Limits and Self-Similarity

  • Heinz-Otto Peitgen
  • Hartmut Jürgens
  • Dietmar Saupe

Abstract

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.

Figure 3.1

The new bread romanesco, a crossing between cauliflower and broccoli, exhibits striking self-similarity.

Keywords

Similarity Transformation Regular Polygon Geometric Series Continue Fraction Expansion 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.

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References

  1. 1.
    Freeman Dyson, Characterizing Irregularity, Science 200 (1978) 677–678.Google Scholar
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    C. R. Wall, Selected Topics in Elementary Number Theory, University of South Carolina Press, Columbia, 1974, page 153.Google Scholar
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    Carl Sagan, Contact, Pocket, Books, Simon & Schuster, New York, 1985.Google Scholar
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    J. Hutchinson, Fractals and self-similarity, Indiana University Journal of Mathematics 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
    • 1
  • Dietmar Saupe
    • 1
  1. 1.Institut für Dynamische SystemeUniversität BremenBremen 33Federal Republic of Germany
  2. 2.Department of MathematicsFlorida Atlantic UniversityBoca RatonUSA

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