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332 chapter 13copy of a frond, and the fractal obtained with (13.10) is still self-affine, yet with adimension that varies in different parts of the figure.13.3.3 Self-affinity in Trees ImplementationNow that you know how to grow ferns, look around and notice the regularity intrees (such as in Figure 13.2 right). Can it be that this also arises from a self-affinestructure? Write a program, similar to the one for the fern, starting at (x 1 ,y 1 )=(0.5, 0.0) and iterating the following self-affine transformation:⎧(0.05x n , 0.6y n ),10% probability,(0.05x n , −0.5y n +1.0),10% probability,⎪⎨ (0.46x n − 0.15y n , 0.39x n +0.38y n +0.6), 20% probability,(x n+1 ,y n+1 )=(0.47x n − 0.15y n , 0.17x n +0.42y n +1.1), 20% probability,(0.43x n +0.28y n , −0.25x n +0.45y n +1.0),⎪⎩(0.42x n +0.26y n , −0.35x n +0.31y n +0.7),20% probability,20% probability.(13.13)13.4 Ballistic Deposition (Problem 3)There are a number of physical and manufacturing processes in which particles aredeposited on a surface and form a film. Because the particles are evaporated froma hot filament, there is randomness in the emission process yet the produced filmsturn out to have well-defined, regular structures. Again we suspect fractals. Yourproblem is to develop a model that simulates this growth process and compareyour produced structures to those observed.13.4.1 Random Deposition AlgorithmThe idea of simulating random depositions was first reported in [Vold 59], whichincludes tables of random numbers used to simulate the sedimentation of moistspheres in hydrocarbons. We shall examine a method of simulation [Fam 85] thatresults in the deposition shown in Figure 13.3. Consider particles falling onto andsticking to a horizontal line of length L composed of 200 deposition sites. All particlesstart from the same height, but to simulate their different velocities, we assumethey start at random distances from the left side of the line. The simulation consistsof generating uniform random sites between 0 and L and having a particle stick tothe site on which it lands. Because a realistic situation may have columns of aggregatesof different heights, the particle may be stopped before it makes it to the line,or it may bounce around until it falls into a hole. We therefore assume that if thecolumn height at which the particle lands is greater than that of both its neighbors,−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 332