Space-Filling CurvesSpringer Science & Business Media, 6 déc. 2012 - 194 pages The subject of space-filling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Working in this area is like skating on the edge of reason. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant effort by W. Sierpiriski in 1912. At that time, the subject was still in its infancy and the most interesting and perplexing results were still to come. Besides, Sierpiriski's paper was written in Polish and published in a journal that is not readily accessible (Sierpiriski [2]). Most of the early literature on the subject is in French, German, and Polish, providing an additional raison d'etre for a comprehensive treatment in English. While there was, understandably, some intensive research activity on this subject around the turn of the century, contributions have, nevertheless, continued up to the present and there is no end in sight, indicating that the subject is still very much alive. The recent interest in fractals has refocused interest on space filling curves, and the study of fractals has thrown some new light on this small but venerable part of mathematics. This monograph is neither a textbook nor an encyclopedic treatment of the subject nor a historical account, but it is a little of each. While it may lend structure to a seminar or pro-seminar, or be useful as a supplement in a course on topology or mathematical analysis, it is primarily intended for self-study by the aficionados of classical analysis. |
Table des matières
Preface vii | 1 |
Hilberts SpaceFilling Curve 992 | 8 |
Sierpińskis SpaceFilling Curve | 27 |
Peanos SpaceFilling Curve | 31 |
Lebesgues SpaceFilling Curve | 69 |
Continuous Images of a Line Segment | 84 |
Schoenbergs SpaceFilling Curve | 119 |
Jordan Curves of Positive Lebesgue Measure | 131 |
Fractals 145 | 144 |
Appendix | 169 |
176 | |
187 | |
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Expressions et termes fréquents
accumulation point approximating polygons attractor set binary Cantor curve Cantor set Cauchy sequence closed intervals compact set connected set construction continuous image converges curve of Section defined definition denote differentiable endpoint finite fractal Hahn Hausdorff dimension Heighway dragon Hence Hilbert curve infinitely initial set initial triangle iterated function system joins Jordan curve Knopp Koch curve Lebesgue measure Lebesgue's space-filling curve Lemma limiting arc line segment locally connected mathematics Mazurkiewicz continuum modified Sierpiński triangle N₁ nodal points obtain open intervals open set Osgood curve partition Peano curve preimages Program proof quaternary reduction ratio removed representation Sagan Schoenberg curve set in Fig Show sidelength Sierpiński curve Sierpiński gasket Sierpiński sponge Sierpiński triangle Sierpiński-Knopp curve similarity dimension similarity transformations space-filling curve square step subcube subintervals subsquares subtriangles surjective T₁ ternary Theorem topological dimension two-dimensional Lebesgue measure uniformly unique