Fractals and Multifractals

  • Stephen Lynch
Chapter
First Online:

Abstract

Aims and Objectives

• To provide a brief introduction to fractals.

• To introduce the notion of fractal dimension.

• To provide a brief introduction to multifractals and define a multifractal formalism.

• To consider some very simple examples.

On completion of this chapter the reader should be able to

• plot early-stage generations of certain fractals using either graph paper, pencil, and rule, or MATLAB;

• determine the fractal dimension of some mathematical fractals;

• estimate the fractal dimension using simple box-counting techniques;

• distinguish between homogeneous and heterogeneous fractals;

• appreciate how multifractal theory is being applied in the real world;

• construct multifractal Cantor sets and Koch curves and plot graphs of their respective multifractal spectra.

Keywords

Fractal Dimension Chaotic Attractor Multifractal Analysis Multifractal Spectrum Graph Paper 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    P.S. Addison, Fractals and Chaos: An Illustrated Course (Institute of Physics, London, 1997)CrossRefMATHGoogle Scholar
  2. 2.
    M. Alber, J. Peinke, Improved multifractal box-counting algorithm, virtual phase transitions, and negative dimensions. Phys. Rev. E 57(5), 5489–5493 (1998)CrossRefGoogle Scholar
  3. 3.
    S. Blacher, F. Brouers, R. Fayt, P. Teyssié, Multifractal analysis. A new method for the characterization of the morphology of multicomponent polymer systems. J. Polym. Sci. B Polym. Phys. 31, 655–662 (1993)Google Scholar
  4. 4.
    L.E. Calvet, A.J. Fisher, Multifractal Volatility: Theory, Forecasting, and Pricing (Academic, New York, 2008)Google Scholar
  5. 5.
    A.B. Chhabra, C. Meneveau, R.V. Jensen, K.R. Sreenivasan, Direct determination of the f(α) singularity spectrum and its application to fully developed turbulence. Phys. Rev. A 40(9), 5284–5294 (1989)CrossRefGoogle Scholar
  6. 6.
    R.M. Crownover, Introduction to Fractals and Chaos (Jones and Bartlett Publishers, London, 1995)Google Scholar
  7. 7.
    K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, New York, 2003)CrossRefGoogle Scholar
  8. 8.
    K. Falconer, Fractals: A Very Short Introduction (Oxford University Press, Oxford, 2013)CrossRefGoogle Scholar
  9. 9.
    K.J. Falconer, B. Lammering, Fractal properties of generalized Sierpiński triangles. Fractals 6(1), 31–41 (1998)Google Scholar
  10. 10.
    J. Grazzini, A. Turiel, H. Yahia, I. Herlin, A Multifractal Approach for Extracting Relevant Textural Areas in Satellite Meteorological Images (An Article From: Environmental Modelling and Software) (HTML, Digital) (Elsevier, New York, 2007)Google Scholar
  11. 11.
    T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shraiman, Fractal measures and their singularities. Phys. Rev. A 33, 1141 (1986)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    D. Harte, Multifractals: Theory and Applications (Chapman and Hall, London, 2001)CrossRefGoogle Scholar
  13. 13.
    L. Hua, D. Ze-jun, W. Ziqin, Multifractal analysis of the spatial distribution of secondary-electron emission sites. Phys. Rev. B 53(24), 16631–16636 (1996)CrossRefGoogle Scholar
  14. 14.
    N. Lesmoir-Gordon, Introducing Fractal Geometry, 3rd edn. (Totem Books, Lanham, 2006)Google Scholar
  15. 15.
    J. Mach, F. Mas, F. Sagués, Two representations in multifractal analysis. J. Phys. A Math. Gen. 28, 5607–5622 (1995)CrossRefMATHGoogle Scholar
  16. 16.
    B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Co., New York, 1983)Google Scholar
  17. 17.
    S.L. Mills, G.C. Lees, C.M. Liauw, S. Lynch, An improved method for the dispersion assessment of flame retardant filler/polymer systems based on the multifractal analysis of SEM images. Macromol. Mater. Eng. 289(10), 864–871 (2004)CrossRefGoogle Scholar
  18. 18.
    S.L. Mills, G.C. Lees, C.M. Liauw, R.N. Rothon, S. Lynch, Prediction of physical properties following the dispersion assessment of flame retardant filler/polymer composites based on the multifractal analysis of SEM images. J. Macromol. Sci. B Phys. 44(6), 1137–1151 (2005)CrossRefGoogle Scholar
  19. 19.
    J. Muller, O.K. Huseby, A. Saucier, Influence of multifractal scaling of pore geometry on permeabilities of sedimentary rocks. Chaos Soliton Fract. 5(8), 1485–1492 (1995)CrossRefGoogle Scholar
  20. 20.
    H-O. Peitgen (ed.), E.M. Maletsky, H. Jürgens, T. Perciante, D. Saupe, L. Yunker, Fractals for the Classroom: Strategic Activities, vol. 1 (Springer, New York, 1991)Google Scholar
  21. 21.
    H-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals (Springer, Berlin, 1992)Google Scholar
  22. 22.
    N. Sarkar, B.B. Chaudhuri, Multifractal and generalized dimensions of gray-tone digital images. Signal Process. 42, 181–190 (1995)CrossRefMATHGoogle Scholar
  23. 23.
    L. Seuront, Fractals and Multifractals in Ecology and Aquatic Science (CRC Press, New York, 2009)CrossRefGoogle Scholar
  24. 24.
    V. Silberschmidt, Fractal and multifractal characteristics of propagating cracks. J. Phys. IV 6, 287–294 (1996)Google Scholar
  25. 25.
    H.F. Stanley, P. Meakin, Multifractal phenomena in physics and chemistry. Nature 335, 405–409 (1988)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Stephen Lynch
    • 1
  1. 1.Department of Computing and MathematicsManchester Metropolitan University School of Computing, Mathematics & Digital TechnologyManchesterUK

Personalised recommendations