Abstract
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.
Nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of natural patterns is for all practical purposes infinite.
Benoit B. Mandelbrot1
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Benoit B. Mandelbrot, The Fractal Geometry of Nature, Freeman, 1982.
B. B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 155 (1967) 636–638.
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.
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.
J. Feder, Fractals, Plenum Press, New York 1988.
K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, New York 1990.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media New York
About this chapter
Cite this chapter
Peitgen, HO., Jürgens, H., Saupe, D. (1992). Length, Area and Dimension: Measuring Complexity and Scaling Properties. In: Fractals for the Classroom. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2172-0_4
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2172-0_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-2174-4
Online ISBN: 978-1-4757-2172-0
eBook Packages: Springer Book Archive