Abstract
In recent years, a wide range of complex structures of interest to scientists, engineers and physicians have been quantitatively characterized using the idea of a fractal dimension: a dimension that corresponds in a unique fashion to the geometrical shape under study, and often is not an integer [1.1–17]. The key to this progress is the recognition that many random structures obey a symmetry as striking as that obeyed by regular structures. This “scale symmetry” has the implication that objects look the same on many different scales of observation.
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: The Fractal Geometry of Nature (Freeman, San Francisco 1982)
F. Family, D.P. Landau, eds.: Kinetics of Aggregation and Gelation (Elsevier, Amsterdam 1984)
H.E. Stanley, N. Ostrowsky, eds.: On Growth and Form: Fractal and Non-Fractal Patterns in Physics (Martinus Nijhoff, Dordrecht 1985)
L. Pietronero, E. Tosatti, eds.: Fractals in Physics (Elsevier, Amsterdam 1986)
N. Boccara, M. Daoud, eds.: Physics of Finely Divided Matter (Springer, Berlin, Heidelberg 1986)
R. Pynn, A. Skjeltorp, eds.: Scaling Phenomena in Disordered Systems (Plenum, New York 1986);
R. Pynn, T. Riste, eds.: Time Dependent Effects in Disordered Materials (Plenum, New York 1987)
J. Feder: Fractals (Plenum, New York 1988)
H.E. Stanley, N. Ostrowsky, eds.: Random Fluctuations and Pattern Growth: Experiments and Theory (Kluwer, Dordrecht 1988)
T. Vicsek: Fractal Growth Phenomena (World Scientific, Singapore 1989)
D. Avnir, ed.: The Fractal Approach to Heterogeneous Chemistry (Wiley, Chichester 1989)
J.P. Hulin, A.M. Cazabat, E. Guyon, F. Carmona, eds.: Hydrodynamics of Dispersed Media, Random Materials and Processes, Vol. 1 (North-Holland, Amsterdam 1990)
H.J. Herrmann, S. Roux, eds.: Statistical Models for the Fracture of Disordered Media, Random Materials and Processes, Vol. 2 (North-Holland, Amsterdam 1990)
H.E. Stanley, N. Ostrowsky, eds.: Correlations and Connectivity: Geometric Aspects of Physics, Chemistry and Biology (Kluwer, Dordrecht 1990)
A. Aharony, J. Feder, eds.: Fractals in Physics: Essays in Honor of B.B. Mandelbrot (North-Holland, Amsterdam 1990)
L. Pietronero, ed.: Fractals: Physical Origin and Properties (Plenum, London 1990);
H. Takayasu: Fractals in the Physical Sciences (Manchester Univ. Press, Manchester 1990)
D. Stauffer, H.E. Stanley: From Newton to Mandelbrot: A Primer in Theoretical Physics (Springer, Berlin, Heidelberg 1990)
E. Guyon, H.E. Stanley: Fractal Forms (Elsevier, Amsterdam 1991)
H.E. Stanley: J. Stat. Phys. 36, 843 (1984)
T. A. Witten, L.M. Sander: Phys. Rev. Lett. 47, 1400 (1981);
T.A. Witten, L.M. Sander: Phys. Rev. B 27, 5686 (1983);
L.M. Sander: Nature 332, 789 (1986)
H.E. Stanley, A. Bunde, S. Havlin, J. Lee, E. Roman, S. Schwarzer: Physica A 168, 23 (1990)
R.M. Brady, R.C. Ball: Nature 309, 225 (1984);
M. Matsushita, M. Sano, Y. Hakayawa, H. Honjo, Y. Sawada: Phys. Rev. Lett. 53, 286 (1984)
E. Ben-Jacob, R. Godbey, N.D. Goldenfeld, J. Koplik, H. Levine, T. Mueller, L.M. Sander: Phys. Rev. Lett. 55, 1315 (1985);
E. Ben-Jacob, P. Garik: Nature 343, 523 (1990);
J.S. Langer: Science 243, 1150 (1989);
A. Buka, J. Kertész, T. Vicsek: Nature 323, 424 (1986)
J. Nittmann, H.E. Stanley: Nature 321, 663 (1986)
J. Nittmann, H.E. Stanley: J. Phys. A 20, L981 (1987)
J. Nittmann, H.E. Stanley: J. Phys. A 20, L1185 (1987);
see also F. Family, D. Platt, T. Vicsek: J. Phys. A 20, L1177 (1987)
L. Niemeyer, L. Pietronero, H.J. Wiesmann: Phys. Rev. Lett. 52, 1033 (1984);
L. Pietronero, H.J. Wiesmann: J. Stat. Phys. 36, 909 (1984)
H. S. Hele-Shaw: Nature 58, 34 (1898);
P. G. Saffman, G. I. Taylor: Proc. R. Soc. London A 245, 312 (1958);
J. Nittmann, G. Daccord, H.E. Stanley: Nature 314, 141 (1985);
J.D. Chen, D. Wilkinson: Phys. Rev. Lett. 55, 1892 (1985);
K. J. Måløy, J. Feder, T. Jøssang: Phys. Rev. Lett. 55, 2688 (1985);
H. Van Damme, F. Obrecht, P. Levitz, L. Gatineau, C. Laroche: Nature 320, 731 (1986);
G. Daccord, J. Nittmann, H.E. Stanley: Phys. Rev. Lett. 56, 336 (1986);
U. Oxaal, M. Murat, F. Boger, A. Aharony, J. Feder, T. Jøssang: Nature 329, 32 (1987);
P. Meakin, M. Murat, A. Aharony, J. Feder, T. Jøssang: Physica A 155, 1 (1989)
G. Daccord: Phys. Rev. Lett. 58, 479(1987);
G. Daccord, R. Lenormand: Nature 325, 41 (1987)
A.D. Fowler, H.E. Stanley, G. Daccord: Nature 341, 134 (1989)
M. Matsushita, H. Fujikawa: Physica A 168, 498 (1990)
F. Family, B.R. Masters, D.E. Platt: Physica D 38, 98 (1989)
H.E. Stanley: Bull. Am. Phys. Soc. 34, 716 (1989);
F. Caserta, H.E. Stanley, W. Eldred, G. Daccord, R.E. Hausman, J. Nittmann: Phys. Rev. Lett. 64, 95 (1990)
D. Kleinfeld, F. Raccuia-Behling, G.E. Blonder: Phys. Rev. Lett. 65, 3064 (1990)
P. Meakin: CRC Critical Rev. in Solid State and Materials Sciences 13, 143 (1987);
S. Tolman, P. Meakin: Phys. Rev. A 40, 428 (1989);
P. Meakin, R.C. Ball, P. Ramanlal, L.M. Sander: Phys. Rev. A 35, 5233 (1987);
J-P Eckmann, P. Meakin, I. Procaccia, R. Zeitak: Phys. Rev. A 39, 3185 (1990)
H.E. Stanley: Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, London 1971)
D. Stauffer: Introduction to Percolation Theory (Taylor & Francis, London 1985)
J. Lee, H.E. Stanley: Phys. Rev. Lett. 61, 2945 (1988);
M. Jensen, P. Cvitanovic, T. Bohr: Europhys. Lett. 6, 445 (1988);
J. Lee, P. Alstrøm, H.E. Stanley: Phys. Rev. A 39, 6545 (1989);
T.C. Halsey, in: Fractals: Physical Origin and Properties, ed. by L. Pietronero (Plenum, London 1990); J. Lee, P. Alstrøm, H.E. Stanley: ibid.
B.B. Mandelbrot: J. Fluid Mech. 62, 331 (1974); B.B. Mandelbrot, in [1.8]
P. Grassberger: Physics Lett. 97A, 227 (1983);
P. Grassberger, I. Procaccia: Physica D 13, 34 (1984);
P. Grassberger, Phys. Lett. 107A, 101 (1985)
H.G.E. Hentschel, I. Procaccia: Physica D 8, 435 (1983)
U. Frisch, G. Parisi: In Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, ed. by M. Ghil, R. Benzi, G. Parisi (North-Holland, Amsterdam 1985)
G. Paladin, A. Vulpiani: Phys. Rep. 156, 147 (1987);
R. Benzi, G. Paladin, G. Parisi, A. Vulpiani: J. Phys. A 17, 3521 (1984);
R. Benzi, G. Paladin, G. Parisi, A. Vulpiani: J. Phys. A 18, 2157 (1985);
R. Badii, A. Politi: J. Stat. Phys. 40, 725 (1985);
A. Crisanti, G. Paladin, A. Vulpiani: Phys. Lett. 126A, 120 (1987)
L. de Arcangelis, S. Redner, A. Coniglio: Phys. Rev. B 31, 4725 (1985);
R. Rammal, C. Tannous, P. Breton, A.M.S. Tremblay: Phys. Rev. Lett. 54, 1718 (1985);
L. de Arcangelis, S. Redner, A. Coniglio: Phys. Rev. B 34, 4656 (1986)
A. Coniglio, in [1.3]; A. Coniglio, in: Statistical Physics: Proc. STATPHYS-16, ed. by H.E. Stanley (North-Holland, Amsterdam 1986)
P. Meakin, H.E. Stanley, A. Coniglio, T.A. Witten: Phys. Rev. A 32, 2364 (1985);
T.C. Halsey, P. Meakin, I. Procaccia: Phys. Rev. Lett. 56, 854 (1986)
T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B. Shraiman: Phys. Rev. A 33, 1141 (1986);
P. Meakin, A. Coniglio, H.E. Stanley, T.A. Witten: Phys. Rev. A 34, 3325 (1986)
C. Amitrano, A. Coniglio, F. di Liberto: Phys. Rev. Lett. 57, 1016 (1987);
J. Nittmann, H.E. Stanley, E. Touboul, G. Daccord: Phys. Rev. Lett. 58, 619 (1987);
Y. Hayakawa, S. Sato, M. Matsushita: Phys. Rev. A 36 1963 (1987);
C. Amitrano, P. Meakin, H.E. Stanley: Phys. Rev. A 40, 1713–1716 (1989)
T. Nagatani: Phys. Rev. A 36, 5812 (1987);
T. Nagatani: J. Phys. A 20, L381 (1987);
J. Lee, A. Coniglio, H.E. Stanley: Phys. Rev. A 41, 4589 (1990);
T. Nagatani, H.E. Stanley: Phys. Rev. A 41, 3263 (1990);
T. Nagatani, H.E. Stanley: Phys. Rev. A 42, 3512 (1990);
T. Nagatani, H.E. Stanley: Phys. Rev. A 42, 4838 (1990);
T. Nagatani, J. Lee, H.E. Stanley: Phys. Rev. Lett. 66, 616 (1991)
A.E. Ferdinand, M.E. Fisher: Phys. Rev. 185, 832 (1969)
B.B. Mandelbrot: Physica A 168, 95 (1990)
R. Blumenfeld, A. Aharony: Phys. Rev. Lett. 62, 2977 (1989);
J. Lee, P. Alstrøm, H.E. Stanley: Phys. Rev. Lett. 62, 3013 (1989)
S. Havlin, B.L. Trus, A. Bunde, H.E. Roman: Phys. Rev. Lett. 63, 1189 (1989);
J. Lee, H.E. Stanley: Phys. Rev. Lett. 63, 1190 (1989)
P. Trunfio, P. Alstrøm: Phys. Rev. B 41, 896 (1990)
A.B. Harris, M. Cohen: Phys. Rev. A 41, 971 (1990);
A.B. Harris: Phys. Rev. A 39, 7292 (1989)
S. Schwarzer, J. Lee, A. Bunde, S. Havlin, H.E. Roman, H.E. Stanley: Phys. Rev. Lett. 65, 603 (1990)
J. Lee, S. Havlin, H.E. Stanley, J. Kiefer: Phys. Rev. A 42, 4832 (1990)
S. Schwarzer, J. Lee, S. Havlin, H. E. Stanley, P. Meakin: Phys. Rev. A 43, 1134 (1991)
B.B. Mandelbrot, T. Vicsek: J. Phys. A 20, L377 (1989)
B.B. Mandelbrot, C.J.G. Evertsz: Nature 348, 143 (1990)
B.B. Mandelbrot, C.J.G. Evertsz, Y. Hayakawa: Phys. Rev. A 42, 4528 (1990)
S. Havlin, B.L. Trus: J. Phys. A 21, L731 (1988)
A. Coniglio, M. Zannetti: Physica A 163, 325 (1990); C. Amitrano, A. Coniglio, P. Meakin, preprint
W. Lenz: Phys. Z. 21, 613 (1920)
E. Ising: Ann. Physik 31, 253 (1925)
A. Geiger, H.E. Stanley: Phys. Rev. Lett. 49, 1895 (1982)
E.T. Gawlinski, H.E. Stanley: J. Phys. A 14, L291 (1981)
A. Geiger, H.E. Stanley: Phys. Rev. Lett. 49, 1749 (1982)
R. Ball: Physica 140A, 62 (1986)
C. Tang: Phys. Rev. A. 31 1977 (1985)
J. Kertész, T. Vicsek: J. Phys. A 19, L257 (1986)
H.E. Stanley: Physica A 163, 334 (1990)
U. Nakaya: Snow Crystals (Harvard University Press, Cambridge 1954)
E.R. LaChapelle: Field Guide to Snow Crystals (University of Washington Press, Seattle 1969)
W.A. Bentley, W.J. Humphreys: Snow Crystals (Dover, New York 1962)
A. Dougherty, P.D. Kaplan, J. P. Gollub: Phys. Rev. Lett. 58, 1652 (1987)
H.E. Stanley, in [1.3]
P.G. de Gennes: La Recherche 7, 919 (1976)
P. Meakin, H.E. Stanley: Phys. Rev. Lett. 51, 1457 (1983)
H.E. Stanley, in: Scaling Phenomena in Disordered Systems, ed. by R. Pynn, A. Skjeltorp (Plenum, New York 1986)
S. Alexander, R. Orbach: J. de Phys. 43, L625 (1982);
R. Rammal, G. Toulouse: J. de Phys. 44, L13 (1983)
R.B. Pandey, D. Stauffer: Phys. Rev. Lett. 51, 527 (1983)
S. Havlin, D. Ben-Avraham, J. Phys. A 16, L483 (1983)
P. Meakin, H.E. Stanley: J. Phys. A 17, L173 (1984);
K. Rang, S. Redner: Phys. Rev. Lett. 52, 955 (1984)
F. Ley vraz, H.E. Stanley: Phys. Rev. Lett. 51, 2048 (1983)
H.E. Stanley, I. Majid, A. Margolina, A. Bunde: Phys. Rev. Lett. 53, 1706 (1984)
S. Havlin, D. Ben-Avraham: Adv. Phys. 36, 695 (1987);
A. Bunde: Adv. Solid State Phys. 26, 113 (1986);
J.-P. Bouchaud, A. Georges: Phys. Rep. 195, 127 (1990)
A.L. Ritzenberg, R.I. Cohen: Phys. Rev. B 30, 4038 (1984)
K.M. Middlemiss, S.G. Whittington, D.C. Gaunt: J. Phys. A 13, 1835 (1980)
Z. Alexandrowicz: Phys. Lett. A 80, 284 (1980)
R. Pike, H.E. Stanley: J. Phys. A 14, L169 (1981)
S. Havlin, R. Nossal: J. Phys. A 17, L427 (1984)
H.J. Herrmann, H.E. Stanley: J. Phys. A 21, L829 (1988)
S. Havlin, Z. Djordjevic, I. Majid, H.E. Stanley, G.H. Weiss: Phys. Rev. Lett. 53, 178 (1984)
P. Meakin, I. Majid, S. Havlin, H.E. Stanley: J. Phys. A 17, L975 (1984)
H.E. Stanley, D. Stauffer, J. Kertész, H.J. Herrmann: Phys. Rev. Lett. 59, 2326 (1987);
B. Derrida, G. Weisbuch: Europhys. Lett. 4, 657 (1987);
A. Coniglio, L. de Arcangelis, H.J. Herrmann, N. Jan: Europhys. Lett. 8, 315 (1989); L. de Arcangelis, in [1.13]; N. Jan, in [1.13]
P. Grassberger: J. Phys. A 18, L215 (1985)
M. Barma: J. Phys. A 18, L277 (1985)
H. Kunz, B. Souillard: J. Stat. Phys. 19, 77 (1978)
B. Sapoval, M. Rosso, J.F. Gouyet: J. de Phys. Lett. 46, L149 (1985);
R.S. Voss: J. Phys. A 17, L373 (1984)
H. Saleur, B. Duplantier: Phys. Rev. Lett. 58, 2325 (1987)
A. Coniglio, H.E. Stanley: Phys. Rev. Lett. 52, 1068 (1984)
A. Bunde, H.J. Herrmann, A. Margolina, H.E. Stanley: Phys. Rev. Lett. 55, 653 (1985);
A. Bunde, H.J. Herrmann, H.E. Stanley: J. Phys. A 18, L523 (1985);
H.J. Herrmann, H.E. Stanley: Z. Phys. 60, 165 (1985)
F. Family, T. Vicsek: J. Phys. A 18, L75 (1985);
Y.C. Zhang: J. de Phys. 51, 2113 (1990);
J. Krug: J. de Phys. I 1, 9 (1991);
S.V. Buldyrev, S. Havlin, J. Kertesz, H.E. Stanley, T. Vicsek: Phys. Rev. A 43, (15 June 1991); S. Havlin, S.V. Buldyrev, H.E. Stanley, G. H. Weiss: 1991 preprint
H.E. Stanley: J. Phys. A 10, L211 (1977);
A. Coniglio: Phys. Rev. Lett. 46, 250 (1981);
A. Coniglio: J. Phys. A 15, 3829 (1982);
T. Grossman, A. Aharony: J. Phys. A 19, L745 (1986);
T. Grossman, A. Aharony: J. Phys. A 20, L1193 (1987);
B. Duplantier, H. Saleur: Phys. Rev. Lett. 62, 1368 (1989)
A. Coniglio, N. Jan, I. Majid, H.E. Stanley: Phys. Rev. B 35, 3617 (1987);
I. Majid, N. Jan, A. Coniglio, H.E. Stanley: Phys. Rev. Lett. 52, 1257 (1984);
B. Duplantier, H. Saleur: Phys. Rev. Lett. 59, 539 (1987);
P.H. Poole, A. Coniglio, N. Jan, H.E. Stanley: Phys. Rev. Lett. 60, 1203 (1988);
P.H. Poole, A. Coniglio, N. Jan, H.E. Stanley: Phys. Rev. B 39, 495 (1989)
S. Kirkpatrick: AIP Conf Proc 40, 99 (1978);
G. Shlifer, W. Klein, P.J. Reynolds, H.E. Stanley: J. Phys. A 12, L169 (1979)
H.E. Stanley, R.J. Birgeneau, P.J. Reynolds, J.F. Nicoll: J. Phys. C 9, L553 (1976)
H.J. Herrmann, H.E. Stanley: Phys. Rev. Lett. 53, 1121 (1984)
G. Huber, H.E. Stanley: 1991 preprint
H.J. Herrmann, D. Hong, H.E. Stanley: J. Phys. A 17, L261 (1984)
H.E. Stanley: Physica D 38, 330 (1989)
H.E. Stanley, P. Meakin: Nature 335, 405 (1988)
S. Redner: Am. J. Phys. 58, 267 (1990)
S. Havlin, R. Selinger, M. Schwartz, H.E. Stanley, A. Bunde: Phys. Rev. Lett. 61, 1438 (1988);
S. Havlin, M. Schwartz, R. Blumberg Selinger, A. Bunde, H.E. Stanley: Phys. Rev. A 40, 1717 (1989);
R. Blumberg Selinger, S. Havlin, F. Leyvraz, M. Schwartz, H.E. Stanley: Phys. Rev. A 40, 6755 (1989); C.K. Peng, S. Havlin, M. Schwartz, H.E. Stanley: 1991 preprint; C.K. Peng, S. Havlin, M. Schwartz, H.E. Stanley, G.H. Weiss: 1991 preprint
A. Bunde, S. Havlin, H.E. Roman: Phys. Rev. A 42, 6274 (1990);
S. Havlin, A. Bunde: Physica D 38, 184 (1989)
D. Ruelle: Statistical Mechanics, Thermodynamic Formalism (Addison-Wesley, Reading MA 1975);
M.J. Feigenbaum: J. Stat. Phys. 46, 919 (1987)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Stanley, H.E. (1991). Fractals and Multifractals: The Interplay of Physics and Geometry. In: Bunde, A., Havlin, S. (eds) Fractals and Disordered Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51435-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-51435-7_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-51437-1
Online ISBN: 978-3-642-51435-7
eBook Packages: Springer Book Archive