Length, Area and Dimension: Measuring Complexity and Scaling Properties

  • Heinz-Otto Peitgen
  • Hartmut Jürgens
  • Dietmar Saupe


Geometry has always had two sides, and both together have played very important roles. There has been the analysis of patterns and forms on the one hand; and on the other, the measurement of patterns and forms. The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number π. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus.


Fractal Dimension Line Segment Head Size Infinite Length Logarithmic Spiral 
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  1. 1.
    Benoit B. Mandelbrot, The Fractal Geometry of Nature, Freeman, 1982.Google Scholar
  2. 3.
    B. B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 155 (1967) 636–638.Google Scholar
  3. 15.
    From M. Sernetz, B. Gelléri, F. Hofman, The Organism as a Bioreactor, Interpretation of the Reduction Law of Metabolism in terms of Heterogeneous Catalysis and Fractal Structure, Journal Theoretical Biology 117 (1987) 209–230.Google Scholar
  4. 16.
    See B. B. Mandelbrot, An introduction to multifractal distribution functions, in: Fluctuations and Pattern Formation, H. E. Stanley and N. Ostrowsky (eds.), Kluwer Academic, Dordrecht, 1988.Google Scholar
  5. J. Feder, Fractals, Plenum Press, New York 1988.Google Scholar
  6. K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, New York 1990.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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