## Abstract

So far, we have discussed two extreme ends of fractal geometry. We have explored fractal monsters, such as the Cantor set, the Koch curve, and the Sierpinski gasket; and we have argued that there are many fractals in natural structures and patterns, such as coastlines, blood vessel systems, and cauliflowers. We have discussed features, such as self-similarity, scaling properties, and fractal dimensions shared by those natural structures and the monsters; but we have not yet seen that they are close relatives in the sense that maybe a cauliflower is just a ‘mutant’ of a Sierpinski gasket, and a fern is just a Koch curve ‘let loose’. Or phrased as a question, is there a framework in which a natural structure, such as a cauliflower, and an artificial structure, such as a Sierpinski gasket, are just examples of one unifying approach; and if so, what is it? Believe it or not, there is such a theory, and this chapter is devoted to it. It goes back to Mandelbrot’s book, *The Fractal Geometry of Nature*, and a beautiful paper by the Australian mathematician Hutchinson.^{2} Barnsley and Berger have extended these ideas and advocated the point of view that they are very promising for the encoding of images.^{3} In fact, this will be the focus of the appendix on image compression.

## Keywords

Target Image Fractal Geometry Hausdorff Distance Initial Image Encode Image## Preview

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## Reference

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