![]() With this Laplacian, the heat and wave equations can be defined on fractals. Graphs corresponding to protofractal approximations of the equilateral Sierpinski gasket. The analysis is subtle, involving some deep number theory.įIGURE 26. This could not be the Hausdorff dimension, but Lapidus (1995) showed that it is the Minkowski–Bouligard dimension. 51–53) postulated that some form of the dimension of ∂ D appears in the second term in the expansion of N(λ), therefore can be recovered from the eigenvalues. If the boundary ∂ D is a fractal, Berry (1979, pp. We remark that if in the construction of univariate fractal functions on I having the interpolation property with respect to Δ : =. Throughout this section let I denote the unit interval ⊂ R. Some of the material presented here can also be found in Ref. In this section we establish that the known Peano curves may be generated-on occasion with a slight modification-with IFSs and that they are projections of isodimensional fractal functions. ![]() Several other geometric constructions appeared in the literature, until Knopp showed that all Peano curves can be generated with a simple geometric principle. Hilbert replaced Peano’s arithmetic definition of p by a simple geometric construction. ![]() It came as a surprise that such a continuous vector-valued function p : → X exists and it proved that continuity in the component functions of p is not enough to ensure bijectivity. Space-filling curves have been known in mathematics since the latter part of the 19th century, when Peano showed that it is possible to map the interval continuously onto a compact subset of X ⊂ R 2. Massopust, in Fractal Functions, Fractal Surfaces, and Wavelets (Second Edition), 2016 9 Peano curves ![]()
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