FractionalGeometry-Chap2

In our minds this process can be continued forever

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Unformatted text preview: tion of a portion of the middle. In our minds, this process can be continued forever. Figure 2.1: Successive magnifications of portions of the gasket. The reappearance, under increasingly high magnification, of copies of the whole shape, suggests two things. First, some sort of limiting process is involved here. Second, it is not the limiting process familiar from calculus, where under magnification a smooth curve approaches its tangent line. Something different is happening here: the level of complexity remains about constant under magnification. So we must determine what sort of limit produces fractals. 25 26 CHAPTER 2. ITERATED FUNCTION SYSTEMS 2.1 Iterated function system formalism Based on work of Mandelbrot [102] and Hutchinson [81], and popularized by Barnsley [5], iterated function systems are a formalism for generating fractals and for compressing images. First we recall some background on transformations of the plane. In the plane, a general linear transformation plus translation can be writ...
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