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CHAOS RULES! Robert L Justby lookingat the Sierpinski triangle,we can read off the rules of the game we played to produce it. CHAOS RULES! Robert L. Devaney BostonUniversity The "Classical" Chaos Game The "chaos game" and itsmultitude of variations provides a wonderful opportunity to combine elementary ideas fromgeometry, linear algebra, probability, and topology with some quite contemporarymathematics. The easiest chaos game to understand is played as follows. Startwith threepoints 2a and 2b: The and at the vertices of an equilateral triangle.Color one vertex red, Figure Sierpinski hexagon carpet. one green, and one blue. Take a die and color two sides red, two sides green, and two sides blue. Then pick any point what Other Chaos Games soever in the triangle; this is the seed. Now roll the die. Depending upon which color comes up, move the seed half the For a different example of a chaos game, put six points at distance to the similarly colored vertex. Then repeat this pro the vertices of a regular hexagon. Number them one through cedure, each timemoving the previous point half the distance six and erase the colors on the die. We change the rules a bit to the vertex whose color turnsup when the die is rolled. After here: instead of moving the point half the distance to the a dozen rolls, startmarking where these points land. appropriate vertex after each roll, we now "compress the dis When you repeat thisprocess many thousands of times, the tance by a factor of three."By thiswe mean we move the point pattern thatemerges is a surprise: it is not a "random mess," as so that the resulting distance from themoved point to the cho most first-time players would expect. Rather, the image that sen vertex is one third the original distance. We say that the unfolds is one of themost famous fractals of all, the Sierpinski compression ratio for this game is three. are no triangle shown in Figure 1.Notice that there points in Again we get a surprise: after rolling the die thousands of we the "missing" triangles in this set. This iswhy did not plot times the resulting image is a "Sierpinski hexagon" as shown the firstfew points when we rolled the die. inFigure 2a. And again we can go backwards: this image con Now here is the observation that leads to the geometry: the sists of six self-similar pieces, each of which is exactly one Sierpinski triangle consists of three self-similar pieces, each third the size of the full Sierpinski hexagon?the same num of which is exactly one half the size of the original triangle in bers we used to design the game. By theway, there ismuch terms of the lengths of the sides. These are precisely the num more to this picture thanmeets the eye at first: notice that the bers thatwe used to play the game: three vertices and move interiorwhite regions of thisfigure are all bounded by thewell half the distance to the vertex after each roll. So we can go known Koch snowflake fractal! we can backwards. Just by looking at the Sierpinski triangle, The Sierpinski triangle and hexagon show that the objects read off the rules of the game we played to produce it. that result from playing these chaos games have interesting topology. Here is an even more intriguing example of this illustration.Play the chaos game with eight vertices: four at the corners of a square and four at themidpoints of the sides. When a compression ratio of three is used, the result is the equally famous Sierpinski carpet shown inFigure 2b. Topolo gists know that this object contains a homeomorphic copy of no every planar, one-dimensional, compact, connected set, matter how complicated that set is.What thismeans is that, roughly speaking, every bounded curve, containing any num as as set can Figure V.The Sierpinski triangle. The original red, green, and ber of branch points, long the is one-dimensional, to blue vertices are located at the vertices of this image. be deformed fit in inside the Sierpinski carpet. WWW.MAA.ORG/MATHH0RIZ0NS 11 This content downloaded from 208.58.72.65 on Fri, 30 Aug 2013 20:13:24 PM All use subject to JSTOR Terms and Conditions MATH HORIZONS half the distance to that vertex when that vertex is called. For the top vertex, the rule is: firstmove the point half the distance to that vertex, and then rotate the point 90 degrees about the vertex in the clockwise direction. The result of this chaos game is shown in Figure 3a: note that there are basically three self similar pieces in the fractal, each of which is half the size of the original, but the top one is rotated by 90 degrees in the clockwise direction. Again, as before, we can go backwards and determine the rules of the chaos game that produced the and 3c: with rotations. Figure 3a, 3b, Sierpinski image. Changing the rotation at the top vertex to 180 degrees yields the image in Figure 3b. This time, the top self-similar piece is Here now is a "reverse surprise" (and also an example that rotated 180 degrees. For the fractal in Figure 3c, we rotated does not involve the name Sierpinski). Play the chaos game twentydegrees in the clockwise direction around the lower left with four vertices at the corners of a square and a compression vertex, twenty degrees in the counterclockwise direction ratio of two. After the previous examples, the result of this around the lower rightvertex, and therewas no rotation around game is?surprise!?a square. But this is not really a surprise, the top vertex. since the square consists of four self-similar subsquares, each Determining the rules of a chaos game thatproduced a cer of which is exactly one half the size of the original (in length tain image is not easy. In Figure 4 we give you the opportuni and width). While the square is not a fractal, it is indeed a self ty to tryyour hand at this.You must determine the number and similar object. An applet called Fractalina can be used to cre locations of the vertices, the compression ratio, and the rota ate similar chaos game images. It is available at the Boston tions involved in each case. University Dynamical Systems and Technology website Another great source of fun is fractalmovie-making. Once (math.bu.edu/DYSYS). you know how to create a single fractal pattern via the chaos game, you can slowly vary some of the rotations, compression Fractals ratios, or locations of the vertices to create a fractalmovie. We Clearly, self-similarity is only one component in the defini challenge our students tomake a movie that is "beautiful" tion of a fractal.A line segment and a square are self-similar where the underlying rule is hard to figure out. Our students sets, but theyare definitely not fractals. The missing ingredient often work for hours tomake these animations. Of course, here is fractal dimension: a fractal set must also have fractal beautiful here means "with a lot of symmetry," so there really dimension that exceeds the set's topological dimension. With is a lot of geometry in this activity.While a magazine may not out going into details, topological dimension is the "usual" be the best place to display a movie, several frames from the dimension of a set; it is always a nonnegative integer.Fractal movie "Dancing Sierpinski" are displayed inFigure 5. Anoth dimension gives finer informationabout the roughness or com er applet called Fractanimate is available tomake thesemovies plexity of a set. Sets like the Sierpinski triangle, hexagon, and at theBoston University Dynamical Systems and Technology carpet have intricate geometries and therefore have fractal website. A number of fractalmovies created by students are dimension larger than one, which is the topological dimension also posted at this site. of all three.For more details, consult Fractals Everywhere by Barnsley, or Fractals: A Toolkit of Dynamics Activities by Choate, Devaney, and Foster. Incidentally, many people believe that a fractal is a setwhose fractal dimension is not an integer.This is incorrect: there aremany fractals thathave inte ger fractal dimension. The Sierpinski tetrahedron (a tetrahe dral analog of the triangle) has fractal dimension two (but topological dimension one). i; ; 2b 2a 3a I Jt Rotations Now let's add rotations to themix. This iswhere the geom etryof transformationsbecomes more important.Start with the vertices of a triangle as in the case of the Sierpinski triangle. Figure 4: Challenging chaos game images. For the bottom two vertices, the rules are as before: just move 12 NOVEMBER 2004 This content downloaded from 208.58.72.65 on Fri, 30 Aug 2013 20:13:24 PM All use subject to JSTOR Terms and Conditions MATH HORIZONS AA ?k &k &k 4k & p> $r ? J? Jtiik&k^foi^ik ^bJ?& ^Bkjjfitik ^ ^ ^ 4^ ^ ^ i^^ys^k (iS^ij^k jkiSu^iik^bJ^ik fibjbk^^^^kk^ftk jf^^A J^^J^^k i^^^^Si il^^^^k Figure 5: How did we produce this fractal "movie" ? contraction.When we allow this, the output of the chaos game Probability and Linear Algebra produces a much richer collection of fractals, including not to now, all of the ratios we have used in a Up compression only fractals from geometry, but also fractals from nature. The chaos game have been the same. When these numbers given fractal fern in Figure 7 was produced using just four affine it often becomes necessary to the change, change probability transformationswhich, after some algebraic simplifications, of a certain vertex.
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