# 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 York
5. 5.
Vicsek T (1989)Fractal growth phenomena. World Scientific, Singapore
6. 6.
Avnir D (1992) Thefractal approach to heterogeneous chemistry. Wiley, New YorkGoogle Scholar
7. 7.
Barnsley M (1988)Fractals everywhere. Academic Press, San Diego
8. 8.
Takayasu H (1990)Fractals in the physical sciences. Manchester University Press,Manchester
9. 9.
Schuster HG (1984)Deterministic chaos – An introduction. Physik Verlag,Weinheim
10. 10.
Peitgen H-O,Richter PH (1986) The beauty of fractals. Springer, Heidelberg
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,Heidelberg
14. 14.
Gouyet J-F (1992)Physique et structures fractales. Masson, Paris
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 19:25
18. 18.
Grassberger P(1981) On the Hausdorff dimension of fractal attractors. J Stat Phys26:173
19. 19.
Mandelbrot BB,Given J (1984) Physical properties of a new fractal model of percolation clusters. PhysRev Lett 52:1853
20. 20.
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 Lett58:2325
26. 26.
Arapaki E,Argyrakis P, Bunde A (2004) Diffusion‐driven spreading phenomena: The structure ofthe hull of the visited territory. Phys Rev E 69:031101
27. 27.
Witten TA, SanderLM (1981) Diffusion‐limited aggregation, a kinetic critical phenomenon. Phys RevLett 47:1400
28. 28.
Meakin P (1983)Diffusion‐controlled cluster formation in two, three, and four dimensions. PhysRev A 27:604,1495
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 A191:85
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:L377
34. 34.
Schwarzer S, LeeJ, Bunde A, Havlin S, Roman HE, Stanley HE (1990) Minimum growth probability ofdiffusion‐limited aggregates. Phys Rev Lett 65:603
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, Boston
38. 38.
Grimmet GR (1989)Percolation. Springer, New YorkGoogle Scholar
39. 39.
Song C, Havlin S,Makse H (2005) Self‐similarity of complex networks. Nature433:392
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 Rep213:311
43. 43.
Kaye BH (1989)A random walk through fractal dimensions. Verlag Chemie,Weinheim
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 Res5:321–340
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:W01202
49. 49.
Livina VL,Ashkenazy Y, Braun P, Monetti A, Bunde A, Havlin S (2003) Nonlinear volatility of riverflux fluctuations. Phys Rev E 67:042101
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 Geophys40:91
52. 52.
Talkner P, WeberRO (2000) Power spectrum and detrended fluctuation analysis: Application to dailytemperatures. Phys Rev E 62:150–160
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 345:713–721
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 E60:1390
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 Lett70:1343–1346
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 Lett85:3736
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):L06718
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 Lett89:028501
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:L10206
66. 66.
Monetti A, HavlinS, Bunde A (2003) Long‐term persistence in the sea surface temperaturefluctuations. Physica A 320:581–589
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.
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 330:1
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 94:048701
74. 74.
Eichner J,Kantelhardt JW, Bunde A, Havlin S (2006) Extreme value statistics in records withlong‐term persistence. Phys Rev E 73:016130
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 81:69001
77. 77.
Corral A (2004)Long‐term clustering, scaling, and universality in the temporal occurrence ofearthquakes. Phys Rev Lett 92:108501
78. 78.
Stanley HE, MeakinP (1988) Multifractal phenomena in physics and chemistry. Nature355:405
79. 79.
Ivanov PC,Goldberger AL, Havlin S, Rosenblum MG, Struzik Z, Stanley HE (1999) Multifractality in humanheartbeat dynamics. Nature 399:461
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 99:240601
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. Preprint