## 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.

## Keywords

Line Segment Sierpinski Gasket Hilbert Curve Classical Fractal Binary Expansion## Preview

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## Reference

- 11.B. Bondarenko,
*Generalized Triangles and Pyramids of Pascal*,*Their Fractals*,*Graphs and Applications*, Tashkent, Fan, 1990, in Russian.Google Scholar - 12.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.Google Scholar
- 13.
*H. von Koch*Sur une courbe continue sans tangente obtenue par une construction géometrique élémentaire*Arkiv 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 planes*Acta Mathematica 30 (1906) 145–174*.Google Scholar - 14.G. Peano,
*Sur une courbe*,*qui remplit toute une aire plane*, Mathematische Annalen 36 (1890) 157–160.MathSciNetzbMATHCrossRefGoogle Scholar - 15.D. Hilbert,
*Über die stetige Abbildung einer Linie auf ein Flächenstück*,Mathematische Annalen 38 (1891) 459–460.^{15}Hilbert 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.Google Scholar - 18.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.Google Scholar
- 19.L. Velho, J. de Miranda Gomes,
*Digital Halftoning with Space-Filling Curves*, Computer Graphics 25. 4 (1991) 81–90.Google Scholar - 20.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.CrossRefGoogle Scholar - 22.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}).Google Scholar - 23.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)*.Google Scholar - 24.C. Kuratowski,
*Topologie**11*, PWN, 1961. R. Engelking,*Dimension Theory*, North Holland, 1978.Google Scholar - 25.Open’ means that we consider a disk (resp. ball) without the bounding circle (resp. sphere) or, more generally. unions of such disks (resp. balls).Google Scholar
- 26.For more details about dimensions we refer to Gerald E. Edgar,
*Measure, Topology and Fractal Geometry*, Springer-Verlag, New York, 1990.Google Scholar - 27.W. Sierpinski,
*Sur une courbe cantorienne qui contient une image biunivoques et continue detoute courbe donnée*C. R. Acad. Paris 162 (1916) 629–632.zbMATHGoogle Scholar - 28.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.Google Scholar - 29.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.Google Scholar - 30.Formally, for any compact one-dimensional set A there is a compact subset
*B*of the Menger sponge which is homeomorphic to A.Google Scholar - 32.Formally, we need that
*X*is a compact metric space.Google Scholar - 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.Google Scholar - 35.This is rather remarkable and it is therefore very instructive to try to construct a spider with five arms and understand the obstruction!Google Scholar
- A picture is worth a thousand words.Google Scholar
- 39.B. Mandelbrot,
*The Fractal Geometry of Nature*, Freeman, New York, 1982.zbMATHGoogle Scholar - 41.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.Google Scholar