nature having (random) fractal geometry and are formed by various natural activities. Fractal dimension is a measure of degree of geometric irregularity present in the coastline. A novel multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia. The reliability of the coastline length of Australia is addressed by recovering the power law from our computational results. For simulations, the algorithm is implemented on a parallel computer for multi-core processing using the QGIS software, R-programming language and Python codes. From clouds to mountains, snowfakes to river networks, broccoli to blood vessels, fractals can be noticed every- where in nature. Te application of fractals can be seen in fractal antennas, digital imaging, computer graphics, computational geometry, geology and many other felds. Man made fractals include the Cantor set, Sierpinski triangle, and the Mandelbrot set etc. Coastlines (the boundary between land and water) and other natural boundaries have been a subject of human fascination since long. Teir construction is random as compared to deterministic fractals such as the Mandel- brot set (which is formed through repeated iterations of a simple mathematical equation). Te irregularities and variations in coastlines have defed attempts of characterization using methods which are based on Euclidean geometry. Measuring the length of smooth curves is a simple process of successive approximations by line seg- ments. For more accurate measurements, smaller segments can be used. As the segment size approaches zero, the sum of the lengths of the segments will approach the length of the curve. However, this naive application of Euclidean geometry to natural curves fails because the limit may not exist. Tere are several possibilities for this failure such as sloppy length measurements, use of diferent data sets by diferent researchers (some of which may be erroneous), and use of diferent measurement methods, leading to diferences in measured lengths etc. Te variation in the lengths of coastlines at diferent scales intrigued the scientist Lewis Fry Richardson in the 1920s. He examined the coastlines of several countries including Great Britain, South Africa, and Australia as well as the border of Spain and Portugal 35. He chose to measure a coastline length by walking a divider of a specifc length along the coastline to see how many dividers were needed to cover the entire coastline and calculated the length of the coastline by multiplying the number of dividers by the length of each divider. Tere are various numbers, associated with fractals, which provide an objective means for comparing fractals. Tey are generally referred to as fractal dimensions. Tese numbers attempt to quantify how densely the fractal occupies the space in which it lies. Many variations of fractal dimension have been defned, and for some sets, these diferent dimensions yield diferent values. Fractal dimension which itself is a class of diferent dimensions (such as the Hausdorf dimension, similarity dimension, box-counting dimension and the divider dimension) which are all equal for exactly self-similar fractals like the Koch curve. We refer the reader to the classical text books by Barnsley 1, Falconer5, and the book by Frame et al.13 for a comprehensive study of various types of fractal dimensions and methods to calculate these. Te recent book by Fernández-Martínez et al.11 is a good reference from both theoretical and applied viewpoints which is focused on the calculation of the fractal dimension of an object in more general settings. Te inspiration to calculate the fractal dimension of coastlines came with the landmark paper of Mandelbrot “How long is the coast of Britain? Statistical Similarity and Fractal Dimension”28. Te answer depends on how closely you look at it, or how long is your measuring stick. Mandelbrot’s research 28–30 showed that the fractal dimension of a coastline is constant, whereas the length varies in accordance with diferent measurement scales. Afer the pioneering work of Mandelbrot, the fractal dimensions of diferent coastlines have been calculated by many researchers, including Mandelbrot 28, Goodchild17, Kappraf22, Philips (1986), Feder 6, Longley and Batty26, Carr and Benzer2, Jay and Xia20, Paar et al.33, Jiang and Plotnick21, Zhu42, Ma et al.27 and others. Applications of fractal dimension have also been investigated by a number of researchers and we refer to the papers by Cosandy 1Department of Applied Sciences, BML Munjal University, Gurgaon, Haryana 122413, India. 2School of Engineering and Technology, BML Munjal University, Gurgaon, Haryana 122413, India. 3Department of Mechanical Engineering, College of Engineering, Qassim University, Buraidah, Al Qassim 51452, Saudi Arabia. *email: [email protected] Scientifc Reports | (2021) 11:6304 | https://doi.org/10.1038/s41598-021-85405-0 1 Vol.:(0123456789) www.nature.com/scientificreports/ Figure 1. Te Mandelbrot set. (2002), Dimri3, Fernández-Martínez et al.10, Gonzato16, Hayward et al.19, Khoury and Wenger 23, Li et al.25, Wu et al.40, Zhang et al. (2002) and the books by Korvin 24, Turcott38 for applications of fractal dimension in many other exciting felds of research. We also refer to the interesting paper by Fernández-Martínez et al.7 for models of fractal dimension calculation in non-Euclidean contexts. In addition, use of GIS tools (e.g. ArcGIS, QGIS etc.) for calculating fractal dimensions have proven to be generous and a number of researchers have used these tools for estimating fractal dimension. In this article, we calculate the fractal dimension of coastline of Australia using the box-counting method. For simulations we have designed a novel, scalable, multicore parallel processing algorithm and implemented the algorithm on a parallel computer using QGIS sofware (Pi-version) and Python codes to speedup the pro- cessing time and efciency of computations. We also discuss the reliability of the coastline length of Australia, which is in use at present, by recovering the inverse power law in computing the fractal dimension using the data from simulations. Te only available results for fractal dimension of Australia are from the earliest works by Mandelbrot28 and Richardson35 using the divider method. Te algorithm can be used to compute fractal dimen- sion of any coastline and other natural objects such as mountains, rivers, glaciers etc. Tis makes the study and the algorithm important and exciting. Tis paper appears to be the frst to obtain box-counting dimension of Australia and also the frst to introduce a multi-core parallel processing algorithm for computing box-counting dimension in the context of coastlines and otherwise too. Fractal dimension: a brief review Fractals model broken, jagged, complex, wiggly and rough shapes. But some shapes are rougher and more com- plex (e.g. coastlines) than others. Te fractal dimension is used to quantify this roughness and complexity. Te higher the dimension, the higher the roughness, the higher the complexity. Te earliest known measure of roughness of an object is the Hausdorf dimension (also known as Hausdorf- Besicovitch dimension) introduced by Felix Hausdorf in 1918 even before the term fractal was coined, and much of the early work on the subject was done by A.S. Besicovitch. It is now considered one of the fractal dimensions because it has been useful in describing the properties of many fractal sets. For a complete treatment of Hausdorf dimension we refer to the book by Edgar 4. A computational approach to calculate the Hausdorf dimension of compact Euclidean subsets was given by Fernández-Martínez and Sánchez-Granero9 and we refer to the recent work by Fernández-Martínez et al.12 for calculation of the Hausdorf dimension in higher dimensional Euclidean spaces. About his exploratory work on the fractal dimension of coastlines 28, B.B. Mandelbrot quoted in his famous book Te Fractal Geometry of Nature (1982), “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line”. Mandelbrot discovered and plotted one of the most interesting and popular fractal known as the Mandelbrot set (see Fig. 1) at IBM on March 1, 1980. Shishikura36 proved the intriguing conjecture by Mandelbrot32 which states that the Hausdorf dimension of the boundary of the Mandelbrot set is 2. Shishikura’s proof was based on the concept of bifurcation of parabolic periodic points. Let us begin by introducing the formal defnition of fractal dimension. Defnition 1 Let (X, d) be a metric space and A be a compact subset of X. Let ǫ>0 (a small number) be given and defne Nǫ(A) = minimum number of closed balls which covers A n i.e. A ⊂ Bǫ(xi) , where Bǫ(xi) is a ball of radius ǫ centered at xi . We say that A has fractal dimension D if i=1 −D Nǫ(A) ≈ Cǫ (1) where C > 0 is a constant. Scientifc Reports | (2021) 11:6304 | https://doi.org/10.1038/s41598-021-85405-0 2 Vol:.(1234567890) www.nature.com/scientificreports/ f (ǫ) In the above defnition ≈ means that if f (ǫ), g(ǫ) be two functions of ǫ then f ≈ g if and only if lim = 1 . ǫ→0 g(ǫ) From Eq. (1), we obtain log (Nǫ(A)) D = lim ǫ→0 1 (2) log ǫ provided the limit exists. We recall two important results from Barnsley 1. Te frst result gives existence of fractal dimension and the second result replaces the continuous variable ǫ in Eq. (2) by a discrete variable to simplify the process of computations. Theorem 1 (Teorem 2.1, Chapter 4) Let m be a positive integer. Consider the space (Rm, Euclidean). Ten D(A) exists for all A ∈ H(Rm).
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