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Unformatted text preview: sequence of cat images in Fig. 2.3 suggests the general convergence of the
determinstic IFS algorithm. Look at the picture; what more do you need to
know? Again and again, Mandelbrot said that one of the main lessons of fractal geometry is the importance of the eyes to science. This is surely true for
formulating questions, for guiding our curiosity, but absolute certainty requires
careful arguments. For example, in Ch 7 we see many pictures suggesting that
the Mandelbrot set is surrounded by a halo of many – in fact, inﬁnitely many
– small copies of the Mandelbrot set, to all appearances, Mandelbrot islands.
But Douady and Hubbard proved that despite all the visual evidence, the Mandelbrot set is connected. Each putative island is attached to the main body of
the Mandelbrot set by an isthmus of truly tiny Mandelbrot sets, Mandelbrot
noseeums. Cats-into-gasket looks clear, but we really need to prove that our
eyes are not fooled.
We have the tools to do this. The Hausdorﬀ metric is how we measure...
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