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Extra info for Algorithmic Geometry [auth. unkn.]

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I. . , , . I . Figure 8. 5). lie within M2 in these cases. 1 on each axis. ·More Mandelbrot and Julia Sets The algorithms described above can easily be adapted to calculate the Mandelbrot and Julia sets (denoted by Mn and I n , respectively) for the general mapping Z -+ zn + c. For the present n is restricted to 3, 4 and 5, with Zo = (0 , 0) . Figures 9, 10 and 11 show the canonical Mandelbrot sets M 3 , M4 and Ms. 5]. Together with Figure 1, these figures provide an intriguing sequence.

But to examine the dimension of a growing object, the logic already developed can be extended in an alternate although complementary way. In Figure 2b we show the development of a fractal object across four scales, from a unit of development when k = 0 to a structure some 27 times as large when k = 3. In Figure 2b we grow the fractal from its basic seed, which now has a linear dimension ~, to a cluster of seeds which spans the whole of the fixed space with linear dimension L, within which growth takes place.

For a fixed mapping, different choices of Zo will give rise to distorted versions of the 'canonical' set obtained for Zo = (0,0) [Dewd87J. 5, MAXIT = 200 and critical magnitude = 10. In the computational algorithm the square of the critical magnitude is used in order to save repeated use of the SQRT function. In the following black and white figures both M and N are set to 512. The computational time required is typically three quarters of an hour on a SUN 3/50 workstation. Figure 1 shows the canonical Mandelbrot set M2 (0, 0) as calculated using the parameter settings given above.

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