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FractionalGeometry-Chap2 - Chapter 2 Iterated function...

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Chapter 2 Iterated function systems: fractals as limits Often a Frst exposure to fractals consists of googling “fractal geometry,” Fnding 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 Frst question is “How do these rules make that picture?” The answer to that question is the Frst topic of this chapter, if an I±S 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 ±ig. 2.1. This shape, the Sierpinski gasket, is one of the simplest examples of a self-similar fractal. The middle image, a magniFcation of a small portion of the left side, is indistinguishable from the left. The right is a magniFcation of a portion of the middle. In our minds, this process can be continued forever. ±igure 2.1: Successive magniFcations of portions of the gasket. The reappearance, under increasingly high magniFcation, of copies of the whole shape, suggests two things. ±irst, some sort of limiting process is in- volved here. Second, it is not the limiting process familiar from calculus, where under magniFcation a smooth curve approaches its tangent line. Something dif- ferent is happening here: the level of complexity remains about constant under magniFcation. 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 b x y B = b r cos( θ ) s sin( ϕ ) r sin( θ ) s cos( ϕ ) Bb x y B + b e f B (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 2 ), d p T b x 1 y 1 B ,T b x 2 y 2 B P t · d p b x 1 y 1 B , b x 2 y 2 B P , (2.2) and for any number s < t , d p T b x 1 y 1 B b x 2 y 2 B P > s · d p b x 1 y 1 B , b x 2 y 2 B P for at least one pair of points ( x 1 1 ) and ( x 2 2 ). For example, if T ( x,y ) = ( x/ 2 ,y/ 2), then d p T b x 1 y 1 B b x 2 y 2 B P = r ± x 1 2 x 2 2 ² 2 + ± y 1 2 y 2 2 ² 2 = 1 2 d p b x 1 y 1 B , b x 2 y 2 B P for any pair of points. Consequently, this T is a contraction with contraction factor 1 / 2. A slightly more di±cult example is given in Prob. 2.1.2.
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FractionalGeometry-Chap2 - Chapter 2 Iterated function...

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