Springer-Verlag, 1994 - 193 pages
The subject of space-filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by Peano. Cantor, Hilbert, Moore, Knopp, Lebesgue, and Polya are among the prominent mathematicians who have contributed to the field. However, there have been no comprehensive treatments of the subject since Siepinsky's in 1912. Cantor showed in 1878 that the number of points on an interval is the same as the number of points in a square (or cube, or whatever), and in 1890 Peano showed that there is indeed a continuous curve that continuously maps all points of a line onto all points of a square, though the curve exists only as a limit of very convoluted curves. This book discusses generalizations of Peano's solution and the properties that such curves must possess and discusses fractals in this context. The only prerequisite is a knowledge of advanced calculus.
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Hilberts SpaceFilling Curve
Differentiability of the Hilbert Curve
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accumulation point adjacent approximating polygons attractor set beginning point binary Cantor curve Cantor set Cauchy sequence Cesaro Chapter closed intervals compact set compact subset connectedness construction continuous image converges coordinate functions curve of Section defined Definition denote diagonal differentiable endpoint exit point finite four congruent fractal geometric Hahn Hence Hilbert curve integer irz ji isosceles iterated function system jLZZ jL joins Jordan curve Knopp Koch curve Lebesgue measure Lebesgue's space-filling curve Lemma line segment mapping Math Mathematical n-dimensional Netto nodal points non-empty obtain open intervals open set Osgood curve partition Peano curve Polya preimages Program proof quaternary removed represented Sagan Schoenberg curve set in Fig Show sidelength Sierpinski triangle Sierpinski-Knopp curve similarity dimension similarity transformations space space-filling curve square step stochastically independent subcubes subintervals subsquares subtriangles surjective ternary representation Theorem theory three-dimensional topological dimension two-dimensional Lebesgue measure uniformly unique University