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fractals & statistical growth 3434. Compute the density ρ by dividing the number of particles by the number ofsites available in the box (49 in our example).5. Repeat the procedure using larger and larger squares.6. Stop when the cluster is covered.7. The (box counting) fractal dimension d f is estimated from a log-log plot of thedensity ρ versus L. If the cluster is a fractal, then (13.2) tells us that ρ ∝ L d f −2 ,and the graph should be a straight line of slope d f − 2.The graph we generated had a slope of −0.36, which corresponds to a fractal dimensionof 1.66. Because random numbers are involved, the graph you generate willbe different, but the fractal dimension should be similar. (Actually, the structure ismultifractal, and so the dimension varies with position.)13.8 Fractal Structures in aBifurcation Graph (Problem 7)Recollect the project involving the logistics map where we plotted the values ofthe stable population numbers versus the growth parameter µ. Take one of thebifurcation graphs you produced and determine the fractal dimension of differentparts of the graph by using the same technique that was applied to the coastline ofBritain.13.9 Fractals from Cellular AutomataWe have already indicated in places how statistical models may lead to fractals. There isa class of statistical models known as cellular automata that produce complex behaviorsfrom very simple systems. Here we study some.Cellular automata were developed by von Neumann and Ulam in the early 1940s(von Neumann was also working on the theory behind modern computers then).Though very simple, cellular automata have found applications in many branchesof science [Peit 94, Sipp 96]. Their classic definition is [Barns 93]:A cellular automaton is a discrete dynamical system in which space, time, and thestates of the system are discrete. Each point in a regular spatial lattice, called a cell, canhave any one of a finite number of states, and the states of the cells in the lattice areupdated according to a local rule. That is, the state of a cell at a given time dependsonly on its won state one time step previously, and the states of its nearby neighborsat the previous time step. All cells on the lattice are updated synchronously, and sothe state of the entice lattice advances in discrete time steps.The program CellAut.java given on the CD creates a simple 1-D cellular automatonthat grows on your screen. A cellular automaton in two dimensions consistsof a number of square cells that grow upon each other. A famous one, invented by−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 343