Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Fractal Geometry, A Brief Introduction to

  • Armin Bunde
  • Shlomo Havlin
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_218

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 ,cracks ,mountains , andclouds . This means that the coastline is...

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Notes

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.

Bibliography

  1. 1.
    Mandelbrot BB (1977)Fractals: Form, chance and dimension. Freeman, San Francisco; Mandelbrot BB (1982) The fractalgeometry of nature. Freeman, San FranciscoGoogle Scholar
  2. 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, NewYorkGoogle Scholar
  3. 3.
    Peitgen H-O,Jürgens H, Saupe D (1992) Chaos and fractals. Springer, NewYorkGoogle Scholar
  4. 4.
    Feder J (1988)Fractals. Plenum, New YorkzbMATHGoogle Scholar
  5. 5.
    Vicsek T (1989)Fractal growth phenomena. World Scientific, SingaporezbMATHGoogle Scholar
  6. 6.
    Avnir D (1992) Thefractal approach to heterogeneous chemistry. Wiley, New YorkGoogle Scholar
  7. 7.
    Barnsley M (1988)Fractals everywhere. Academic Press, San DiegozbMATHGoogle Scholar
  8. 8.
    Takayasu H (1990)Fractals in the physical sciences. Manchester University Press,ManchesterzbMATHGoogle Scholar
  9. 9.
    Schuster HG (1984)Deterministic chaos – An introduction. Physik Verlag,WeinheimzbMATHGoogle Scholar
  10. 10.
    Peitgen H-O,Richter PH (1986) The beauty of fractals. Springer, HeidelbergzbMATHGoogle Scholar
  11. 11.
    Stanley HE,Ostrowsky N (1990) Correlations and connectivity: Geometric aspects of physics, chemistry andbiology. Kluwer, DordrechtGoogle Scholar
  12. 12.
    Peitgen H-O,Jürgens H, Saupe D (1991) Chaos and fractals. Springer,HeidelbergGoogle Scholar
  13. 13.
    Bunde A,Havlin S (1996) Fractals and disordered systems. Springer,HeidelbergzbMATHGoogle Scholar
  14. 14.
    Gouyet J-F (1992)Physique et structures fractales. Masson, PariszbMATHGoogle Scholar
  15. 15.
    Bunde A,Havlin S (1995) Fractals in science. Springer,HeidelbergGoogle Scholar
  16. 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, CambridgeGoogle Scholar
  17. 17.
    Feigenbaum M(1978) Quantitative universality for a class of non‐lineartransformations. J Stat Phys 19:25MathSciNetADSzbMATHGoogle Scholar
  18. 18.
    Grassberger P(1981) On the Hausdorff dimension of fractal attractors. J Stat Phys26:173MathSciNetADSGoogle Scholar
  19. 19.
    Mandelbrot BB,Given J (1984) Physical properties of a new fractal model of percolation clusters. PhysRev Lett 52:1853ADSGoogle Scholar
  20. 20.
    Douady A, HubbardJH (1982) Itération des polynômes quadratiques complex. CRAS Paris294:123MathSciNetzbMATHGoogle Scholar
  21. 21.
    Weiss GH (1994)Random walks. North Holland, AmsterdamGoogle Scholar
  22. 22.
    Flory PJ (1971)Principles of polymer chemistry. Cornell University Press, NewYorkGoogle Scholar
  23. 23.
    De Gennes PG(1979) Scaling concepts in polymer physics. Cornell University Press,IthacaGoogle Scholar
  24. 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 55:2091Google Scholar
  25. 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 Lett58:2325ADSGoogle Scholar
  26. 26.
    Arapaki E,Argyrakis P, Bunde A (2004) Diffusion‐driven spreading phenomena: The structure ofthe hull of the visited territory. Phys Rev E 69:031101ADSGoogle Scholar
  27. 27.
    Witten TA, SanderLM (1981) Diffusion‐limited aggregation, a kinetic critical phenomenon. Phys RevLett 47:1400ADSGoogle Scholar
  28. 28.
    Meakin P (1983)Diffusion‐controlled cluster formation in two, three, and four dimensions. PhysRev A 27:604,1495MathSciNetADSGoogle Scholar
  29. 29.
    Meakin P (1988)In: Domb C, Lebowitz J (eds) Phase transitions and critical phenomena, vol 12. Academic Press,New York, p 335Google Scholar
  30. 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 100:337Google Scholar
  31. 31.
    Pietronero L(1992) Fractals in physics: Applications and theoretical developments. Physica A191:85ADSGoogle Scholar
  32. 32.
    Meakin P, Majid I,Havlin S, Stanley HE (1984) Topological properties of diffusion limited aggregation andcluster‐cluster aggregation. Physica A 17:L975Google Scholar
  33. 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:L377ADSGoogle Scholar
  34. 34.
    Schwarzer S, LeeJ, Bunde A, Havlin S, Roman HE, Stanley HE (1990) Minimum growth probability ofdiffusion‐limited aggregates. Phys Rev Lett 65:603ADSGoogle Scholar
  35. 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 53:1653Google Scholar
  36. 36.
    Stauffer D,Aharony A (1992) Introduction to percolation theory. Taylor and Francis,LondonGoogle Scholar
  37. 37.
    Kesten H (1982)Percolation theory for mathematicians. Birkhauser, BostonzbMATHGoogle Scholar
  38. 38.
    Grimmet GR (1989)Percolation. Springer, New YorkGoogle Scholar
  39. 39.
    Song C, Havlin S,Makse H (2005) Self‐similarity of complex networks. Nature433:392ADSGoogle Scholar
  40. 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 Lett61:1438Google Scholar
  41. 41.
    Voss RF (1985) In:Earshaw RA (ed) Fundamental algorithms in computer graphics. Springer, Berlin,p 805Google Scholar
  42. 42.
    Coleman PH,Pietronero L (1992) The fractal structure of the universe. Phys Rep213:311ADSGoogle Scholar
  43. 43.
    Kaye BH (1989)A random walk through fractal dimensions. Verlag Chemie,WeinheimzbMATHGoogle Scholar
  44. 44.
    Turcotte DL (1997)Fractals and chaos in geology and geophysics. Cambridge University Press,CambridgeGoogle Scholar
  45. 45.
    Hurst HE, BlackRP, Simaika YM (1965) Long‐term storage: An experimental study. Constable,LondonGoogle Scholar
  46. 46.
    Mandelbrot BB,Wallis JR (1969) Some long‐run properties of geophysical records. Wat Resour Res5:321–340ADSGoogle Scholar
  47. 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 J322:120–137Google Scholar
  48. 48.
    Mudelsee M (2007)Long memory of rivers from spatial aggregation. Wat Resour Res43:W01202ADSGoogle Scholar
  49. 49.
    Livina VL,Ashkenazy Y, Braun P, Monetti A, Bunde A, Havlin S (2003) Nonlinear volatility of riverflux fluctuations. Phys Rev E 67:042101ADSGoogle Scholar
  50. 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 81:729–732Google Scholar
  51. 51.
    Pelletier JD,Turcotte DL (1999) Self‐affine time series: Application and models. Adv Geophys40:91ADSGoogle Scholar
  52. 52.
    Talkner P, WeberRO (2000) Power spectrum and detrended fluctuation analysis: Application to dailytemperatures. Phys Rev E 62:150–160ADSGoogle Scholar
  53. 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 68:046133Google Scholar
  54. 54.
    Király A,Bartos I, Jánosi IM (2006) Correlation properties of daily temperature anormalies overland. Tellus 58A(5):593–600Google Scholar
  55. 55.
    Santhanam MS,Kantz H (2005) Long‐range correlations and rare events in boundary layer windfields. Physica A 345:713–721ADSGoogle Scholar
  56. 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 E60:1390ADSGoogle Scholar
  57. 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 Lett70:1343–1346ADSGoogle Scholar
  58. 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 Lett85:3736ADSGoogle Scholar
  59. 59.
    Leland WE, TaqquMS, Willinger W, Wilson DV (1994) On the self‐similar nature of Ethernettraffic. IEEE/Transactions ACM Netw 2:1–15Google Scholar
  60. 60.
    Kantelhardt JW,Koscielny‐Bunde E, Rego HA, Bunde A, Havlin S (2001) Detectinglong‐range correlations with detrended fluctuation analysis. Physica A295:441Google Scholar
  61. 61.
    Rybski D,Bunde A, Havlin S, Von Storch H (2006) Long‐term persistence in climate and thedetection problem. Geophys Res Lett 33(6):L06718ADSGoogle Scholar
  62. 62.
    Rybski D,Bunde A (2008) On the detection of trends in long‐term correlatedrecords. Physica AGoogle Scholar
  63. 63.
    Giese E, Mossig I,Rybski D, Bunde A (2007) Long‐term analysis of air temperature trends in CentralAsia. Erdkunde 61(2):186–202Google Scholar
  64. 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 Lett89:028501ADSGoogle Scholar
  65. 65.
    Vjushin D, ZhidkovI, Brenner S, Havlin S, Bunde A (2004) Volcanic forcing improves atmosphere‐oceancoupled general circulation model scaling performance. Geophys Res Lett31:L10206ADSGoogle Scholar
  66. 66.
    Monetti A, HavlinS, Bunde A (2003) Long‐term persistence in the sea surface temperaturefluctuations. Physica A 320:581–589ADSGoogle Scholar
  67. 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 111:1106Google Scholar
  68. 68.
    Bunde A,Kropp J, Schellnhuber H-J (2002) The science of disasters – climate disruptions,heart attacks, and market crashes. Springer, BerlinGoogle Scholar
  69. 69.
    Pfisterer C (1998)Wetternachhersage, 500 Jahre Klimavariationen und Naturkatastrophen 1496–1995. VerlagPaul Haupt, BernGoogle Scholar
  70. 70.
    Glaser R (2001)Klimageschichte Mitteleuropas. Wissenschaftliche Buchgesellschaft,DarmstadtGoogle Scholar
  71. 71.
    Mudelsee M,Börngen M, Tetzlaff G, Grünwald U (2003) No upward trends in the occurrence ofextreme floods in Central Europe. Nature 425:166Google Scholar
  72. 72.
    Bunde A,Eichner J, Havlin S, Kantelhardt JW (2003) The effect of long‐term correlations on thereturn periods of rare events. Physica A 330:1MathSciNetADSzbMATHGoogle Scholar
  73. 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 94:048701ADSGoogle Scholar
  74. 74.
    Eichner J,Kantelhardt JW, Bunde A, Havlin S (2006) Extreme value statistics in records withlong‐term persistence. Phys Rev E 73:016130ADSGoogle Scholar
  75. 75.
    Yamasaki K,Muchnik L, Havlin S, Bunde A, Stanley HE (2005) Scaling and memory in volatility returnintervals in financial markets. PNAS 102:26 9424–9428Google Scholar
  76. 76.
    Lennartz S, LivinaVN, Bunde A, Havlin S (2008) Long‐term memory in earthquakes and the distributionof interoccurence times. Europ Phys Lett 81:69001ADSGoogle Scholar
  77. 77.
    Corral A (2004)Long‐term clustering, scaling, and universality in the temporal occurrence ofearthquakes. Phys Rev Lett 92:108501ADSGoogle Scholar
  78. 78.
    Stanley HE, MeakinP (1988) Multifractal phenomena in physics and chemistry. Nature355:405ADSGoogle Scholar
  79. 79.
    Ivanov PC,Goldberger AL, Havlin S, Rosenblum MG, Struzik Z, Stanley HE (1999) Multifractality in humanheartbeat dynamics. Nature 399:461ADSGoogle Scholar
  80. 80.
    Bogachev MI,Eichner JF, Bunde A (2007) Effect of nonlinear correlations on the statistics of returnintervals in multifractal data sets. Phys Rev Lett 99:240601ADSGoogle Scholar
  81. 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. PreprintADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Armin Bunde
    • 1
  • Shlomo Havlin
    • 2
  1. 1.Institut für Theoretische PhysikGießenGermany
  2. 2.Institute of Theoretical PhysicsBar-Ilan‐UniversityRamat GanIsrael