Springer New York, 2 sept. 1994 - 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 ). 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.
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Hilberts SpaceFilling Curve
Peanos SpaceFilling Curve
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accumulation point adjacent angle appeared apply approximating polygons assume bounded called Cantor set Chapter compact complex condition congruent connected connected set construction contains continuous image converges coordinate defined Definition denote differentiable example exists finite formula four fractal function geometric Hahn Hausdorff Hence Hilbert curve independent infinitely initial interval iterated joins Jordan curve Knopp Lebesgue measure Lemma length lies limit line segment locally connected mapping Math Mathematical Mazurkiewicz means Note obtain original partition Peano curve positive possible preceding preimages Problem Program proof proved published quaternary ratio reader remaining removed representation represented result Schoenberg Section sequence Show Sierpiński similarity transformations space space-filling curve square step subset subsquares surjective ternary Theorem theory topological topological dimension triangle two-dimensional uniformly unique unit University