Fibonacci's Rabbits: Spring 2024 PDF

Fibonacci’s Rabbits Sequences of Numbers Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 2 Let’s make a poll: Which of the following photos looks more pleasing? IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz K...

Fibonacci’s Rabbits Sequences of Numbers Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 2 Let’s make a poll: Which of the following photos looks more pleasing? IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 3 Let’s make a poll: Which of the following photos looks more pleasing? IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 4 Let’s make a poll: Which of the following photos looks more pleasing? IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 5 Let’s make a poll: Which of the following photos looks more pleasing? IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Why are the ones on the...............-hand side LEFT of these photos more pleasing to many? 6 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Part I: Fibonacci & Liber Acci 7 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Fibonacci (aka Leonardo Bonacci) (AD. 1170 – 1240) Fibonacci was an Italian mathematician from the Republic of Pisa. He was son of an Italian merchant and customs official who directed a trading post in Bugia, the capital of the Hammadid empire. Fibonacci traveled with him as a young boy around the Mediterranean coast, met with many merchants and learnt about their systems of doing arithmetic. 8 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 9 Liber Abaci (”The Book of Calculation”) In Bugia (Algeria), merchants were using Hindu–Arabic numeral system. Fibonacci soon realised many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time in Europe, allowed easy calculation using a place-value system. In 1202, he completed the Liber Abaci (Book of Abacus or The Book of Calculation), which popularized Hindu–Arabic numerals in Europe. IŞIK UNIVERSITY Liber Abaci (”The Book of Calculation”) In the Liber Abaci (1202), Fibonacci introduced Hindu–Arabic numeral system. The manuscript advocated numeration with ten digits including a zero and positional notation. The book showed the practical use and value of this by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, and other applications. ”Fibonacci’s Rabbits” by Deniz Karlı 10 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 11 Liber Abaci (”The Book of Calculation”) The book was well-received throughout educated Europe and had a profound impact on European thought. Replacing Roman numerals, its ancient Egyptian multiplication method, and using an abacus for calculations, was an advance in making business calculations easier and faster, which assisted the growth of banking and accounting in Europe. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 12 Liber Abaci (”The Book of Calculation”) Fibonacci introduced new ideas through some problems, one of which is known as ”Fibonacci Sequence” today. He posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions. This problem was known to be described by Indian mathematicians as early as the sixth century. However, Liber Abaci contains the earliest known description of this problem outside of India. Later, this problem presented very interesting results which are used in Mathematics, Arts, Architecture, Visual Arts, Biology and Nature, Finance etc. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Fibonacci’s Rabbits Fibonacci set up a rabbit farm where we have 1 pair of baby rabbits initially. He poses some assumptions on these rabbits:. ∗ Baby rabbits grow to a mature state in 1 month. ∗ A pair means a male and female couple. ∗ Each mature pair of rabbits mate each other every month. ∗ A pair of mated rabbits gives birth to a pair of baby rabbits in month. ∗ There is no death in the farm, and mature pairs mate forever. Question: How many pairs are present in each month? 13 IŞIK UNIVERSITY Fibonacci’s Rabbits: Let’s Count Month 1 ”Fibonacci’s Rabbits” by Deniz Karlı 14 IŞIK UNIVERSITY Fibonacci’s Rabbits: Let’s Count Month 2 ”Fibonacci’s Rabbits” by Deniz Karlı 15 IŞIK UNIVERSITY Fibonacci’s Rabbits: Let’s Count Month 3 ”Fibonacci’s Rabbits” by Deniz Karlı 16 IŞIK UNIVERSITY Fibonacci’s Rabbits: Let’s Count Month 4 ”Fibonacci’s Rabbits” by Deniz Karlı 17 IŞIK UNIVERSITY Fibonacci’s Rabbits: Let’s Count Month 5 ”Fibonacci’s Rabbits” by Deniz Karlı 18 IŞIK UNIVERSITY Fibonacci’s Rabbits: Let’s Count Month 6 ”Fibonacci’s Rabbits” by Deniz Karlı 19 IŞIK UNIVERSITY a a d 1, 1 , E ”Fibonacci’s Rabbits” by Deniz Karlı I The number of rabbits evolves as 2, 3, 5, 8,... ,... , a This collection is called the Fibonacci Sequence. 93 92 91 ay as 93 92 94 93 20... IŞIK UNIVERSITY Fibonacci Sequence in Nature: Sunflower ”Fibonacci’s Rabbits” by Deniz Karlı 21 IŞIK UNIVERSITY Fibonacci Sequence in Nature: Pine Cones ”Fibonacci’s Rabbits” by Deniz Karlı 22 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 23 Fibonacci Sequence in Nature: Pine Cones Number of branches of a tree at any level o a IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Part II: Sequences 24 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 25 Sequences A ”countably infinite” collection of numbers is called a sequence. For example: {1, 2, 3, 4, · · · } {0.7, 2/5, π, e, · · · } {−4, −10, 2, 0.4, · · · } We denote the nth element of a sequence by an or bn or cn etc. The corresponding sequence is denoted by ∞ ∞ {an }∞ n=1 or {bn }n=1 or {cn }n=1 etc. For example, if then {an }∞ n=1 = {0.7, 2/5, π, e, · · · } a1 = 0.7, a2 = 2/5, a3 = π, a4 = e, etc. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 26 Limits of Sequences If terms of a sequence get close to (approach) a number L as much as we like, then this number L is called the limit of the sequence. For example, consider the following sequences. {an }∞ n=1 = {0.1, 0.01, 0.001, 0.0001, · · · } in in limit 00 0 01 6.01 {bn }∞ n=1 = {2.9, 2.99, 2.999, 2.9999, · · · } limit is 21.99 21.9 {cn }∞ n=1 = {1, 2, 3, 4, · · · } no limit here f1 R 3 f 9999 2.999 3 IR IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 27 Limits Notation If terms of a sequence an get close to (approach) a number L as much as we like, then this number L is called the limit of the sequence. In this case, we write lim an = L n→∞ For example, {an }∞ n=1 = {0.1, 0.01, 0.001, 0.0001, · · · } shows that {bn }∞ n=1 = {2.9, 2.99, 2.999, 2.9999, · · · } shows that {cn }∞ n=1 = {1, 2, 3, 4, · · · } shows that lim cn = n→∞ lim an = O lim bn = 3 n→∞ n→∞ 5 b IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Basic Properties of Limits Assume k is a constant number. lim k = k n→∞ kekik kk 1 =0 n→∞ n lim f f f Inşaat 7 fiyat T 28 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 29 Basic Properties of Limits Assume an and bn are two sequences with limits A and B, respectively. lim (an + bn ) = lim an + lim bn = A+B. n→∞ n→∞ n→∞ an 4.414,4 bu a E f bu 4 7,4 1,4 5 tt lim (an − bn ) = lim an − lim bn = A−B. n→∞ EE FE n→∞ limant n→∞ T Iff 1T IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 30 Basic Properties of Limits Assume an and bn are two sequences with limits A and B, respectively. lim an an A n→∞ lim = = n→∞ bn lim bn B lim (an · bn ) = lim an · lim bn = A · B n→∞ E n→∞ Liz O L n→∞ n→∞ E iç ʰI if B ̸= 0 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 31 Basic Properties of Limits Assume an and bn are two sequences with limits A and B, respectively. Assume k is a constant number. lim (k · an ) = k · lim an = k · A n→∞ E his Için Some Examples n→∞ t.tn 0 0 E IE E z EIfif hiç 1 limatta awE HI üç IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Part III: What is the Golden Ratio 32 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 33 Revisit Fibonacci Sequence Recall the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34,... Hence we have a1 = 1, a2 = 1, a3 = 2, a4 = 3, a5 = 5, a6 = 8, a7 = 13,... We observe that lim an = n→∞ α This is not much interesting. However, there is another related sequence which provides more interesting results. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Ratio of Each Term to Previous Term 1,1 2,315,8 34 13,21 a2 a1 a3 a2 a4 a3 a5 a4 a6 a5 a7 a6 a8 a7 a9 a8... ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 1 1 2 1 3 2 5 3 8 5 13 8 21 13 34 21... ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 1 2 1.5 1.666... 1.6 1.625 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ b1 b2 b3 b4 b5 b6 b7 b8... 1.615.... 1.619....... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 35 Ratio of Terms of Fibonacci Sequence 1.615.... 1.619....... 1 2 1.5 1.666... 1.6 1.625 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ b1 b2 b3 b4 b5 b6 b7 b8... Question: What is the limit of this sequence? lim bn =? n→ IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 36 Limit of Ratio Sequence Assume bn has a limit, call this limit value L. That is, assume lim bn = L. n→ Remember that bn = 7 an+1. an Moreover. an+2 = an+1 + an. Since an+1 ̸= 0, divide both sides of the last equality by an+1. an+2 an+1 an = + an+1 an+1 an+1 I ⇒ an+2 1 = 1 + an+1 an+1 an Id Ty ⇒ bn+1 = 1 + b1n. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 37 Limit of Ratio Sequence We have lim bn = L n→ and bn+1 = 1 + 1 bn Next, take limit of both sides of the last equation: lim bn+1 = lim (1 + n→∞ L 1 22 n→∞ 4 1 1 bn ) 22_L 1 0 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 38 Limit of Ratio Sequence Hence, we need to solve the quadratic equation: L2 − L − 1 = 0 Roots D are and 12T 1 172 4.1 C 1 5 Solutions of a quadratic equation ax2 + bx + c = 0 are √ √ −b − ∆ −b + ∆ and 2 2 if ∆ = b2 − 4 · a · c is positive. 1 2_1 L x 1 0 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı The positive solution of the equation L2 − L − 1 = 0 is denoted by φ. It is called the ”Golden Ratio”. φ = 1.61803398875... E 39 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 40 Different ”So-called-Golden Ratios” It is possible for you to create your own ”so-called Golden Ratio” by defining your version of Fibonacci sequence. Remember that the classical Fibonacci sequence has the rule an+2 = an+1 + an with a1 = 1 , a2 = 1. with a1 = 1 , a2 = 1. with a1 = 2 , a2 = 3. For example, try the sequence with the rule an+2 = 2an+1 + an or the sequence with the rule an+2 = an+1 + 3an What are their ”Golden Ratios”? IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Part IV: Applications of Golden Ratio 41 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 42 Golden Rectangle One of the applications is the use of Golden Rectangle. L Golden rectangle is the rectangle where the ratio of the sides is the Golden Ratio. a+b =φ a This rectangle is assumed to be most pleasing rectangle to human eye. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 43 Golden Rectangle 16 9 4 3 Old Tvs were produced in the ratio of 4:3. In the last decade, the ratio became 16:9. This is a more pleasing shape to human eye. Notice that 16 ! 1.77 ≈ φ 9 whereas 4 ! 1.33. 3 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 44 Golden Spiral Another aesthetic use is through Golden Spiral. i To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 45 Another aesthetic use is through Golden Spiral. 1 To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 46 Another aesthetic use is through Golden Spiral. i To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 47 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 48 Another aesthetic use is through Golden Spiral. 5 5 To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 8 8 49 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 13 50 Another aesthetic use is through Golden Spiral. 13 To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 51 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... 21 21 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 52 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 53 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 54 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 55 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 56 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 57 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 58 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 59 Another aesthetic use is through Golden Spiral. To obtain the Golden Spiral, first, draw squares of side length being equal to terms of Fibonacci sequence in clockwise direction. Recall the terms of Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,... Finally connect opposite corners of these squares with arcs so that they form a connected path. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 60 This spiral is called the Golden Spiral. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 61 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı Part V: Golden Ratio in Photography 62 IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 63 One of the most important rules in photography is to create the right composition. This can be obtained in different ways. One basic rule is to use the Golden Ratio in composition. That is, split the scene in two pieces (both in vertical and horizontal directions) a and b so that the ratio a/b = φ. a b IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 64 First divide the frame horizontally into 2 pieces where left piece has length a and the right piece has length b. a = φ = 1.61803398875... and b=1 where a = φ = 1.61803398875... b IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 65 Also divide the frame horizontally into 2 pieces where right piece has length a and the left piece has length b. a = φ = 1.61803398875... and b=1 where a = φ = 1.61803398875... b IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 66 Second divide the frame vertically into 2 pieces where top piece has length a and the bottom piece has length b. a = φ = 1.61803398875... and b=1 where a = φ = 1.61803398875... b IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 67 Also divide the frame vertically into 2 pieces where bottom piece has length a and the top piece has length b. a = φ = 1.61803398875... and b=1 where a = φ = 1.61803398875... b IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 68 Finally we combine all these 4 lines together. a = φ = 1.61803398875... and b=1 where a = φ = 1.61803398875... b IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 69 Intersections of these lines represents possible placements for our objects. a = φ = 1.61803398875... and b=1 where a = φ = 1.61803398875... b IŞIK UNIVERSITY Composition in Photography ”Fibonacci’s Rabbits” by Deniz Karlı 70 In a ”photography for everyone” class, this is taught as ”one-third” rule. For those, who don’t know Golden Ratio, it is explained as partitioning the frame into 3 equal pieces both vertically and horizontally. The grid is formed on these thirds. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 75 Finally let us go back to the photos at the beginning of the lecture. IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 76 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 77 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 78 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 79 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 80 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı 81 IŞIK UNIVERSITY ”Fibonacci’s Rabbits” by Deniz Karlı End of the lecture. 82