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328 chapter 1330010,000 points20010000 100 200 300Figure 13.1 Left: A Sierpiński gasket containing 10,000 points constructed as a statisticalfractal. Each filled part of this figure is self-similar. Right: A Sierpiński gasket constructed bysuccessively connecting the midpoints of the sides of each equilateral triangle. (A–C) The firstthree steps in the process.13.2.1 Sierpiński ImplementationWrite a program to produce a Sierpiński gasket. Determine empirically the fractaldimension of your figure. Assume that each dot has mass 1 and that ρ = CL α .(Youcan have the computer do the counting by defining an array box of all 0 values andthen change a 0 to a 1 when a dot is placed there.)13.2.2 Assessing Fractal DimensionThe topology in Figure 13.1 was first analyzed by the Polish mathematicianSierpiński. Observe that there is the same structure in a small region as there isin the entire figure. In other words, if the figure had infinite resolution, any partof the figure could be scaled up in size and would be similar to the whole. Thisproperty is called self-similarity.We construct a regular form of the Sierpiński gasket by removing an invertedequilateral triangle from the center of all filled equilateral triangles to create thenext figure (Figure 13.1 right). We then repeat the process ad infinitum, scaling upthe triangles so each one has side r =1after each step. To see what is unusual aboutthis type of object, we look at how its density (mass/area) changes with size and−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 328