Limits and Self-Similarity

  • Heinz-Otto Peitgen
  • Hartmut Jürgens
  • Dietmar Saupe


Dyson is referring to mathematicians, like G. Cantor, D. Hilbert, and W. Sierpinski, who have been justly credited with having helped to lead mathematics out of its crisis at the turn of the century by building marvelous abstract foundations on which modern mathematics can now flourish safely. Without question, mathematics has changed during this century. What we see is an ever increasing dominance of the algebraic approach over the geometric. In their striving for absolute truth, mathematicians have developed new standards for determining the validity of mathematical arguments. In the process, many of the previously accepted methods have been abandoned as inappropriate. Geometric or visual arguments were increasingly forced out. While Newton’s Principia Mathematica, laying the fundamentals of modern mathematics, still made use of the strength of visual arguments, the new objectivity seems to require a dismissal of this approach. From this point of view, it is ironic that some of the constructions which Cantor, Hilbert, Sierpinski and others created to perfect their extremely abstract foundations simultaneously hold the clues to understanding the patterns of nature in a visual sense. The Cantor set, Hilbert curve, and Sierpinski gasket all give testimony to the delicacy and problems of modern set theory and at the same time, as Mandelbrot has taught us, are perfect models for the complexity of nature.


Similarity Transformation Irrational Number Regular Polygon Geometric Series Continue Fraction Expansion 


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  1. 1.
    Freeman Dyson, Characterizing Irregularity,Science 200 (1978) 677–678.Google Scholar
  2. 2.
    We quote D’Arcy Thompson’s account from his famous 1917 On Growth and Form (New Edition, Cambridge University Press, 1942, page 27): “[Galileo] said that if we tried building ships, palaces or temples of enormous size, yards, beams andGoogle Scholar
  3. 4.
    See J. B. S. Haldane, On Being the Right Size,1928, for a classic essay on the problem of scale.Google Scholar
  4. 5.
    The data in this table is taken from D’Arcy Thompson, On Growth and Form, New Edition, Cambridge University Press, 1942, page 190.Google Scholar
  5. 8.
    Mathematically it is a continuous curve which is nowhere differentiable. It was invented by Helge von Koch just to provide an example for such a curve, see H. v. Koch, Une méthode géometrique élémentaire par !’étude de certaines questions de la théorie des courbes planes, Acta. Mat. 30 (1906) 145–174.Google Scholar
  6. 13.
    A number x is called algebraic provided that it is the root of a polynomial equation with rational coefficients. A transcendental number is one that is not algebraic.Google Scholar
  7. 14.
    D. Shanks and J. W. Wrench, Jr., Calculation of 71“ to 100,000 Decimals,Mathematics of Computation 16, 77 (1962) 76–99.Google Scholar
  8. 15.
    The program ran about 15 hours on a Macintosh llfx.Google Scholar
  9. 16.
    R. P. Brent, Fast multiple-precision evaluation of elementary functions, Journal Assoc. Comput. Mach. 23 (1976) 242–251. E. Salamin, Computation of yr Using Arithmetic-Geometric Mean, Mathematics of Computation 30, 135 (1976) 565–570.MathSciNetGoogle Scholar
  10. 17.
    See the book J. M. Borwein, P. B. Borwein, Pi and the AGM — A Study in Analytic Number Theory, Wiley, New York, 1987.Google Scholar
  11. 18.
    More precisely, the way the computing requirements grow as the number of digits in the factors of the multiplication is increased is not much worse than the corresponding (linear) growth of computing time for the addition of long numbers. The interested reader is referred to the survey in D. Knuth, The Art of Computer Programming, Volume Two, Seminumerical Algorithms, Addison Wesley, 1981, pages 278–299.Google Scholar
  12. 19.
    Besides Yasumasa Kanada from the University of Tokyo, also David and Gregory Chudnovsky from Columbia University, New York succeeded in computing one billion digits. Their results agree.Google Scholar
  13. 20.
    For a recent update on techniques and algorithms see J. M. Borwein, P. B. Borwein, and D. H. Bailey: Ramanujan, modular equations, and approximations to pi, or how to compute one billion digits of pi, American Mathematical Monthly 96 (1989) 201–219.MathSciNetCrossRefGoogle Scholar
  14. 21.
    In fact, in the 1962 paper by Shanks and Wrench, one instance of such hardware failure was reported, and an auxiliary run of the program was made to correct for the error. Thus, at least in the time about 30 years ago, reliability of the arithmetic was an important practical issue even for the `end user’.Google Scholar
  15. 23.
    Carl Sagan, Contact, Pocket Books, Simon und Schuster, New York, 1985.Google Scholar
  16. 24.
    J. Hutchinson, Fractals and self-similarity, Indiana University Math. J. 30 (1981) 713–747.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Heinz-Otto Peitgen
    • 1
    • 2
  • Hartmut Jürgens
    • 3
  • Dietmar Saupe
    • 4
  1. 1.CeVis and MeVisUniversität BremenBremenGermany
  2. 2.Department of MathematicsFlorida Atlantic UniversityBoca RatonUSA
  3. 3.CeVis and MeVisUniversität BremenBremenGermany
  4. 4.Department of Computer ScienceUniversität FreiburgFreiburgGermany

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