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FractionalGeometry-Chap2

FractionalGeometry-Chap2 - Chapter 2 Iterated function...

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Chapter 2 Iterated function systems: fractals as limits Often a first exposure to fractals consists of googling “fractal geometry,” finding a program to generate fractals, selecting some example from a preset menu, and clicking the RUN button. The image drifts into being, dot after dot dancing across the screen in furious pattern. If the rules generating this fractal can be viewed with the program, the first question is “How do these rules make that picture?” The answer to that question is the first topic of this chapter, if an IFS program had the highest google rank. Before starting this relatively long analysis, a moment’s consideration of a self-similar fractal suggests a general direction for the proof. Look at the fractal shown in the left of Fig. 2.1. This shape, the Sierpinski gasket, is one of the simplest examples of a self-similar fractal. The middle image, a magnification of a small portion of the left side, is indistinguishable from the left. The right is a magnification 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 in- volved here. Second, it is not the limiting process familiar from calculus, where under magnification a smooth curve approaches its tangent line. Something dif- ferent is happening here: the level of complexity remains about constant under magnification. So we must determine what sort of limit produces fractals. 25
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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 written as T bracketleftbigg x y bracketrightbigg = bracketleftbigg r cos( θ ) s sin( ϕ ) r sin( θ ) s cos( ϕ ) bracketrightbiggbracketleftbigg x y bracketrightbigg + bracketleftbigg e f bracketrightbigg (2.1) That is, any real 2 × 2 matrix can be expressed as the 2 × 2 in eq (2.1). (See Prob. 2.1.1.) Let d ( , ) denote the Euclidean distance. A transformation T is a d -contraction with contraction factor t , 0 t< 1, if for all points ( x 1 ,y 1 ) and ( x 2 ,y 2 ), d parenleftBigg T bracketleftbigg x 1 y 1 bracketrightbigg ,T bracketleftbigg x 2 y 2 bracketrightbigg parenrightBigg t · d parenleftBigg bracketleftbigg x 1 y 1 bracketrightbigg , bracketleftbigg x 2 y 2 bracketrightbigg parenrightBigg , (2.2) and for any number s<t , d parenleftBigg T bracketleftbigg x 1 y 1 bracketrightbigg ,T bracketleftbigg x 2 y 2 bracketrightbigg parenrightBigg >s · d parenleftBigg bracketleftbigg x 1 y 1 bracketrightbigg , bracketleftbigg x 2 y 2 bracketrightbigg parenrightBigg for at least one pair of points ( x 1 ,y 1 ) and ( x 2 ,y 2 ).
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