Abstract
Fractal sets have been known for more than a century. However, only in 1975, Mandelbrot gave them the name “fractal” and mathematically defined them as sets whose Hausdorff dimension exceeds the topological dimension. While initially fractals were a pure mathematical phenomenon, afterwards fractal-like properties have been discovered in many natural and artificial objects and processes. The recently developed fractal analysis and theory of fractals have been applied in many different areas, including biology, health care, urban planning, environmental studies, geology, geography, chemistry, ecology, astronomy, computer science, social science, music, literature, art to name a few. This chapter gives a literature survey of the fractal theory and analysis applications to solve different problems in various areas. We describe main concepts and frequently used methods to compute a fractal dimension. Finally, we apply the fractal analysis to study the geography and infrastructure of Kyiv and facilitate decision making in urban planning.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman and Company, San Francisco, CA (1982)
Sponge, M.: https://upload.wikimedia.org/wikipedia/commons/5/52/Menger-Schwamm-farbig.png. Last accessed 01 June 2021
Barbara, D.: Chaotic mining: knowledge discovery using the fractal dimension. In: 1999 ACM SIGMOD Workshop on Research Issues in Data Mining and Knowledge Discovery, Philadelphia, USA (1999)
Barbará, D., Chen, P.: Using the fractal dimension to cluster datasets. In: Proceedings of the Sixth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 260–264. Association for Computing Machinery, Boston, Massachusetts, USA (2000). https://doi.org/10.1145/347090.347145
Harte, D.: Multifractals: Theory and Applications, 1st edn. Chapman & Hall/CRC (2001)
Suki, B., Barabasi, A.-L., Hantos, Z., Petak, F., Stanley, H.: Avalanches and power-law behaviour in lung inflation. Nature 368(6472), 615–618 (1994). https://doi.org/10.1038/368615a0
Warsi, M.A.: The fractal nature and functional connectivity of brain function as measured by BOLD MRI in Alzheimer’s disease. Dissertation, McMaster University (2012)
Stanley, H.E., Amaral, L.A., Goldberger, A.L., Havlin, S., Ivanov, P. Ch., Peng, C.K.: Statistical physics and physiology: monofractal and multifractal approaches. Physica A 270(1–2), 309–324 (1999)
Captur, G., Karperien, A.L., Hughes, A.D., Francis, D.P., Moon, J.C.: The fractal heart—Embracing mathematics in the cardiology clinic. Nat. Rev. Cardiol. 14(1), 56–64 (2017). https://doi.org/10.1038/nrcardio.2016.161
Uahabi, K.L., Atounti, M.: Applications of fractals in medicine. Ann. Univ. Craiova Math. Comput. Sci. Ser. 42(1), 167–174 (2015)
Sugihara, G., May, R.M.: Applications of fractals in ecology. Trends Ecol. Evol. 5(3), 79–86 (1990)
Lovejoy, S., Schertzer, D., Ladoy, P.: Fractal characterization of inhomogeneous geophysical measuring networks. Nature 319(6048), 43–44 (1986)
Ribeiro, M.B., Miguelote, A.Y.: Fractals and the Distribution of galaxies. Braz. J. Phys. 28(2), 132–160 (1998). https://doi.org/10.1590/S0103-97331998000200007
Shen, G.: Fractal dimension and fractal growth of urbanized areas. Int. J. Geogr. Inf. Sci. 16(5), 419–437 (2002)
Giovanni, R., Caglioni, M.: Contribution to fractal analysis of cities: A study of metropolitan area of Milan. Cybergeo: European Journal of Geography (2004).
Chen, Y., Wang, J., Feng, J.: Understanding the fractal dimensions of urban forms through spatial entropy. Entropy 19(11), 600 (2017). https://doi.org/10.3390/e19110600
Bao, L., Ma, J., Long, W., He, P., Zhang, T., Nguyen, A.V.: Fractal analysis in particle dissolution: a review. Rev. Chem. Eng. 30(3), 261–287 (2014)
Kaneko, K., Sato, M., Suzuki, T., Fujiwara, Y., Nishikawa, K., Jaroniec, M.: Surface fractal dimension of microporous carbon fibres by nitrogen adsorption. J. Chem. Soc., Faraday Trans. 87(1), 179–184 (1991). https://doi.org/10.1039/FT9918700179
Sajjipanon, P., Ratanamahatana, C.A.: Efficient time series mining using fractal representation. In: Third International Conference on Convergence and Hybrid Information Technology, pp. 704–709. IEEE, Busan, the Republic of Korea (2008). https://doi.org/10.1109/ICCIT.2008.311
Traina, C., Jr., Traina, A., Wu, L., Faloutsos, C.: Fast feature selection using fractal dimension. J. Inf. Data Manage. 1(1), 3–16 (2010)
Tasoulis, D.K., Vrahatis, M.: Unsupervised clustering using fractal dimension. Int. J. Bifurcation Chaos 16(07), 2073–2079 (2006). https://doi.org/10.1142/S021812740601591X
Wilson, T., Dominic, J., Halverson, J.: Fractal interrelationships in field and seismic data. Technical Report 32158-5437, Department of Geology and Geography, West Virginia University, Morgantown, WV, United States (1997)
Aviles, C.A., Scholz, C.H., Boatwright, J.: Fractal analysis applied to characteristic segments of the San Andreas Fault. J. Geophys. Res. 92(B1), 331–344 (1987). https://doi.org/10.1029/JB092iB01p00331
Chang, Y.-F., Chen, C.-C., Liang, C.-Y.: The fractal geometry of the surface ruptures of the 1999 Chi-Chi earthquake, Taiwan. Geophys. J. Int. 170(1), 170–174 (2007). https://doi.org/10.1111/j.1365-246X.2007.03420.x
Vislenko, A.: Possibilities of fractal analysis application to cultural objects. Observatory Culture 2, 13–19 (2015)
Song, C., Havlin, S., Makse, H.A.: Origins of fractality in the growth of complex networks. Nat. Phys. 2(4), 275–281 (2006). https://doi.org/10.1038/nphys266
Das, A., Das, P.: Classification of different Indian songs based on fractal analysis. Complex Syst. 15(3), 253–259 (2005)
Reljin, N., Pokrajac, D.: Music performers classification by using multifractal features: A case study. Archiv. Acoust. 42(2), 223–233 (2017). https://doi.org/10.1515/aoa-2017-0025
Boon, J.-P., Decroly, O.: Dynamical systems theory for music dynamics. Chaos 5(3), 501–508 (1995)
Gonzato, G., Mulargia, F., Marzocchi, W.: Practical application of fractal analysis: Problems and solutions. Geophys. J. Int. 132(2), 275–282 (1998)
Richardson, L.F.: The problem of contiguity: an appendix of statistics of deadly quarrels. Gen. Syst. Yearbook 6, 139–187 (1961)
Mandelbrot, B.B.: How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156(3775), 636–638 (1967)
Acknowledgements
The presented results were obtained in the National Research Fund of Ukraine project 2020.01/0247 «System methodology-based tool set for planning underground infrastructure of large cities providing minimization of ecological and technogenic risks of urban space».
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Malishevsky, A. (2022). Fractal Analysis and Its Applications in Urban Environment. In: Zgurovsky, M., Pankratova, N. (eds) System Analysis & Intelligent Computing. SAIC 2020. Studies in Computational Intelligence, vol 1022. Springer, Cham. https://doi.org/10.1007/978-3-030-94910-5_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-94910-5_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-94909-9
Online ISBN: 978-3-030-94910-5
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)