Chapter 3Fractals3.1 Examples of FractalsA fractal is a geometric object which is self-s imila r, with structure at all levelsof magnification. Rather than try to tighten down on this definition, it is moreuseful to generate some examples.Example 1: In Fig. 3.1(a) we show an interval of le ngth 1. In going from(a) to (b) we remove the middle half of this interval. This leaves two intervals,each of equal length14. This first step is the generating step. The second step,from (b) to (c), is a repeated application of the generating step. We remove themiddle half of each of the two subintervals. This leaves 4 = 22intervals, all ofequal length116= (14)2. We continue in the obvious way. At the nthstep wehave 2nintervals, each of length (14)n. This process continues forever.3.2 Fractal DimensionA convenient way to define the dimension of a geometric object is to c over itwith boxes whose edge length is ǫ (i.e., small). In Fig. 3.2 we show how thisprocess works for some familiar geometric objects: two points, a smooth curve,and a simple area. In these three examples, the number of boxes, N(ǫ), re quiredto cover the geometric objects behaves like:Geometric Object N(ǫ)Points P ∼ K/ǫ0Smooth Curves C ∼ K/ǫ1Simple Areas A ∼ K/ǫ2where K is an unimportant constant. The number of boxes required to coverthe geometric object behaves like ǫ−d, where d is the dimension of the object.We can turn this observation around, and use this type of computation to definethe dimension of peculiar objects.12 CHAPTER 3. FRACTALSFigure 3.1: A middle ha lf fractal is constructed by repeated application of thefirst, generating step. The middle half of the interval of length one (a) is removed(b). At each succeeding step, the middle half of each interval is removed. Thiscontinues forever.3.2.1 Definition of Dimension (Box Counting)Definition: We define the dimension, d, of a geometric in terms of ǫ and N(ǫ)as follows:d = limǫ→0log N(ǫ)log(1/ǫ)(3.1)Example 1 (Continued): At the nthstep of the generation process ofthe middle half fractal, there are 2nboxes, each of length (14)n. The fractaldimension is therefored = limn→∞log(2n)log(1 /14)n=log 2log 4=123.2.2 Dimension of the Middle 1/p FractalExample 2: We can gener alize this to1pfractals. These are fractals in whichthe middle1pof the interval is removed in the generating step. Each intervalobtained during the generating step has length ǫ =p−12p. Thend =log 2log(2pp−1)3.3. TWO SCALE FRACTALS 3For p = 2, 3, 4, · · · these dimensions arep Dimension2123 log(2)/ log(3)4 log(2)/ log(8/3)5 log(2)/ log(5/2)We plot the fractal dimension, d, as a function of f = 1/p in Fig. 3.3.3.2.3 Direct Product Spaces, Direct Sum DimensionsFractals in higher dimensional spaces can be built up systematically as directproducts of fractals in lower dimensional spaces. If a fractal is a direct productof two fractals with dimensions d1and d2, then its dimension is the (direct) sumof the dimensions of the two fractals:d = d1+ d2As an example, a frac tal in the plane can be constructed as the direct productof the middle half fractal along each of the axes. The dimension of this directproduct frac tal is thend =12+12= 1It is clear from this example that fractals can have integer dimension.3.3 Two Scale Fractals3.3.1 ConstructionAnother way to build up fractals is shown in Fig. 3.4. In the generatingstep, an interval of length 1 is re produced twice, once reduced by the scalefactor λ1, the other time re duced by the scale factor λ2. These reduced in-tervals are shown on the left and right in Fig. 3.4(b). The process is re-peated in the second generation. This pro duces four subintervals, of lengthsλ21, λ1λ2, λ2λ1, λ22, proceeding from left to right. In the third generation thedistribution is λ31, 3λ21λ2, 3λ1λ22, λ32. You can see the binomial distribution oflengths emerg ing from this process, which of course continues forever, as before.3.3.2 DimensionThe dimension of this two scale fractal can be computed as follows. Assumethat at level k, Nk(ǫ) boxes of length ǫ are required to cover the 2kintervals.At the next level k + 1, the structure on the left is a scaled down version of theentire structure at level k. Therefore the number of boxes of length ǫ required4 CHAPTER 3. FRACTALSFigure 3.2: (Ott, p. 70)(a) Two boxes cover two points, no matter how s mallthe boxes are. (b) The number of boxes required to cover a smooth curve isproportional to the length of the curve, and inversely proportional to the boxsize, tha t is, N(ǫ) ∼ 1/ǫ. (c) The number of boxes required to cover the areabehaves like N(ǫ) ∼ 1/ǫ2.3.3. TWO SCALE FRACTALS 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1f = 1/p00.10.20.30.40.50.60.70.80.91Fractal DimensionDimension of Middle 1/p FractalFigure 3.3: The dimension of a middle 1/p fractal is plo tted as a function off = 1/p.6 CHAPTER 3. FRACTALSFigure 3.4: Construction of a two scale frac tal proceeds as shown. Each of thetwo subintervals in the generating stage (a) → (b) is a replica of the original,reduced in scale by the scale factors λ1and λ2. If λ1is negative, −1 < λ1< 0,the orientation of an interval is r eversed when scaled down by λ1.to cover the left ha lf of the structure at level k + 1 is equal to the number oflarger b oxes (of leng th ǫ /λ1) required to cover the structure a t level k:N(k+1)left(ǫ) = Nk(ǫ/λ1)A similar argument holds for the half on the right at level k + 1. Thus we haveNk+1(ǫ) = N(k+1)left(ǫ) + N(k+1)right(ǫ) = Nk(ǫ/λ1) + Nk(ǫ/λ2)If we assume, as us ual, that N(ǫ) ∼ Kǫ−d, then Kǫ−d= K(ǫ/λ1)−d+K(ǫ/λ2)−dleads directly to a simple e xpression defining the fractal dimension d:λd1+ λd2= 1Fractals obtained from three or more scaling transformations in the generatingstep have dimensions determined by similar expressions.3.3.3 Feigenbaum FractalThe Feigenbaum attractor is the fractal which exists at the accumulation pointof the period doubling cas c ade. Figure 3.5(a) shows the locations of orbits o fperiods 1, 2, 4, 8, · · ·. The locations suggest that a scaling exists. This scalingis reinforced in Fig. 3-5(b), which shows the locatio ns of points in the orbits o f2n. These occur alterna tely in the left and the right halves of the return plot,and seem to o bey s c aling 1/α2on the left and 1/α on the right.We now describe how to view the Feigenbaum attractor as a two scale fractal.Begin by c onnecting the two points in the period two orbit by a line. Next,3.3.