WHY the KOCH CURVE LIKE √ 2 1. Introduction This Essay Aims To

√ 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 mathematical objects. The notion of self-similarity will be introduced, leading to a discussion of similarity transformations which are the main mathematical tool used to show this connection. The Koch curve and Koch island (or Koch snowﬂake) are the√ 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,√ 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).

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, ∞ [ 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. √ WHY THE KOCH CURVE LIKE 2 3

We are now ready to compute the area of the Koch island. The√ 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 √ √ 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, ∞ X k−1 A = A0 + 3 · 4 Ak k=1 √ ∞ √ 3 X 3 a 2 = a2 + 3 · 4k−1 · 4 4 3k k=1 √ ∞ √ k−1 3 X 3 4 = a2 + a2 4 12 9 k=1 √ √ ∞ k 3 3 X 4 = a2 + a2 4 12 9 k=0 Figure 3. [2] The first two stages in the construc- √ √ 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. 2√ = 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 |4/9| < 1). Hence, we have shown√ 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 = ∞. k→∞ 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 = ∞. 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. For a geometric series ∞ X k 2 S∞ := ar = a + ar + ar + ..., k=0 we can scale it by a factor r 2 3 rS∞ = ar + ar + ar + .... Therefore, we can express the infinite geometric series in terms of a scaled version of itself

S∞ = a + rS∞. This demonstrates self-similarity of the infinite geometric series, and note that just like fractals, this self-similarity 2 property does not hold for a finite geometric sum. i.e. If S2 = a+ar, then a+rS2 = a+ar+ar 6= S2. Furthermore, we can actually derive the formula for the sum of the infinite geometric series using this self-similarity,

S∞ = a + rS∞ a S = . ∞ 1 − r This is the formula we used to calculate the area of the Koch island.4

4. Irrational numbers In this section we will delve into irrational numbers, looking at some of their unique properties and the various ways of defining them so that we may compare√ them to fractals in the preceding section. The two irrational numbers that we will focus on are, π and 2, two very famous numbers! 4.1. Pi. Pi is irrational, meaning that when expressed in base 10 (or any rational base for that matter), the decimal expansion goes on for ever, or more precisely it can not be expressed as the ratio of two integers. We can go even further and say that π is a transcendental number, which means that it is not the root of any polynomial with integer coefficients. However, π can be defined simply as the ratio of a circle’s circumference to its diameter, and this is how the number was originally discovered. In fact, if it were not for this simple relationship, π probably wouldn’t “exist” at all because the digits of π appear random and it is conjectured to be a ‘normal number’5,

4For an alternative derivation of this formula see appendix item 1. 5 −1 A number is said to be normal in base b if the number of occurrences of each digit appearing in its base b expansion tends√ to b . Although it has been proven that most numbers are normal, it remains to be proven that fundamental constants such as π, e, 2, ln 2 are normal [4]. √ WHY THE KOCH CURVE LIKE 2 5 meaning that no digits appear more frequently than others regardless of the base that it is expressed in (where the base must be a natural number greater than 2). So without this clean definition, π would have no more significance than the infinitude of numbers with random decimal expansions. Pi can also be defined in terms of a limit. There are many ways of achieving a limit definition for π but here we will we consider the polygon approximation for π.6 We begin with an inscribed and circumscribed regular hexagon in a circle of radius r (Figure 6). The arc length s of the circle between two corners of the inscribed hexagon will be 1/6th of its circumference, i.e. s = 2πr/6, and we know that this arc length will be greater than the side length of the inscribed hexagon l and less than the side length of the circumscribed hexagon L. The lengths of the sides of the inscribed and circumscribed hexagons are 360 ◦ 360 ◦ 7 l = 2r sin 2·6 and L = 2r tan 2·6 respectively. Hence, we can find an upper and lower bound for π as follows,

l < s < L 360 ◦ 2πr 360 ◦ 2r sin < < 2r tan 2 · 6 6 2 · 6 180 ◦ 180 ◦ 6 sin < π < 6 tan 6 6 Figure 6. 10 [5] Polygon approximation for π. 3 < π < 3.46. The side lengths l, L of a blue inscribed hexagon 8 and red circumscribed hexagon give an upper and This yields a rather unsatisfactory approximation , but lower bound for the arc length s of the circle sub- we can generalise our formula to gain more precise ap- tended by an angle of 60◦. Trigonometry can be proximations. For an n sided regular polygon, our for- used to find the values of these side lengths in mula becomes terms of the radius r of the circle. 180 ◦ 180 ◦ n sin < π < n tan . n n Here we can make n as large as we want to gain a satisfac- n lower bound upper bound tory approximation (Table 1 shows some approximations 6 3.00000 3.46410 for π using some larger values of n). In particular, we 7 3.03719 3.37102 can express π as the limit of this inequality as n tends to 8 3.06147 3.31371 infinity. i.e. 9 3.07818 3.27573 180 ◦ 180 ◦ 10 3.09017 3.24920 lim n sin = π = lim n tan .9 . . . n→∞ n n→∞ n . . . Therefore we can express π concisely as the limit of an 1000 3.14159 3.14160 expression instead of an infinite decimal expansion, just like we were able to express the area of the Koch island Table 1. Upper and lower bounds for π ≈ as the limit of a geometric series as opposed to an infinite 3.141592 using the polygon approximation method geometric construction. with an n sided regular polygon. The convergence for this method is rather slow. It is not until a 1000 sided regular polygon that we gain an approximation that is accurate within 6 significant figures.

6This method of approximation is was originally used by Archimedes in 260 B.C. 7Why have we not used radians here? As a mathematician, radians are our friends. However, radians are defined in terms of π, so if we are using them to approximate π that would be circular logic. 8Unless you’re an engineer. 9This is an incarnation of the Squeeze Law, or you might like to call it the ‘Sandwich Law’ seeing as though we are also discussing pie. 10In Peitgen, Jürgens& Saup (2004), the figure similar to this one is Figure 3.14 [6]. Coincidence? I think not. 6 XANDA KOLESNIKOW (SID: 480393797)

4.2. The square root of two. √ Similarly to π, at ﬁrst appearance 2 is not a pretty looking number, √ 2 = 1.41421356 .... √ Just like π, 2 is an irrational and normal√ number, but it is not a transcendental number: it is a root of the polynomial x2 − 2. This is one reason why 2 exists.

√ Moreover, there exists a concise continued fraction expansion of 2 . A continued fraction expansion is achieved in the following way: begin with any rational number, say 415/93, we can then express this as a mixed fraction 415/93 = 4 + 43/93. The inverse of 43/93 is an improper fraction which we can also express as a mixed fraction (43/93)−1 = 93/43 = 2 + 7/43. Repeating this procedure with 7/43, we have that (7/43)−1 = 6 + 1/7. The inverse of 1/7 has no mixed fraction expansion, so we are done. This yields the continued fraction expansion 415 43 = 4 + 93 93 1 = 4 + 93 43 1 = 4 + 7 2 + 43 1 = 4 + 1 2 + 43 7 1 = 4 + 1 . 2 + 1 6+ 7 We can summarise this expansion using the notation 415/93 = [4; 2, 6, 7], where the number before the semi-colon denotes the integer outside of the fraction and the numbers after the semicolon denote the integers at each of the denominators in the fraction. It turns out that√ all rational numbers will have a finite continued fraction expansion. 2 But what about√ an irrational number like 2? Here, we begin with the equation x + 2x − 1 = 0, which has solutions −1 ± 2. By rearranging the equation we can write 2x + x2 = 1 x(2 + x) = 1 1 x = 2 + x 1 1 = 1 since x = 2 + 2+x 2 + x 1 = 1 2 + 1 2+ 2+x 1 = 1 . 2 + 1 2+ 2+... √ As 2 − 1 is a solution to this equation, we can write √ 1 x = 2 − 1 = 1 2 + 1 2+ 2+... √ 1 2 = 1 + 1 . 2 + 1 2+ 2+... √ √ ˙ Hence,√ the continued fraction expansion of 2 is [1; 2, 2, 2,... ], or by using familiar decimal notation 2 = [1; 2]. So, 2 has a clean continued fraction expansion. It turns√ out that all irrational√ numbers have an infinite continued fraction expansion, but they are not all as clean as 2. For example, 19 = [4; 2, 1, 3, 1, 2, 8] [7], where the bar denotes that the sequence (2, 1, 3, 1, 2, 8) repeats indefinitely. The continued fraction expansion of π is even worse, √ WHY THE KOCH CURVE LIKE 2 7

11 with no repeating pattern at all π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1,... ][8]. Also√ note the inherent self-similarity of this notation. With the exception of the ﬁrst 1 in the fraction expansion of 2, the denominator√ at any stage is the same as the whole fraction expansion. On the other hand, the fractional expansion√ of 19 would be self-similar 1+ 5 ˙ 12 at every 6th denominator. The fractional expansion of the golden mean ϕ = 2 = [1; 1] has perhaps the simplest fraction expansion and is perfectly self-similar at every stage.

The square root of two can also be expressed as the limit of an iterative procedure.√ We start with the equation x2 − 2 = 0, for which 2 is a solution and rearrange as follows, x2 = 2 n xn xn+1 0 0.2 5.1 x 1 = where we have divided by 2x 1 5.1 2.74608 2 x 2 2.74608 1.73719 x 1 x = + 3 1.73719 1.44424 2 x 4 1.44424 1.41453 1 2 = x + . 5 1.41453 1.41421 2 x √ We can convert this equation to an iterative equation Table 2. Approximations to 2 ≈ 1.414214 us- 1 2 ing the iterative equation xn+1 = xn + whereby we start with some x0 > 0 and iterate on the 2 xn 1 2 13 with x = 0.2. The convergence for this method is equation xn+1 = xn + . This procedure will con- 0 √ 2 xn √ much faster than the polygon approximation for verge to 2 (see Table 2). Therefore, we can express 2 π. It only requires 6 iterations to achieve an ap- in terms of a limit, proximation that is accurate within 6 significant √ figures. lim xn = 2. n→∞ √ Moreover, 2 is invariant under the function√ 1 2 f(x) = 2 x + x , meaning that f(x) = x for x = 2. √ In summary, both π and 2 have a decimal expansion consisting of an infinite string of (conjectured to be) random numbers. However, both of these numbers are significant because√ we can find ways to express them concisely. Pi can be expressed as the limit of a sandwiched inequality, and 2 can be expressed as the solution to an equation, or as a repeated fraction expansion.

5. Fractals as solutions to equations So far we have defined fractals as the limit of a repeated procedure. We also saw that this can be done with irrational numbers. We were then able to define irrational numbers as the solutions to certain equations. i.e. π is the solution to 180 ◦ 180 ◦ lim n sin = π = lim n tan n→∞ n n→∞ n √ and 2 is the solution to x2 − 2 = 0. So the question is: can we define fractals as the solutions to equations too?

5.1. Invariance. An object√ is said to be invariant if it does not change under a certain transformation. For example, we already 1 2 saw that 2 is invariant under the transformation f(x) = 2 x + x . We can also describe fractals as being invariant under certain transformations. These transformations are similarity transformations called Hutchinson operators.14 We will explore a couple of these using the Koch curve and the Cantor set as our examples.

11Not all transcendental numbers have no pattern in their continued fraction expansion. For example, e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8,... ] has a period of three, where the last term in each period increases by two each time [9]. 12For a derivation of this fraction expansion see appendix item 3. 13For an alternative derivation of this equation using Newton’s Method see appendix item 4. 14Named after John E. Hutchinson, who is the Australian mathematician who discovered them [10]. 8 XANDA KOLESNIKOW (SID: 480393797)

5.2. Hutchinson operator for the Koch curve. Here, we describe more precisely the similarity transformations that construct the Koch curve. Let L(x, y) = ([0, 1], 0) be the horizontal unit line segment with endpoints x = 0 and x = 1, the initiator for the Koch curve. We note that the next stage in the construction of the Koch curve consists of 4 line segments of length 1/3, the middle two of which have had a rotation and translation applied to them, and the right most one has just been translated (recall Figure 4). We can then deﬁne the following similarity transformations,

1 1 k1(x, y) := S 1 (x, y) = x, y , 3 3 3 √ √ ! 1 3 1 3 1 k2(x, y) := T( 1 ,0) ◦ R π ◦ S 1 (x, y) = x − y + , + y , 3 3 3 6 6 3 6 6 √ √ √ ! 1 3 1 3 1 3 √ k3(x, y) := T 1 3 ◦ R− π ◦ S 1 (x, y) = x + y + , − x + y + , ( 2 , 6 ) 3 3 6 6 2 6 6 6 1 2 1 k4(x, y) := T 2 ,0 ◦ S 1 (x, y) = x + , y . ( 3 ) 3 3 3 3

Now, define the Hutchinson operator for the Koch curve to be 4 [ K(x, y) = ki(x, y). i=1 Thus, we can describe the object in the first step of the Koch curve construction to be K(L). Since the Koch curve is the infinite repetition of this procedure, we can define the Koch curve to be lim Kn(L), n→∞ where Kn denotes K composed with itself n times. More importantly, the Koch curve is invariant under the operator K due to the fact that K ◦ lim Kn(L) = lim Kn(L) n→∞ n→∞ Figure 7. Invariance of the Koch curve under its (see Figure 7 for a visual representation of this). Note Hutchinson operator. When K is applied to the that this property results in the self-similarity of the Koch set of points in the Koch curve P , the result is the curve. In fact, it can be proven that the Koch curve Koch curve. i.e. K(P ) = P. is the only object that is invariant under K [10]! For example, if we take another set of points, say the set of points in the word ‘Robby’, and apply K to it, we are left with something that looks different to the original word ‘Robby’ (Figure 8). So we can conclude that the set of points in the word ‘Robby’ is not invariant under K. However, if we apply the procedure enough times we end up with something that looks like the Koch curve (see Figure 9). Therefore, the set of points in the word ‘Robby’ converges to the Koch curve. This is directly 1 2 analogous to the iterative equation xn+1 = xn + , √ 2 xn which,√ for x0 > 0, will converge to 2 and for which 2 is invariant under. Hence, fractals may be described as mathematical objects which are left invariant under a Figure 8. K applied to a different set of points. Hutchinson operator, resulting in their self-similarity. When K is applied to the set of points in the word ‘Robby’, we are left with something that does not resemble the initial set of points. √ WHY THE KOCH CURVE LIKE 2 9

Figure 9. ‘Robby’ converges to the Koch curve. After applying K to the set of points in the word ‘Robby’ 4 times, we are left with something that looks like the Koch curve.

5.3. Hutchinson operator for the Cantor set. Here we also start with the unit line segment but are working in 1 dimension so we can define L = [0, 1] to be our initiator for the Cantor set. We can then define the following similarity transformations, 1 c1(x) = S 1 (x) = x, 3 3 1 2 c2(x) = T 2 ◦ S 1 (x) = x + . 3 3 3 3 So our Hutchinson operator for the Cantor set becomes C(x) := c1(x) ∪ c2(x). Similarly to the Koch curve, it can be proven that the Cantor set is the only set of points that is invariant under C [10]. Although proving the uniqueness is difficult, we can provide a formal proof that the Cantor set is invariant under C using its triadic expansion.

Proof. Here we wish to prove that C(A) = A, where A denotes the set of points in the Cantor set. The Cantor set can be defined as the set of all numbers in the interval [0, 1] with triadic expansions such that each digit is 0 or 2. i.e. A = {0.a1a2a3 ... |ai ∈ {0, 2}}. To show that C(A) ⊆ A, we observe that if x = 0.x1x2x3 · · · ∈ A, then c1(x) = 0.0x1x2x3 ... which is in A and c2(x) = 0.2x1x2x3 ... which is also in A since the xi’s are either 0 or 2 by definition of x and we have appended a 0 or 2 to the beginning of the sequence. Therefore, C(A) ⊆ A. Now, to show that A ⊆ C(A), we take y = 0.y1y2y3 · · · ∈ A. Then, if y1 = 0, then y = c1(0.y2y3 ... ), and if y1 = 2, then y = c2(0.y2y3 ... ). Since 0.y2y3 · · · ∈ A by definition of y, we have that y ∈ C(A), and therefore A ⊆ C(A). Since C(A) ⊆ A and A ⊆ C(A) we can conclude that C(A) = A.

6. Conclusion In this essay, we have drawn an unexpected, yet undeniable connection between irrational numbers and fractals. Among the integers and rational numbers, irrational numbers appear ill-defined when looking at their decimal expansion. However, we are able to define irrational numbers in a concise way: as solutions to equations or the limit of a certain procedure. Similarly, fractals also appear ill-defined amongst familiar shapes such as polygons as they have paradoxical properties such as infinite circumference but finite area. However, they may be defined concisely as the unique shapes that are invariant under Hutchinson operators, and can also be defined as the limit of applying certain similarity transformations infinitely. Thus, we may view fractals as the irrational numbers of shapes. 10 REFERENCES

7. Appendix Hence, ϕ = [1; 1, 1,... ] = [1; 1].˙ (1) Here we prove the formula for the limiting sum of an infinite geometric series. We start by finding (4) We can use Newton’s method with the function 2 the sum of a geometric series with n terms, f√(x) = x −2 to derive the iterative equation for 2 n−1 2. i.e. To find the roots of f(x) we can use, Sn := a + ar + ar + ··· + ar . f(xn) Multiplying this expression by the common ratio xn+1 = xn − 0 f (xn) yields, 2 xn − 2 rS = ar + ar2 + ··· + arn−1 + arn. = xn − n 2xn By subtracting these two expressions we arrive 1 2 = xn − xn + . at a formula for the sum of a geometric series, 2 xn √ n 2 Sn − rSn = a − ar As f(x) = x − 2 has roots x = ± 2, we use n x > 0 to find the positive root. Sn(1 − r) = a(1 − r ) 0 a(1 − rn) S = . References n 1 − r This essay is based on Chapter 3 from the textbook n Now, if |r| < 1, then r = 0 as n → ∞. Hence, Fractals and self similarity by Peitgen, Jürgens& Saup we find the limiting sum of an infinite geometric (2004) (See reference [6]). series is a 1. Katie. The healthy dessert blog. Sticky S∞ = lim Sn = . sesame cauliflower online. 2017. https : / / n→∞ 1 − r chocolatecoveredkatie . com / 2017 / 01 / 09 / (2) The Koch island can be described as the union sticky-sesame-cauliflower/. of 3 Koch curves, 2 of which have had a ro- 2. Lockhart, R. Fun with Koch snowlakes tation and translation similarity transformation http : / / demonstrations . wolfram . com / applied to them. If we denote P (x, y) as the set FunWithKochSnowflakes/. of all points in the Koch curve then the perime- 3. Mandelbrot, B. B. How long is the coast of Britain? ter of the Koch island becomes Statistical self-similarity and fractional dimension. h i h i Science 156 (1967). √ √ P ∪ T(1/2,− 3/2)(Rπ/3(P )) ∪ T(1/2,− 3/2)(R2π/3(P )) . 4. Borel, E. The numerable probabilities and their ap- plications. Reports of the Mathematical Circle of (3) By definition, the golden mean is the positive Palermo 27 (1909). root of the equation ϕ2 − ϕ − 1 = 0. By divid- 5. Tucker, J. Archimedes’ approximation of Pi ing by ϕ and rearranging we can obtain the ex- https : / / demonstrations . wolfram . com / 1 ArchimedesApproximationOfPi/. pression ϕ = 1 + ϕ , which can be used to find the continued fractional expansion of the golden 6. Peitgen, H.-O., Jürgens,H. & Saup, D. Chaos and mean as follows, fractals: new frontiers of science isbn: 0387202293 (Springer Science & Business Media, 2004). 1 ϕ = 1 + 7. Sloane, N. J. A. Database code: A010124. Con- ϕ tinued fraction for sqrt(19) https://oeis.org/ 1 A010124. = 1 + 1 1 + ϕ 8. Dincer, S. S. & Schoenfield, J. E. Database code: 1 A133593. ”exact” continued fraction for Pi https: = 1 + 1 //oeis.org/A133593. 1 + 1 1+ ϕ 9. Sloane, N. J. A. Database code: A003417. Contin- 1 ued fraction for e https://oeis.org/A003417. = 1 + . 1 + 1 10. Hutchinson, J. E. Fractals and self similarity. Indi- 1+ 1 1+... ana University Mathematics Journal 30 (1981).