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
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Reference
B. Bondarenko, Generalized Triangles and Pyramids of Pascal, Their Fractals, Graphs and Applications, Tashkent, Fan, 1990, in Russian.
n chapter 8 we will demonstrate how the fractal patterns and self-similarity features can be deciphered by the tools which are the theme of chapter 5.
H. von Koch Sur une courbe continue sans tangente obtenue par une construction géometrique élémentaireArkiv för Matematik 1 (1904) 681–704. Another article is H. von Koch Une méthode géométrique élémentaire pour l’étude de certaines questions de la théorie des courbes planesActa Mathematica 30 (1906) 145–174.
G. Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen 36 (1890) 157–160.
D. Hilbert, Über die stetige Abbildung einer Linie auf ein Flächenstück,Mathematische Annalen 38 (1891) 459–460. 15Hilbert introduced his example in Bremen, Germany during the annual meeting of the Deutsche Gesellschafi für Natur-forscher und Ärzte,which was the meeting at which he and Cantor were instrumental in founding the Deutsche Mathematiker Vereinigung,the German mathematical society.
SIGGRAPH is the Special Interest Group Graphics of the Association for Computing Machinery (ACM). Their yearly conventions draw about 30,000 professionals from the field of computer graphics.
L. Velho, J. de Miranda Gomes, Digital Halftoning with Space-Filling Curves, Computer Graphics 25. 4 (1991) 81–90.
R. J. Stevens, A. F. Lehar, F. H. Preston, Manipulation and Presentation of Multidimensional Image Data Using the Peano Scan, IEEE Transactions on Pattern Analysis and Machine Intelligence 5 (1983) 520–526.
Two objects X and Y (topological spaces) are homeomorphic if there is a homeomorphism h: X Y (i.e., a continuous one-to-one and onto mapping that has a continuous inverse h— I).
The notion `onto’ means here that for every point z of the unit square there is one point x in the unit interval that is mapped to z = f (x).
C. Kuratowski, Topologie 11, PWN, 1961. R. Engelking, Dimension Theory, North Holland, 1978.
Open’ means that we consider a disk (resp. ball) without the bounding circle (resp. sphere) or, more generally. unions of such disks (resp. balls).
For more details about dimensions we refer to Gerald E. Edgar, Measure, Topology and Fractal Geometry, Springer-Verlag, New York, 1990.
W. Sierpinski, Sur une courbe cantorienne qui contient une image biunivoques et continue detoute courbe donnée C. R. Acad. Paris 162 (1916) 629–632.
Compactness is a technical requirement which can be assumed to be true for any drawing on a sheet of paper. For instance, a disk in the plane without its boundary would not he compact, or a line going to infinity would also not be compact. Technically, compactness for a set X in the plane (or in space) means that it is bounded, i.e. it lies entirely within some sufficiently large disk in the plane (or ball in space) and that every convergent sequence of points from the set converges to a point from the set.
K. Menger, Allgemeine Räume und charakteristische Räume, Zweite Mitteilung: „Über umfassendste n-dimensionale Men-gen“, Proc. Acad. Amsterdam 29, (1926) 1125–1128. See also K. Menger, Dimensionstheorie, Leipzig 1928.
Formally, for any compact one-dimensional set A there is a compact subset B of the Menger sponge which is homeomorphic to A.
Formally, we need that X is a compact metric space.
A formal definition goes like this. Let a be a cardinal number. Then one defines ordx(p) a, provided for any E 0 there is a neighborhood U of p with a diameter diam(U) E and such that the cardinality of the boundary DU of U in X is not greater than a, card(DU) a. Moreover, one defines ordx(p) = a, provided ordx (p) a and additionally there is e0 0, such that for all neighborhoods U of x with diameter less than 60 the cardinality of the boundary of U is greater or equal to a, card(DU) a.
This is rather remarkable and it is therefore very instructive to try to construct a spider with five arms and understand the obstruction!
A picture is worth a thousand words.
B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982.
E. E. Kummer, Über Ergänzungssätze zu den allgemeinen Reziprozitätsgesetzen, J. reine angew. Math. 44 (1852) 93–146. S. Wilson was probably the first who gave a rigorous explanation for the Sierpinski gasket appearing in Pascal’s triangle, however, not using Kummer’s result. See S. Wilson, Cellular automata can generate fractals, Discrete Appl. Math. 8 (1984) 91–99.
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). Classical Fractals and Self-Similarity. In: Chaos and Fractals. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4740-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4757-4740-9_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-4742-3
Online ISBN: 978-1-4757-4740-9
eBook Packages: Springer Book Archive