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Fibonacci Numbers and the Golden Ratio Fibonacci Numbers and the Golden Ratio Lecture Notes for Jeffrey R. Chasnov The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon Hong Kong Copyright ○c 2016-2019 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. Preface View the promotional video on YouTube These are my lecture notes for my online Coursera course, Fibonacci Numbers and the Golden Ratio. These lecture notes are divided into chapters called Lectures, and each Lecture corresponds to a video on Coursera. I have also uploaded the Coursera videos to YouTube, and links are placed at the top of each Lecture. Most of the Lectures also contain problems for students to solve. Less experienced students may find some of these problems difficult. Do not despair! The Lectures can be read and watched, and the material understood and enjoyed without actually solving any problems. But mathematicians do like to solve problems and I have selected those that I found to be interesting. Try some of them, but if you get stuck, full solutions can be read in the Appendix. My aim in writing these lecture notes was to place the mathematics at the level of an advanced high school student. Proof by mathematical induction and matrices, however, may be unfamiliar to a typical high school student and I have provided a short and hopefully readable discussion of these topics in the Appendix. Although all the material presented here can be considered elementary, I suspect that some, if not most, of the material may be unfamiliar to even professional mathematicians since Fibonacci numbers and the golden ratio are topics not usually covered in a University course. So I welcome both young and old, novice and experienced mathematicians to peruse these lecture notes, watch my lecture videos, solve some problems, and enjoy the wonders of the Fibonacci sequence and the golden ratio. For your interest, here are the links to my other online courses. If you are studying differential equations, you may want to browse Differential Equations for Engineers If your interests are matrices and elementary linear algebra, have a look at Matrix Algebra for Engineers And for a course on multivariable calculus, try Vector Calculus for Engineers iii Contents I Fibonacci: It’s as Easy as 1, 1, 2, 3 1 1 The Fibonacci sequence5 2 The Fibonacci sequence redux7 Practice quiz: The Fibonacci numbers9 3 The golden ratio 11 4 Fibonacci numbers and the golden ratio 13 5 Binet’s formula 15 Practice quiz: The golden ratio 19 II Identities, Sums and Rectangles 21 6 The Fibonacci Q-matrix 25 7 Cassini’s identity 29 8 The Fibonacci bamboozlement 31 Practice quiz: The Fibonacci bamboozlement 35 9 Sum of Fibonacci numbers 37 10 Sum of Fibonacci numbers squared 39 Practice quiz: Fibonacci sums 41 11 The golden rectangle 43 12 Spiraling squares 45 III The Most Irrational Number 49 13 The golden spiral 53 v vi CONTENTS 14 An inner golden rectangle 57 15 The Fibonacci spiral 61 Practice quiz: Spirals 63 16 Fibonacci numbers in nature 65 17 Continued fractions 67 18 The golden angle 71 19 The growth of a sunflower 75 Practice quiz: Fibonacci numbers in nature 77 Appendix 78 A Mathematical induction 81 B Matrix algebra 83 B.1 Addition and Multiplication.................................... 83 B.2 Determinants............................................ 84 C Problem and practice quiz solutions 87 Part I Fibonacci: It’s as Easy as 1, 1, 2, 3 1 3 In this week’s lectures, we learn about the Fibonacci numbers, the golden ratio, and their relationship. We conclude the week by deriving the celebrated Binet’s formula, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprical. 4 Lecture 1 The Fibonacci sequence View this lecture on YouTube Fibonacci published in the year 1202 his now famous rabbit puzzle: A man put a male-female pair of newly born rabbits in a field. Rabbits take a month to mature before mating. One month after mating, females give birth to one male-female pair and then mate again. No rabbits die. How many rabbit pairs are there after one year? To solve, we construct Table 1.1. At the start of each month, the number of juvenile pairs, adult pairs, and total number of pairs are shown. At the start of January, one pair of juvenile rabbits is introduced into the population. At the start of February, this pair of rabbits has matured. At the start of March, this pair has given birth to a new pair of juvenile rabbits. And so on. month J F M A M J J A S O N D J juvenile 1 0 1 1 2 3 5 8 13 21 34 55 89 adult 0 1 1 2 3 5 8 13 21 34 55 89 144 total 1 1 2 3 5 8 13 21 34 55 89 144 233 Table 1.1: Fibonacci’s rabbit population. We define the Fibonacci numbers Fn to be the total number of rabbit pairs at the start of the nth month. The number of rabbits pairs at the start of the 13th month, F13 = 233, can be taken as the solution to Fibonacci’s puzzle. Further examination of the Fibonacci numbers listed in Table 1.1, reveals that these numbers satisfy the recursion relation Fn+1 = Fn + Fn−1. (1.1) This recursion relation gives the next Fibonacci number as the sum of the preceeding two numbers. To start the recursion, we need to specify F1 and F2. In Fibonacci’s rabbit problem, the initial month starts with only one rabbit pair so that F1 = 1. And this initial rabbit pair is newborn and takes one month to mature before mating so F2 = 1. The first few Fibonacci numbers, read from the table, are given by 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . and has become one of the most famous sequences in mathematics. 5 6 LECTURE 1. THE FIBONACCI SEQUENCE Problems for Lecture 1 1. The Fibonacci numbers can be extended to zero and negative indices using the relation Fn = Fn+2 − Fn+1. Determine F0 and find a general formula for F−n in terms of Fn. Prove your result using mathematical induction. 2. The Lucas numbers are closely related to the Fibonacci numbers and satisfy the same recursion relation Ln+1 = Ln + Ln−1, but with starting values L1 = 1 and L2 = 3. Determine the first 12 Lucas numbers. 3. The generalized Fibonacci sequence satisfies fn+1 = fn + fn−1 with starting values f1 = p and f2 = q. Using mathematical induction, prove that fn+2 = Fn p + Fn+1q. (1.2) 4. Prove that Ln = Fn−1 + Fn+1. (1.3) 5. Prove that 1 F = (L − + L + ) . n 5 n 1 n 1 6. The generating function for the Fibonacci sequence is given by the power series ¥ n f (x) = ∑ Fnx . n=1 Assuming the power series converges, prove that x f (x) = . 1 − x − x2 Solutions to the Problems Lecture 2 The Fibonacci sequence redux View this lecture on YouTube We can solve another puzzle that also leads to the Fibonacci sequence: How many ways can one climb a staircase with n steps, taking one or two steps at a time? Any single climb can be represented by a string of ones and twos which sum to n. We define an as the number of different strings that sum to n. In Table 1, we list the possible strings for the first five values of n. It appears that the an’s form the beginning of the Fibonacci sequence. To derive a relationship between an and the Fibonacci numbers, consider the set of strings that sum to n. This set may be divided into two nonoverlapping subsets: those strings that start with one and those strings that start with two. For the subset of strings that start with one, the remaining part of the string must sum to n − 1; for the subset of strings that start with two, the remaining part of the string must sum to n − 2. Therefore, the number of strings that sum to n is equal to the number of strings that sum to n − 1 plus the number of strings that sum to n − 2. The number of strings that sum to n − 1 is given by an−1 and the number of strings that sum to n − 2 is given by an−2, so that an = an−1 + an−2. And from the table we have a1 = 1 = F2 and a2 = 2 = F3, so that an = Fn+1 for all positive integers n. n strings an 1 1 1 2 11, 2 2 3 111, 12, 21 3 4 1111, 112, 121, 211, 22 5 5 11111, 1112, 1121, 1211, 2111, 122, 212, 221 8 Table 2.1: Strings of ones and twos that add up to n. 7 8 LECTURE 2. THE FIBONACCI SEQUENCE REDUX Problems for Lecture 2 1. Consider a string consisting of the first n natural numbers, 123 . n. For each number in the string, allow it to either stay fixed or change places with one of its neighbors. Define an to be the number of different strings that can be formed. Examples for the first four values of n are shown in Table 2.2.
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