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Contraction Mapping & the Collage Theorem CONTRACTION MAPPING & THE COLLAGE THEOREM Porter 1 Abstract The Collage Theorem was given by Michael F. Barnsley who wanted to show that it was possible to reconstruct images using a set of functions. This idea was to reduce the space needed on a computer to save an image or video. An image/video file can be quite large, while saving only a set of functions that can duplicate the original image/video frees up extra space – and during the late 80's- early 90's when computers were still somewhat primitive, this was an important development. This paper introduces the idea of fractals with an emphasis on the Collage Theorem, using a special case of the Contraction Mapping Theorem. The Collage theorem says that we can approximate an image by using a set of iterated function systems with a specific attractor that will yield the desired image regardless of the initial set. CONTRACTION MAPPING & THE COLLAGE THEOREM Porter 2 Fractal Geometry – Contraction Mapping & The Collage Theorem “Why am I learning this?” “Where am I ever going to use this outside of school?” These are questions that every student of mathematics has asked at some point in his or her life. Often, this question is valid, because what we work with in the classroom, generally, are “nice” equations, graphs, problems, etc. Rarely in real life do we find ourselves presented with a perfect circle, a single-variable equation, or a straight line. For example, when we travel, we can find on a map roughly how many miles there are between where we are and our destination, but the question is “Is there a perfectly straight road from here to there?” Unless you're driving to the end of the block, generally the answer to will be no. We can translate this idea from the distance between two points to the perimeter of a shape. Let's use the example of measuring the coastline of Australia. Most maps have a key with a scale, so let's use a 500-kilometer measuring stick. By dividing the coastline into 500-kilometer chunks, we get a total of 13,000 kilometers. But what happens if we use, say, a 100-kilometer measuring stick? The length should not change very much, right? Well, with a that smaller interval, the coastline is now 15,500 kilometers, an almost 20% increase! Where did the extra 2,500 kilometers come from? What would happen with an even smaller measuring stick? The CIA World Factbook website gives the length of Australia's coastline as 25,760 kilometers – which is a 66% increase from the second measurement. What happens is that as the intervals decrease in length, the length of the coastline will continue to increase to infinity because there are always going to be smaller crevices to be measured. This is called the Coastline Paradox, and it is an example of an application of fractals. The definition of the word “fractal” varies slightly depending on who gives it. The word, who we have thanks to Beniot B. Mandelbrot, comes from the Latin word frāct or frāctus, meaning “broken” or “uneven.” One of the more basic definitions of “fractal” is “the roughness of an object or space.” From the coastline example, we can see how these ideas apply. However, we are still missing one of the most important pieces of what it means to be a fractal – and that word is “self-similarity.” CONTRACTION MAPPING & THE COLLAGE THEOREM Porter 3 Wolfram MathWorld defines a fractal as “an object that displays self-similarity, in a somewhat technical sense, on all scales.” According to this definition, these objects do not need to match perfectly on every scale, but that similar structures are present on all scales. However, self-similarity, in and of itself, is not a sufficient definition. What makes a fractal is its dimension, a fractional (or fractal) dimension. We begin by recalling what our idea of dimension is from our middle-school/high-school years, which is that a point will have dimension 0, a line dimension 1, a circle/square/etc dimension 2, a sphere/cube/etc dimension 3, and so on. There is a formula that allows us to calculate the dimension of certain objects, which says: log(number of self −similar pieces) dimension= (1) log(magnification factor) While we can show this for any shape, the 2-dimensional square, arguably, gives the best depiction of this formula. Figure 1: Defining Dimension # of self-similar pieces: 1 # of self-similar pieces: 4 # of self-similar pieces: 9 Magnification Factor: 1 Magnification Factor: 2 Magnification Factor 3 CONTRACTION MAPPING & THE COLLAGE THEOREM Porter 4 Following from the figure above, we see that when a square is divided into 1, 4, and 9 self-similar pieces we can reconstruct the original square by multiplying any one of those self-similar pieces by a magnification factor of 1, 2, and 3 (respectively). So using (1) we get log(1)2 2log(1) log(4) log(2)2 log(9) log(3)2 = = 2 = = 2 = = 2 log(1) log(1) log(2) log(2) log(3) log(3) We can now generalize this to a square divided into N 2 self-similar pieces with a magnification factor of N. From this example, we can imagine that a line divided into N self-similar pieces will have a magnification factor of N, so log(N ) = 1 log(N ) . Also, a cube divided into N 3 self-similar pieces and magnification factor of N will have dimension log(N )3 = 3 log(N ) which confirms our knowledge of dimension. But this begs the question: What dimension does a fractal have? Intuition tells us that fractal (or fractional) dimension should be between integers. It is perhaps easier to see this in an example. The Classical Cantor Set, or Cantor Comb, is one such example of self-similarity and fractional dimension. The iterations of this set say to translate down the previous level and remove the middle third of each piece (See Figure 2 below). Figure 2: The Cantor Set I 0 I 1 I 2 I 3 I 4 I 5 CONTRACTION MAPPING & THE COLLAGE THEOREM Porter 5 With interval for I0 being [0,1], we see that 1 2 3 I =[0, ]∪[ ] 1 3 3, 3 1 2 3 6 7 8 9 I =[0, ]∪[ , ]∪[ , ]∪[ , ] 2 9 9 9 9 9 9 9 ⋮ I N =take away the middle open third for each interval in I N −1 ⋮ On this scale, after the fifth iteration there is no visual difference from the previous iteration; however, this process continues on to infinity. We see that if we magnify, say, the first piece of I1 by a factor of 3 we will get the original line. The Cantor set is not the entire picture, but instead is the limit C=∩∞ I . of this picture – which, in this case, is n=1 n So as we continue to take the middle third out of each step above, we will get smaller and smaller pieces to the point where this fractal will contain no intervals, but will have infinitely many points around each point. This implies C is a perfect set, which means it is a closed subset of R where every point of the set is a limit point. Our intuition tells us that this particular fractal should have a dimension between that of a point (zero) and a line (one). Using formula (1) to calculate the dimension of the Cantor Set, we get log(2) ≈0.63 log(3) and our intuition holds true. This is the basis of fractal geometry – if you take any magnification of a section of the whole fractal, it will look very similar to, if not exactly like, the larger image, and its dimension is between positive integers. Before we get into the Contraction Mapping Theorem or the Collage Theorem, we need to look at several defninitions. Recall from Real Analysis, Definition 1. A metric space is a set S with a global distance funtion (the metric d) that, for every two points x,y in S, gives the distance between them as a nonnegative real number. A metric space must also satisfy: i. d(x,y) = 0 iff x=y CONTRACTION MAPPING & THE COLLAGE THEOREM Porter 6 ii. d(x,y) = d(y,x) iii. The triangle inequality d(x,y) + d(y,z) ≥ d(x,z). (Wolfram Alpha). The idea of compactness, as well as completeness, will be vital to the rest of this paper, so we will give the definitions here. For completeness, we first need to define what it means for a sequence to be Cauchy. Definition 2. A sequence a1, a2 , … is a Cauchy sequence if the metric d(am,an) satisfies (Wolfram Alpha). lim min (m , n)→ ∞ [d (am ,an)]=0. See Figure 3 below for a visual example of a Cauchy and non-Cauchy sequence. Figure 3: Cauchy vs non-Cauchy Sequences Cauchy Sequence non-Cauchy Sequence Images from: http://en.wikipedia.org/wiki/Cauchy_sequence Definition 3. A metric space, X, is complete (or a complete metric space) if every Cauchy sequence is convergent. (Wolfram Alpha). lim n →∞ Sn=S Definition 4. A sequence Sn converges to the limit S if, for any ε > 0, there exists an N such that |Sn – S| < ε for n > N. If Sn does not converge, it is said to diverge. (Wolfram Alpha). Definition 5. A metric space, X, is compact if every open cover of X has a finite subcover. (Wolfram Alpha). Another definition of compact spaces is that they are closed and bounded.
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