Abstract
Mandelbrot is often characterized as the father of fractal geometry. Some people, however, remark that many of the fractals and their descriptions go back to classical mathematics and mathematicians of the past like Georg Cantor (1872), Giuseppe Peano (1890), David Hilbert (1891), Helge von Koch (1904), Waclaw Sierpinski (1916), Gaston Julia (1918), or Felix Hausdorff (1919), to name just a few. Yes, indeed, it is true that the creations of these mathematicians played a key role in Mandelbrot’s concept of a new geometry. But at the same time it is true that they did not think of their creations as conceptual steps towards a new perception or a new geometry of nature. Rather, what we know so well as the Cantor set, the Koch curve, the Peano curve, the Hilbert curve and the Sierpinski gasket, were regarded as exceptional objects, as counter examples, as ‘mathematical monsters’. Maybe this is a bit overemphasized. Indeed, many of the early fractals arose in the attempt to fully explore the mathematical content and limits of fundamental notions (e.g., ‘continuous’ or ‘curve’). The Cantor set, the Sierpinski carpet and the Menger sponge stand out in particular because of their deep roots and essential role in the development of early topology.
The art of asking the right questions in mathematics is more important than the art of solving them.
Georg Cantor
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Peitgen, HO., Jürgens, H., Saupe, D. (2004). Classical Fractals and Self-Similarity. In: Chaos and Fractals. Springer, New York, NY. https://doi.org/10.1007/0-387-21823-8_3
Download citation
DOI: https://doi.org/10.1007/0-387-21823-8_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-9396-2
Online ISBN: 978-0-387-21823-6
eBook Packages: Springer Book Archive