p WHY THE KOCH CURVE LIKE 2 XANDA KOLESNIKOW (SID: 480393797) 1. Introduction This essay aims to elucidate a connection between fractals and irrational numbers, two seemingly unrelated math- ematical objects. The notion of self-similarity will be introduced, leading to a discussion of similarity transfor- mations which are the main mathematical tool used to show this connection. The Koch curve and Koch island (or Koch snowflake) are thep two main fractals that will be used as examples throughout the essay to explain this connection, whilst π and 2 are the two examples of irrational numbers that will be discussed. Hopefully,p by the end of this essay you will be able to explain to your friends and family why the Koch curve is like 2! 2. Self-similarity An object is self-similar if part of it is identical to the whole. That is, if you are to zoom in on a particular part of the object, it will be indistinguishable from the entire object. Many objects in nature exhibit this property. For example, cauliflower appears self-similar. If you were to take a picture of a whole cauliflower (Figure 1a) and compare it to a zoomed in picture of one of its florets, you may have a hard time picking the whole cauliflower from the floret. You can repeat this procedure again, dividing the floret into smaller florets and comparing appropriately zoomed in photos of these. However, there will come a point where you can not divide the cauliflower any more and zoomed in photos of the smaller florets will be easily distinguishable from the whole cauliflower. Hence, we can see that the cauliflower is not perfectly self-similar. This means that in order for an object to be perfectly self-similar, it must be able to be divided infinitely, and at each step of division it must look identical to the whole. Although the Standard Model of Matter renders this impossible in the physical world (because all matter is comprised of indivisible elementary particles), we can invent theoretical mathematical objects which have this property. For example, Sierpinski's triangle is an example of a fractal that is perfectly self-similar (Figure 1b). What I mean by this is, if you choose to zoom in anywhere on the triangle, at every stage of zooming, the zoomed version will look like the entire fractal. This property motivates a limit definition of fractals which will be discussed later in the essay. (a) [1] (b) Figure 1. Examples of self similarity. A cauliflower (a) is an example of a self-similar object found in nature whilst Sierpinski's triangle (b) is a theoretical mathematical object that is perfectly self-similar. Date: June 2, 2019. 1 2 XANDA KOLESNIKOW (SID: 480393797) 2.1. Similarity transformations. Self-similarity extends the idea of similarity. Two shapes can be described as similar if corresponding angles are equal and corresponding sides are in the same proportion to one another. A similarity transformation is a transformation that changes a shape to a similar version of itself, and can comprise of scaling, rotation and translation. For all points (x; y) in a two dimensional shape, we can describe these three transformations as follows: ( the shape is enlarged if s > 1 • Scaling: Ss(x; y) = (sx; sy), where : the shape shrinks if s < 1 • Rotation: Rθ(x; y) = (x cos θ − y sin θ; x sin θ + y cos θ): • Translation: T(u;v)(x; y) = (x + u; y + v): We can then compose these transformations to arrive at a general similarity transformation H, H(x; y) = (T(u;v) ◦ Rθ ◦ Ss)(x; y) = (sx cos θ − sy sin θ + u; sx sin θ + sy cos θ + v): S2 R30° T(0.5,0.25) Figure 2. Similarity transformations applied to a triangle. Composition of similarity transformations corresponding to ◦ scaling by 2 (S2), anti-clockwise rotation by 30 (R30◦ ), translation to the right by 0.5 and up 0.25 (T(0:5;0:25)), results in a triangle that is similar to the untransformed triangle. 3. Fractals and geometric series 3.1. Area and circumference of the Koch island. To find the area of the Koch curve we begin by describing the similarity transformations that construct it. Let P (x; y) be the set of all points inside the equilateral triangle with side length a. We start by scaling this triangle by a factor of 1=3, and appending one of these scaled triangles to each of the 3 sides of the triangle as shown in the top panel of Figure 3. Our set then becomes P [ 3 × S1=3(P ) , where S1=3 denotes the similarity transformation associated with a scaling factor of 1=3. We then scale our original triangle by 1=32 and append one of these scaled triangles to each of the 3·4 sides of the shape (Figure 3). Our set is now P [ 3 × S1=3(P ) [ 3 · 4 × S1=3(S1=3(P )) . By repeating this process infinitely we obtain the set, 1 [ k−1 k P 3 · 4 S1=3(P ); k=1 k where S1=3 denotes the composition of S1=3 with itself k times. p WHY THE KOCH CURVE LIKE 2 3 We are now ready to compute the area of the Koch island. pThe area of an equilateral triangle with side length a is 3=4·a2. Thus, the area of one of the equilateral triangles that has been scaled by 1/3 k times is p p k 2 2 2k Ak = 3=4 · (a=3 ) = 3=4 · a =3 : Therefore, as the Koch island is the union of all of these triangles, the area will be the infinite sum of the areas. This can be computed as follows, 1 X k−1 A = A0 + 3 · 4 Ak k=1 p 1 p 3 X 3 a 2 = a2 + 3 · 4k−1 · 4 4 3k k=1 p 1 p k−1 3 X 3 4 = a2 + a2 4 12 9 k=1 p p 1 k 3 3 X 4 = a2 + a2 4 12 9 k=0 Figure 3. [2] The first two stages in the construc- p p tion of the Koch island. At each stage the Koch 3 2 3 2 1 = a + a island becomes the union of scaled versions of the 4 12 1 − 4 9 set of points in the original equilateral triangle P: 2p = 3a2; 5 where we have used the sum of an infinite geometric series formula in the penultimate line (note that this is valid because the absolute value of the common ratio is j4=9j < 1). Hence, we have shownp that the area of the 2 2 Koch island is finite and equal to 5 3a . We now consider the length of the Koch curve. Begin- ning with a line segment of length l, we remove the middle 1 third and replace it with two line segments of length 3 l, resulting in a total of 4 line segments (Figure 4). There- Figure 4. The first two stages in the construction fore, the length of the Koch curve in the fist step of its of the Koch curve. At each stage we remove the construction is L = 4=3 · l. We then remove the middle middle third of all line segments and replace it 1 with two scaled down line segments. third of each of these line segments, and replace it with 1 2 two line segments of length 32 l (a total of 16 = 4 line 2 segments), resulting in a length of L2 = (4=3) l. Hence, k at the kth step of its construction we have Lk = (4=3) l: As the Koch curve is defined as the limit of this procedure repeated infinitely, the length of the Koch curve is 4k L = lim l = 1: k!1 3 Now, as the circumference Koch island is the union of three Koch curves which have had a rotation and translation similarity transformation applied to them1 (Figure 5), the circumference of the Koch island is 3L = 1. This means that the Koch island has a seemingly paradoxical property: it has infinite length but a finite area!2,3 1For a more detailed approach to this construction see appendix item 2. 2This bizarre property can actually be observed in the real world. Coastlines of continents exhibit a similar behaviour, in that the length of the coastline is ill-defined in some sense. For example, if you wanted to measure the coastline of Australia, the more accurate your measurement, the larger your coastline length would be. This is because the more estuaries and bays you take into account when measuring the coastline, the larger the length will be. This raises the question: What level of precision do I stop at? This problem is actually the topic of Mandelbrot's paper How long is the coast line of Britain?[3]. 3A three dimensional analogue of this is Gabriel's Horn which has finite volume but infinite surface area. 4 XANDA KOLESNIKOW (SID: 480393797) Figure 5. Construction of the circumference of the Koch island from 3 Koch curves. The set of points in the circumference of the Koch island is the union of the set of points in three Koch curves with rotation and translation similarity transformations applied to them. 3.2. Self-similarity of the infinite geometric series. In the previous section we used the infinite geometric series formula to compute the area of the Koch island. It turns out that the infinite geometric series shares some properties with fractals.