Skip to main content
SpringerLink
Log in

Fractal Geometry, A Brief Introduction to

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

In this chapter we present some definitions related to the fractal concept as well asseveral methods for calculating the fractal dimension and other relevant exponents. Thepurpose is to introduce the reader to the basic properties of fractals and self‐affinestructures so that this book will be self contained. We do not give references to most of theoriginal works, but, we refer mostly to books and reviews on fractal geometry where theoriginal references can be found.

Fractal geometry isa mathematical tool for dealing with complex systems that have no characteristic lengthscale. A well‐known example is the shape ofa coastline . When we see two picturesof a coastline on two different scales, with 1 cm corresponding for example to0.1 km or 10 km, we cannot tell which scale belongs to which picture: both lookthe same, and this features characterizes also many other geographical patterns likerivers...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Bibliography

  1. Mandelbrot BB (1977)Fractals: Form, chance and dimension. Freeman, San Francisco; Mandelbrot BB (1982) The fractalgeometry of nature. Freeman, San Francisco

    Google Scholar 

  2. Jones H (1991) Part1: 7 chapters on fractal geometry including applications to growth, image synthesis, andneutral net. In: Crilly T, Earschaw RA, Jones H (eds) Fractals and chaos. Springer, NewYork

    Google Scholar 

  3. Peitgen H-O,Jürgens H, Saupe D (1992) Chaos and fractals. Springer, NewYork

    Google Scholar 

  4. Feder J (1988)Fractals. Plenum, New York

    MATH  Google Scholar 

  5. Vicsek T (1989)Fractal growth phenomena. World Scientific, Singapore

    MATH  Google Scholar 

  6. Avnir D (1992) Thefractal approach to heterogeneous chemistry. Wiley, New York

    Google Scholar 

  7. Barnsley M (1988)Fractals everywhere. Academic Press, San Diego

    MATH  Google Scholar 

  8. Takayasu H (1990)Fractals in the physical sciences. Manchester University Press,Manchester

    MATH  Google Scholar 

  9. Schuster HG (1984)Deterministic chaos – An introduction. Physik Verlag,Weinheim

    MATH  Google Scholar 

  10. Peitgen H-O,Richter PH (1986) The beauty of fractals. Springer, Heidelberg

    MATH  Google Scholar 

  11. Stanley HE,Ostrowsky N (1990) Correlations and connectivity: Geometric aspects of physics, chemistry andbiology. Kluwer, Dordrecht

    Google Scholar 

  12. Peitgen H-O,Jürgens H, Saupe D (1991) Chaos and fractals. Springer,Heidelberg

    Google Scholar 

  13. Bunde A,Havlin S (1996) Fractals and disordered systems. Springer,Heidelberg

    MATH  Google Scholar 

  14. Gouyet J-F (1992)Physique et structures fractales. Masson, Paris

    MATH  Google Scholar 

  15. Bunde A,Havlin S (1995) Fractals in science. Springer,Heidelberg

    Google Scholar 

  16. Havlin S,Ben‐Avraham D (1987) Diffusion in disordered media. Adv Phys 36:695; Ben‐AvrahamD, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. CambridgeUniversity Press, Cambridge

    Google Scholar 

  17. Feigenbaum M(1978) Quantitative universality for a class of non‐lineartransformations. J Stat Phys 19:25

    MathSciNet  ADS  MATH  Google Scholar 

  18. Grassberger P(1981) On the Hausdorff dimension of fractal attractors. J Stat Phys26:173

    MathSciNet  ADS  Google Scholar 

  19. Mandelbrot BB,Given J (1984) Physical properties of a new fractal model of percolation clusters. PhysRev Lett 52:1853

    ADS  Google Scholar 

  20. Douady A, HubbardJH (1982) Itération des polynômes quadratiques complex. CRAS Paris294:123

    MathSciNet  MATH  Google Scholar 

  21. Weiss GH (1994)Random walks. North Holland, Amsterdam

    Google Scholar 

  22. Flory PJ (1971)Principles of polymer chemistry. Cornell University Press, NewYork

    Google Scholar 

  23. De Gennes PG(1979) Scaling concepts in polymer physics. Cornell University Press,Ithaca

    Google Scholar 

  24. Majid I, Jan N,Coniglio A, Stanley HE (1984) Kinetic growth walk: A new model for linear polymers. PhysRev Lett 52:1257; Havlin S, Trus B, Stanley HE (1984) Cluster‐growth model for branchedpolymers that are “chemically linear”. Phys Rev Lett 53:1288; Kremer K, Lyklema JW(1985) Kinetic growth models. Phys Rev Lett 55:2091

    Google Scholar 

  25. Ziff RM, CummingsPT, Stell G (1984) Generation of percolation cluster perimeters by a random walk.J Phys A 17:3009; Bunde A, Gouyet JF (1984) On scaling relations in growthmodels for percolation clusters and diffusion fronts. J Phys A 18:L285; Weinrib A,Trugman S (1985) A new kinetic walk and percolation perimeters. Phys Rev B 31:2993;Kremer K, Lyklema JW (1985) Monte Carlo series analysis of irreversible self‐avoidingwalks. Part I: The indefinitely‐growing self‐avoiding walk(IGSAW). J Phys A 18:1515; Saleur H, Duplantier B (1987) Exact determination of thepercolation hull exponent in two dimensions. Phys Rev Lett58:2325

    ADS  Google Scholar 

  26. Arapaki E,Argyrakis P, Bunde A (2004) Diffusion‐driven spreading phenomena: The structure ofthe hull of the visited territory. Phys Rev E 69:031101

    ADS  Google Scholar 

  27. Witten TA, SanderLM (1981) Diffusion‐limited aggregation, a kinetic critical phenomenon. Phys RevLett 47:1400

    ADS  Google Scholar 

  28. Meakin P (1983)Diffusion‐controlled cluster formation in two, three, and four dimensions. PhysRev A 27:604,1495

    MathSciNet  ADS  Google Scholar 

  29. Meakin P (1988)In: Domb C, Lebowitz J (eds) Phase transitions and critical phenomena, vol 12. Academic Press,New York, p 335

    Google Scholar 

  30. Muthukumar M(1983) Mean‐field theory for diffusion‐limited cluster formation. Phys Rev Lett50:839; Tokuyama M, Kawasaki K (1984) Fractal dimensions for diffusion‐limitedaggregation. Phys Lett A 100:337

    Google Scholar 

  31. Pietronero L(1992) Fractals in physics: Applications and theoretical developments. Physica A191:85

    ADS  Google Scholar 

  32. Meakin P, Majid I,Havlin S, Stanley HE (1984) Topological properties of diffusion limited aggregation andcluster‐cluster aggregation. Physica A 17:L975

    Google Scholar 

  33. Mandelbrot BB(1992) Plane DLA is not self‐similar; is it a fractal that becomes increasinglycompact as it grows? Physica A 191:95; see also: Mandelbrot BB, Vicsek T (1989) Directedrecursive models for fractal growth. J Phys A 22:L377

    ADS  Google Scholar 

  34. Schwarzer S, LeeJ, Bunde A, Havlin S, Roman HE, Stanley HE (1990) Minimum growth probability ofdiffusion‐limited aggregates. Phys Rev Lett 65:603

    ADS  Google Scholar 

  35. Meakin P (1983)Formation of fractal clusters and networks by irreversible diffusion‐limitedaggregation. Phys Rev Lett 51:1119; Kolb M (1984) Unified description of static and dynamicscaling for kinetic cluster formation. Phys Rev Lett 53:1653

    Google Scholar 

  36. Stauffer D,Aharony A (1992) Introduction to percolation theory. Taylor and Francis,London

    Google Scholar 

  37. Kesten H (1982)Percolation theory for mathematicians. Birkhauser, Boston

    MATH  Google Scholar 

  38. Grimmet GR (1989)Percolation. Springer, New York

    Google Scholar 

  39. Song C, Havlin S,Makse H (2005) Self‐similarity of complex networks. Nature433:392

    ADS  Google Scholar 

  40. Havlin S,Blumberg‐Selinger R, Schwartz M, Stanley HE, Bunde A (1988) Random multiplicativeprocesses and transport in structures with correlated spatial disorder. Phys Rev Lett61:1438

    Google Scholar 

  41. Voss RF (1985) In:Earshaw RA (ed) Fundamental algorithms in computer graphics. Springer, Berlin,p 805

    Google Scholar 

  42. Coleman PH,Pietronero L (1992) The fractal structure of the universe. Phys Rep213:311

    ADS  Google Scholar 

  43. Kaye BH (1989)A random walk through fractal dimensions. Verlag Chemie,Weinheim

    MATH  Google Scholar 

  44. Turcotte DL (1997)Fractals and chaos in geology and geophysics. Cambridge University Press,Cambridge

    Google Scholar 

  45. Hurst HE, BlackRP, Simaika YM (1965) Long‐term storage: An experimental study. Constable,London

    Google Scholar 

  46. Mandelbrot BB,Wallis JR (1969) Some long‐run properties of geophysical records. Wat Resour Res5:321–340

    ADS  Google Scholar 

  47. Koscielny‐Bunde E,Kantelhardt JW, Braun P, Bunde A, Havlin S (2006) Long‐term persistence andmultifractality of river runoff records: Detrended fluctuation studies. Hydrol J322:120–137

    Google Scholar 

  48. Mudelsee M (2007)Long memory of rivers from spatial aggregation. Wat Resour Res43:W01202

    ADS  Google Scholar 

  49. Livina VL,Ashkenazy Y, Braun P, Monetti A, Bunde A, Havlin S (2003) Nonlinear volatility of riverflux fluctuations. Phys Rev E 67:042101

    ADS  Google Scholar 

  50. Koscielny‐Bunde E,Bunde A, Havlin S, Roman HE, Goldreich Y, Schellnhuber H-J (1998) Indication ofa universal persistence law governing athmospheric variability. Phys Rev Lett 81:729–732

    Google Scholar 

  51. Pelletier JD,Turcotte DL (1999) Self‐affine time series: Application and models. Adv Geophys40:91

    ADS  Google Scholar 

  52. Talkner P, WeberRO (2000) Power spectrum and detrended fluctuation analysis: Application to dailytemperatures. Phys Rev E 62:150–160

    ADS  Google Scholar 

  53. Eichner JF,Koscielny‐Bunde E, Bunde A, Havlin S, Schellnhuber H-J (2003) Power‐lawpersistence and trends in the atmosphere: A detailed study of long temperaturerecords. Phys Rev E 68:046133

    Google Scholar 

  54. Király A,Bartos I, Jánosi IM (2006) Correlation properties of daily temperature anormalies overland. Tellus 58A(5):593–600

    Google Scholar 

  55. Santhanam MS,Kantz H (2005) Long‐range correlations and rare events in boundary layer windfields. Physica A 345:713–721

    ADS  Google Scholar 

  56. Liu YH, Cizeau P,Meyer M, Peng C-K, Stanley HE (1997) Correlations in economic time series. Physica A245:437; Liu YH, Gopikrishnan P, Cizeau P, Meyer M, Peng C-K, Stanley HE (1999) Statisticalproperties of the volatility of price fluctuations. Phys Rev E60:1390

    ADS  Google Scholar 

  57. Peng C-K, MietusJ, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long‐range anticorrelationsand non‐gaussian behavior of the heartbeat. Phys Rev Lett70:1343–1346

    ADS  Google Scholar 

  58. Bunde A,Havlin S, Kantelhardt JW, Penzel T, Peter J-H, Voigt K (2000) Correlated and uncorrelatedregions in heart‐rate fluctuations during sleep. Phys Rev Lett85:3736

    ADS  Google Scholar 

  59. Leland WE, TaqquMS, Willinger W, Wilson DV (1994) On the self‐similar nature of Ethernettraffic. IEEE/Transactions ACM Netw 2:1–15

    Google Scholar 

  60. Kantelhardt JW,Koscielny‐Bunde E, Rego HA, Bunde A, Havlin S (2001) Detectinglong‐range correlations with detrended fluctuation analysis. Physica A295:441

    Google Scholar 

  61. Rybski D,Bunde A, Havlin S, Von Storch H (2006) Long‐term persistence in climate and thedetection problem. Geophys Res Lett 33(6):L06718

    ADS  Google Scholar 

  62. Rybski D,Bunde A (2008) On the detection of trends in long‐term correlatedrecords. Physica A

    Google Scholar 

  63. Giese E, Mossig I,Rybski D, Bunde A (2007) Long‐term analysis of air temperature trends in CentralAsia. Erdkunde 61(2):186–202

    Google Scholar 

  64. Govindan RB,Vjushin D, Brenner S, Bunde A, Havlin S, Schellnhuber H-J (2002) Global climate modelsviolate scaling of the observed atmospheric variability. Phys Rev Lett89:028501

    ADS  Google Scholar 

  65. Vjushin D, ZhidkovI, Brenner S, Havlin S, Bunde A (2004) Volcanic forcing improves atmosphere‐oceancoupled general circulation model scaling performance. Geophys Res Lett31:L10206

    ADS  Google Scholar 

  66. Monetti A, HavlinS, Bunde A (2003) Long‐term persistence in the sea surface temperaturefluctuations. Physica A 320:581–589

    ADS  Google Scholar 

  67. Kantelhardt JW,Koscielny‐Bunde E, Rybski D, Braun P, Bunde A, Havlin S (2006) Long‐termpersistence and multifractality of precipitation and river runoff records. Geophys J ResAtmosph 111:1106

    Google Scholar 

  68. Bunde A,Kropp J, Schellnhuber H-J (2002) The science of disasters – climate disruptions,heart attacks, and market crashes. Springer, Berlin

    Google Scholar 

  69. Pfisterer C (1998)Wetternachhersage, 500 Jahre Klimavariationen und Naturkatastrophen 1496–1995. VerlagPaul Haupt, Bern

    Google Scholar 

  70. Glaser R (2001)Klimageschichte Mitteleuropas. Wissenschaftliche Buchgesellschaft,Darmstadt

    Google Scholar 

  71. Mudelsee M,Börngen M, Tetzlaff G, Grünwald U (2003) No upward trends in the occurrence ofextreme floods in Central Europe. Nature 425:166

    Google Scholar 

  72. Bunde A,Eichner J, Havlin S, Kantelhardt JW (2003) The effect of long‐term correlations on thereturn periods of rare events. Physica A 330:1

    MathSciNet  ADS  MATH  Google Scholar 

  73. Bunde A,Eichner J, Havlin S, Kantelhardt JW (2005) Long‐term memory: A natural mechanismfor the clustering of extreme events and anomalous residual times in climate records. Phys RevLett 94:048701

    ADS  Google Scholar 

  74. Eichner J,Kantelhardt JW, Bunde A, Havlin S (2006) Extreme value statistics in records withlong‐term persistence. Phys Rev E 73:016130

    ADS  Google Scholar 

  75. Yamasaki K,Muchnik L, Havlin S, Bunde A, Stanley HE (2005) Scaling and memory in volatility returnintervals in financial markets. PNAS 102:26 9424–9428

    Google Scholar 

  76. Lennartz S, LivinaVN, Bunde A, Havlin S (2008) Long‐term memory in earthquakes and the distributionof interoccurence times. Europ Phys Lett 81:69001

    ADS  Google Scholar 

  77. Corral A (2004)Long‐term clustering, scaling, and universality in the temporal occurrence ofearthquakes. Phys Rev Lett 92:108501

    ADS  Google Scholar 

  78. Stanley HE, MeakinP (1988) Multifractal phenomena in physics and chemistry. Nature355:405

    ADS  Google Scholar 

  79. Ivanov PC,Goldberger AL, Havlin S, Rosenblum MG, Struzik Z, Stanley HE (1999) Multifractality in humanheartbeat dynamics. Nature 399:461

    ADS  Google Scholar 

  80. Bogachev MI,Eichner JF, Bunde A (2007) Effect of nonlinear correlations on the statistics of returnintervals in multifractal data sets. Phys Rev Lett 99:240601

    ADS  Google Scholar 

  81. Bogachev MI,Bunde A (2008) Memory effects in the statistics of interoccurrence times between largereturns in financial records. Phys Rev E 78:036114; Bogachev MI, Bunde A (2008)Improving risk extimation in multifractal records: Applications to physiology andfinancing. Preprint

    ADS  Google Scholar 

Download references

Acknowledgments

We like to thank all our coworkers in this field, in particular Eva Koscielny‐Bunde, Mikhail Bogachev, Jan Kantelhardt, Jan Eichner, Diego Rybski, Sabine Lennartz, Lev Muchnik, Kazuko Yamasaki, John Schellnhuber and Hans von Storch.

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, Gießen, Germany

    Armin Bunde

  2. Institute of Theoretical Physics, Bar-Ilan‐University, Ramat Gan, Israel

    Shlomo Havlin

Authors
  1. Armin Bunde
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shlomo Havlin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

Bunde, A., Havlin, S. (2009). Fractal Geometry, A Brief Introduction to. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_218

Download citation

Publish with us

Policies and ethics

Search

Navigation