Advertisement

Frustration on Fractals

  • M. Cieplak
  • J. R. Banavar
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 54)

Abstract

MANDELBROT [1] has observed that shapes occurring in nature often look highly fragmented. Such shapes, known as fractals, can be attributed to coastlines, the mammalian brains, and so on. The length or volume of such objects depends on the length scale used. Consider, for instance, the Sierpinski gasket. To construct the 2-dimensional gasket we begin with a triangle. The midpoints of its edges are connected, creating 4 triangles. The central triangle is removed and the same procedure is continued for each of the new triangles. In the n-th step of the construction the length scale is εn = 2−n and the area within the gasket is proportional to ε n 2−D where D = ln3/ln2. If no triangles were removed, the area would be scale invariant and D = 2. Note that the number of triangles in the gasket grows as ε n −D . On going from one length scale to the other the gasket looks the same, i.e. it is self-similar. Other fractals are usually self-similar in a statistical sense, but still they can be characterized by a fractal dimensionality D. In order to determine D of an object in a d-dimensional space, one may pick a length scale, a, and find the corresponding measure of the object within the volume (aL)d. For large L’s the measure will go as LD.

Keywords

Spin Glass Exchange Coupling Triangular Lattice Fractal Dimensionality Sierpinski Gasket 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. B. Mandelbrot: Fractals: Form, Chance and Dimension ( Freeman, San Francisco 1977 )MATHGoogle Scholar
  2. 2.
    S. Kirkpatrick: in Ill-condensed matter, ed.R.Balian, R.Maynard, and G. Toulouse ( North Holland, Amsterdam 1979 )Google Scholar
  3. 3.
    Y. Gefen, A. Aharony, B. B. Mandelbrot, and S. Kirkpatrik: Phys.Rev.Lett. 47 1771 (1981)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Kapitulnik, A. Aharony, G. Deutscher, and D. Stauffer: J. Phys. A 16, L269 (1983)ADSCrossRefGoogle Scholar
  5. 5.
    M. Z. Cieplak and M. Cieplak: Phys.Lett. (to be published)Google Scholar
  6. 6.
    T. A. Witten Jr. and L. M. Sander: Phys.Rev.Lett. 47, 1400 (1981)ADSCrossRefGoogle Scholar
  7. 7.
    P.Meakin: Phys.Rev. A 27, 604 (1983)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Muthukumar: Phys.Rev.Lett. 50, 839 (1983)ADSCrossRefGoogle Scholar
  9. 9.
    J. R. Banavar and M. Muthukumar (submitted for publication)Google Scholar
  10. 10.
    S. Alexander and R. Orbach: J. de Physique-Lett. 43, 625 (1982)CrossRefGoogle Scholar
  11. 11.
    P. J. Ford: Contemp.Phys. 23, 141 (1982)ADSCrossRefGoogle Scholar
  12. 12.
    K. Binder: Z.Phys. B 48, 319 (1982)ADSCrossRefGoogle Scholar
  13. 13.
    K. H. Fischer: Phys.Stat.Sol. (b) 116, 357 (1983)ADSCrossRefGoogle Scholar
  14. 14.
    L. Lundgren, P.Svedlindh, and O.Beckman: Phys. Rev. B 26, 3990 (1982)ADSCrossRefGoogle Scholar
  15. 15.
    S. F. Edwards and P. W. Anderson, J.Phys. F 5, 965 (1975)ADSCrossRefGoogle Scholar
  16. 16.
    J. R. Banavar and M. Cieplak: Phys.Rev.Lett. 48, 832 (1982)ADSCrossRefGoogle Scholar
  17. J. R. Banavar and M. Cieplak J.Phys.O 16, L755 (1983)ADSGoogle Scholar
  18. J. R. Banavar and M. Cieplak: Phys.Rev.B 26, 2662 (1982)ADSCrossRefGoogle Scholar
  19. 17.
    M. Cieplak and J. R. Banavar: Phys.Rev.B 27, 293 (1983)ADSCrossRefGoogle Scholar
  20. M. Cieplak and J. R. Banavar: Phys.Rev.B 29, 469 (1984)ADSCrossRefGoogle Scholar
  21. 18.
    J. R. Banavar, M. Cieplak, and M. Z. Cieplak: Phys.Rev.B 26, 2432 (1982)ADSCrossRefGoogle Scholar
  22. P-t.Z.Cieplak and M.Cieplak: J.Phys. (to be published)Google Scholar
  23. M. Cieplak and M. Z. Cieplak (submitted for publication)Google Scholar
  24. 19.
    J. T. Edwards and D. J. Thouless: J.Phys. 0 5, 807 (1972)ADSGoogle Scholar
  25. 20.
    M. E. Fisher, M. N. Barber, and D. Jasnow: Phys.Rev.A 8, 1111 (1973)ADSCrossRefGoogle Scholar
  26. 21.
    J. R. Banavar and M. Cieplak: Phys.Rev.B 28, 3813 (1983)ADSCrossRefGoogle Scholar
  27. 22.
    G. Grinstein, A. N. Berker, J. Chalupa, and M.Jortis: Phys.Rev.Lett. 36, 1508 (1976)ADSCrossRefGoogle Scholar
  28. 23.
    Y. Gefen, B. B. Plandelbrot, and A. Aharony: Phys.Rev.Lett. 45, 855 (1980)ADSMathSciNetCrossRefGoogle Scholar
  29. 24.
    T. Niemeijer and J. M. J. Van Leeuwen: Physica 71, 17 (1974)ADSCrossRefGoogle Scholar
  30. 25.
    I. Morgenstern and K. Binder: Phys.Rev.Lett. 43, 1615 (1979)ADSCrossRefGoogle Scholar
  31. 26.
    J. R. Banavar, M. Cieplak, and M. Muthukumar: (unpublished)Google Scholar
  32. 27.
    R. J. Glauber: J. Math.Phys.: 4, 294 (1963)MATHADSMathSciNetCrossRefGoogle Scholar
  33. 28.
    W. Kinzel: Phys.Rev.B 26, 6303 (1982)ADSCrossRefGoogle Scholar
  34. 29.
    W. L.Mc Millan: Phys.Rev.B 28, 5216 (1983)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • M. Cieplak
    • 1
  • J. R. Banavar
    • 2
  1. 1.Institute of Theoretical PhysicsWarsaw UniversityWarsawPoland
  2. 2.Schlumberger-Doll ResearchRidgefieldUSA

Personalised recommendations