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fractals & statistical growth 329then apply (13.2) to determine its fractal dimension. Assume that each triangle hasmass m and assign unit density to the single triangle:ρ(L = r) ∝ M r 2 = m r 2def= ρ 0 (Figure 13.1A)Next, for the equilateral triangle with side L =2, the densityρ(L =2r) ∝(M =3m)(2r) 2 =34mr 2 = 3 4 ρ 0 (Figure 13.1B)We see that the extra white space in Figure 13.1B leads to a density that is 3 4that ofthe previous stage. For the structure in Figure 13.1C, we obtain( ) 2(M =9m)ρ(L =4r) ∝(4r) 2 = (34) 2 m 3r 2 = ρ 0 . (Figure 13.1C)4We see that as we continue the construction process, the density of each newstructure is 3 4that of the previous one. Interesting. Yet in (13.2) we derived thatρ ∝ CL d f −2 . (13.4)Equation (13.4) implies that a plot of the logarithm of the density ρ versus thelogarithm of the length L for successive structures yields a straight line of slopeAs applied to our problem,d f =2+d f − 2= ∆ log ρ∆ log L . (13.5)∆ log ρ(L)∆ log L =2+log 1 − log 3 4log1 − log 2≃ 1.58496. (13.6)As is evident in Figure 13.1, as the gasket grows larger (and consequently moremassive), it contains more open space. So even though its mass approaches infinityas L →∞, its density approaches zero! And since a 2-D figure like a solidtriangle has a constant density as its length increases, a 2-D figure has a slopeequal to 0. Since the Sierpiński gasket has a slope d f − 2 ≃−0.41504, it fills space toa lesser extent than a 2-D object but more than a 1-D object does; it is a fractal withdimension ≤1.6.13.3 Beautiful Plants (Problem 2)It seems paradoxical that natural processes subject to chance can produce objectsof high regularity and symmetry. For example, it is hard to believe that somethingas beautiful and graceful as a fern (Figure 13.2 left) has random elements in it.Nonetheless, there is a clue here in that much of the fern’s beauty arises from thesimilarity of each part to the whole (self-similarity), with different ferns similar butnot identical to each other. These are characteristics of fractals. Your problem is todiscover if a simple algorithm including some randomness can draw regular ferns.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 329