Abstract
In this chapter we present some definitions related to the fractal concept as well as several methods for calculating the fractal dimension. The purpose is to introduce the reader to the basic properties of fractals so that this book will be self contained. Because of space constraints, we do not give references to most of the original works. We refer mostly to books and reviews on fractal geometry where the original references can be found.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B.B. Mandelbrot: Fractals: Form, Chance and Dimension (Freeman, San Francisco 1977)
B.B. Mandelbrot: The Fractal Geometry of Nature (Freeman, San Francisco 1982)
H. Jones, in: Fractals and Chaos ,ed. by T. Crilly, R.A. Earschaw, H. Jones (Springer, New York 1991)
H.-O. Peitgen, H. Jürgens, D. Saupe: Chaos and Fractals (Springer Verlag, New York 1992)
J. Feder: Fractals (Plenum, New York 1988)
T. Vicsek: Fractal Growth Phenomena (World Scientific, Singapore 1989)
D. Avnir, ed.: The Fractal Approach to Heterogeneous Chemistry (John Wiley, New York 1992)
M. Barnsley: Fractals Everywhere (Academic Press, San Diego 1988)
H. Takayasu: Fractals in the Physical Sciences (Manchester University Press, Manch-ester 1990)
H.G. Schuster: Deterministic Chaos -An Introduction (Physik Verlag, Weinheim 1984)
H.-O. Peitgen, P.H. Richter: The Beauty of Fractals (Springer Verlag, Heidelberg 1986)
H.E. Stanley, N. Ostrowsky, eds.: Correlations and Connectivity: Geometric Aspects of Physics, Chemistry and Biology (Kluwer, Dordrecht 1990)
H.-O. Peitgen, H. Jürgens, D. Saupe: Chaos and Fractals (Springer Verlag, Heidelberg 1991)
A. Bunde and S. Havlin, eds.: Fractals and Disordered Systems (Springer Verlag, Hei-delberg 1991)
J.-F. Gouyet: Physique et Structures Fractales (Masson, Paris 1992)
S. Havlin, D. ben-Avraham: Adv. in Phys. 36, 695 (1987)
M. Feigenbaum: J. Stat. Phys. 19 , 25 (1978)
P. Grassberger: J. Stat. Phys. 26, 173 (1981)
B.B. Mandelbrot, J. Given: Phys. Rev. Lett. 52, 1853 (1984)
A. Douady and J. H. Hubbard: CRAS Paris, 294, 123 (1982) ; see also [1.10]
G.H. Weiss: Random Walks (North Holland, Amsterdam 1994)
P.J. Flory: Principles of Polymer Chemistry (Cornell University Press, New York 1971)
P.G. de Gennes: Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca 1979)
I. Majid, N. Jan, A. Coniglio, H.E. Stanley: Phys. Rev. Lett. 52, 1257 (1984)
S. Havlin, B. Trus, H.E. Stanley: Phys. Rev. Lett. 53, 1288 (1984)
K. Kremer, J.W. Lyklema: Phys. Rev. Lett. 55, 2091 (1985)
R.M. Ziff, P.T. Cummings, G. Stell: J. Phys. A 17, 3009 (1984)
A. Bunde, J.F. Gouyet: J. Phys. A 18, L285 (1984)
A. Weinrib, S. Trugman: Phys. Rev. B 31, 2993 (1985)
K. Kremer, J.W. Lyklema: J. Phys. A 18, 1515 (1985)
H. Saleur, B. Duplantier: Phys. Rev. Lett. 58, 2325 (1987)
T. A. Witten, L.M. Sander: Phys. Rev. Lett. 47, 1400 (1981)
P. Meakin: Phys. Rev. A 27, 604, 1495 (1983)
P. Meakin, in: Phase Transitions and Critical Phenomena ,Vol.12, ed. by C. Domb and J. Lebowitz (Academic Press, New York 1988) p. 335
M. Muthukumar: Phys. Rev. Lett. 50, 839 (1983)
M. Tokuyama, K. Kawasaki: Phys. Lett. A 100, 337 (1984)
L. Pietronero: Physica A 191, 85 (1992)
P. Meakin, I. Majid, S. Havlin, H.E. Stanley: Physica A 17, L975 (1984)
R.F. Voss, in: Fundamental Algorithms in Computer Graphics ,ed. by R.A. Earshaw (Springer, Berlin 1985) p. 805
B.B. Mandelbrot: Physica A 191, 95 (1992)
also B.B. Mandelbrot, T. Vicsek: J. Phys. A 20, L377 (1989)
S. Schwarzer, J. Lee, A. Bunde, S. Havlin, H.E. Roman, H.E. Stanley: Phys. Rev. Lett. 65, 603 (1990)
P. Meakin: Phys. Rev. Lett. 51, 1119 (1983)
M. Kolb: Phys. Rev. Lett. 53, 1653 (1984)
D. Stauffer, A. Aharony: Introduction to Percolation Theory (Taylor and Francis, Lon-don 1992)
H. Kesten: Percolation Theory for Mathematicians (Birkhauser, Boston 1982)
G.R. Grimmet: Percolation (Springer Verlag, New York 1989)
S. Havlin, R. Blumberg-Selinger, M. Schwartz, H.E. Stanley, A. Bunde: Phys. Rev. Lett. 61, 1438 (1988)
C.-K. Peng, S. Havlin, M. Schwartz, H.E. Stanley: Phys. Rev. A 44, R2239 (1991)
H. Inaoka, H. Takayasu: Phys. Rev. E 47, 899 (1993)
B.H. Kaye: A Random Walk Through Fractal Dimensions (Verlag Chemie, Weinheim 1989)
L.S. Liebovitch, I.M. Sullivan: Biophys. J. 52, 979 (1987)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bunde, A., Havlin, S. (1994). A Brief Introduction to Fractal Geometry. In: Bunde, A., Havlin, S. (eds) Fractals in Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77953-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-77953-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-77955-8
Online ISBN: 978-3-642-77953-4
eBook Packages: Springer Book Archive