Showing papers on "Fractal dimension published in 1981"
TL;DR: In this paper, it was shown that the Hausdorf-besicovitch dimension of a linear fractal function may vary between 1 (completely differentiate) and 2 (so rough and irregular that it effectively takes up the whole of a 2D topological space).
Abstract: Mandelbrot1 has introduced the term ‘fractal’ specifically for temporal or spatial phenomena that are continuous but not difierentiable, and that exhibit partial correlations over many scales. The term fractal strictly defined refers to a series in which the Hausdorf–Besicovitch dimension exceeds the topological dimension. A continuous series, such as a polynomial, is differentiable because it can be split up into an infinite number of absolutely smooth straight lines. A non-differentiable continuous series cannot be so resolved. Every attempt to split it up into smaller parts results in the resolution of still more structure or roughness. For a linear fractal function, the Hausdorf–Besicovitch dimension D may vary between 1 (completely differentiate) and 2 (so rough and irregular that it effectively takes up the whole of a two-dimensional topological space). For surfaces, the corresponding range for D lies between 2 (absolutely smooth) and 3 (infinitely crumpled). Because the degree of roughness of spatial data is important when trying to make interpolations from point data such as by least-squares fitting or kriging2, it is worth examining them beforehand to see if the data contain evidence of variation over different scales, and how important these scales might be. Mandelbrot's work1 suggests that the fractal dimensions of coastlines and other linear natural phenomena are of the order of D = 1.2–1.3, implying that long range effects dominate. I show here that published data on many environmental variables suggest that not only are they fractals, but that they may have a wide range of fractal dimensions, including values that imply that interpolation mapping may not be appropriate in certain cases.
562 citations
TL;DR: Estimating surface and volume density of subcellular membrane systems at different magnifications yield different results, which may explain, at least in part, the large discrepancy in the estimates of the surface of cytomembranes found in the literature.
Abstract: Estimating surface and volume density of subcellular membrane systems at different magnifications yield different results. As the magnification is increased from x 18,000 to x 130,000 the estimates of surface density of endoplasmic reticulum and inner mitochondrial membranes increase by a factor of 3, whereas that for outer mitochondrial membranes increase only by 20%. The estimate of volume density of endoplasmic reticulum also increases by a factor of 3. No further increase is observed at magnifications above x 130,000 which is therefore called critical magnification. The findings are interpreted on the basis of the concept of fractals proposed by Mandelbrot, and the fractal dimensions of the membrane systems considered are estimated. This can lead to the derivation of resolution correction factors which permit measurements obtained at any magnification to be converted to estimates at critical magnification. These findings may explain, at least in part, the large discrepancy in the estimates of the surface of cytomembranes found in the literature.
145 citations
TL;DR: Simulations of chemical reactions in small systems show that placing the reactive site at a central location may be favored at an early stage of growth but, as the system evolves in size, a location on the boundary becomes favored.
Abstract: The effect of dimensionality and spatial extent on the dynamics of an irreversible reaction confined to a finite system was studied by a Monte Carlo simulation. Stochastic flows on surfaces of integral and fractal dimensions and the consequences of reducing the dimensionality of the reaction space are described. As regards the timing and efficiency of chemical reactions in small systems, our simulations show that placing the reactive site at a central location may be favored at an early stage of growth but, as the system evolves in size, a location on the boundary becomes favored. The possible relevance of these calculations to the problem of morphogenesis is brought out.
11 citations