55 The solution of Fibonacci s rabbit problem is examined in Chapter, pages The Fibonacci Sequence and the Golden Ratio The Fibonacci Sequence One of the most famous problems in elementary mathematics comes from the book Liber Abaci, written in 0 by Leonardo of Pisa, aka Fibonacci The problem is as follows: A man put a pair of rabbits in a cage During the first month the rabbits produced no offspring, but each month thereafter produced one new pair of rabbits If each new pair thus produced reproduces in the same manner, how many pairs of rabbits will there be at the end of one year? The solution of this problem leads to a sequence of numbers known as the Fibonacci sequence Here are the first fifteen terms of the Fibonacci sequence:,,, 3, 5, 8, 3,, 34, 55, 89, 44, 33, 377, 60 Notice the pattern established in the sequence After the first two terms (both ), each term is obtained by adding the two previous terms For example, the third term is obtained by adding to get, the fourth term is obtained by adding to get 3, and so on This can be described by a mathematical formula known as a recursion formula If F n represents the Fibonacci number in the nth position in the sequence, then F F F n F n F n for n 3 *For an interesting general discussion, see The Mathematics of Identification Numbers, by Joseph A Gallian in The College Mathematics Journal, May 99, p 94
55 The Fibonacci Sequence and the Golden Ratio 33 The Fibonacci Association is a research organization dedicated to investigation into the Fibonacci sequence and related topics Check your library to see if it has the journal Fibonacci Quarterly The first two journals of 963 contain a basic introduction to the Fibonacci sequence The Fibonacci sequence exhibits many interesting patterns, and by inductive reasoning we can make many conjectures about these patterns However, as we have indicated many times earlier, simply observing a finite number of examples does not provide a proof of a statement Proofs of the properties of the Fibonacci sequence often involve mathematical induction (covered in college algebra texts) Here we simply observe the patterns and do not attempt to provide such proofs As an example of one of the many interesting properties of the Fibonacci sequence, choose any term of the sequence after the first and square it Then multiply the terms on either side of it, and subtract the smaller result from the larger The difference is always For example, choose the sixth term in the sequence, 8 The square of 8 is 64 Now multiply the terms on either side of 8: 5 3 65 Subtract 64 from 65 to get 65 64 This pattern continues throughout the sequence EXAMPLE Find the sum of the squares of the first n Fibonacci numbers for n,, 3, 4, 5, and examine the pattern Generalize this relationship The following program for the TI-83 Plus utilizes the Binet form of the n th Fibonacci number (see Exercises 33 38) to determine its value PROGRAM: FIB : Clr Home : Disp WHICH TERM : Disp OF THE : Disp SEQUENCE DO : Disp YOU WANT? : Input N : 5 l A : 5 l B : A N B N 5 l F : Disp F This screen indicates that the twentieth Fibonacci number is 6765 F F F F 3 6 3 F 3 F 4 3 5 3 5 F 4 F 5 3 5 40 5 8 F 5 F 6 The sum of the squares of the first n Fibonacci numbers seems to always be the product of F n and F n This has been proven to be true, in general, using mathematical induction There are many other patterns similar to the one examined in Example, and some of them are discussed in the exercises of this section An interesting property of the decimal value of the reciprocal of 89, the eleventh Fibonacci number, is examined in Example EXAMPLE Observe the steps of the long-division process used to find the first few decimal places for 89 035 89000000 89 0 89 0 78 30 67 530 445 850 Notice that after the 0 in the tenths place, the next five digits are the first five terms of the Fibonacci sequence In addition, as indicated in color in the process, the digits,,, 3, 5, 8 appear in the division steps Now, look at the digits next to the ones in color, beginning with the second ; they, too, are,,, 3, 5,
34 CHAPTER 5 Number Theory M F M F F M F F M F M F F M F M F F M F FIGURE 6 If the division process is continued past the final step shown on the preceding page, the pattern seems to stop, since to ten decimal places, 89 035955 (The decimal representation actually begins to repeat later in the process, since 89 is a rational number) However, the sum below indicates how the Fibonacci numbers are actually hidden in this decimal 0 00 000 00003 000005 0000008 0000003 0000000 0000000034 00000000055 000000000089 89 0359550 Fibonacci patterns have been found in numerous places in nature For example, male honeybees hatch from eggs which have not been fertilized, so a male bee has only one parent, a female On the other hand, female honeybees hatch from fertilized eggs, so a female has two parents, one male and one female Figure 6 shows several generations of a male honeybee (In this case, we plot generations of ancestors downward rather than generations of descendants) Notice that in the first generation there is bee, in the second there is bee, in the third there are bees, and so on These are the terms of the Fibonacci sequence Furthermore, beginning with the second generation, the numbers of female bees form the sequence, and beginning with the third generation, the numbers of male bees also form the sequence Successive terms in the Fibonacci sequence also appear in plants For example, the photo (below left) shows the double spiraling of a daisy head, with clockwise spirals and 34 counterclockwise spirals These numbers are successive terms in the sequence Most pineapples (see the photo on the right) exhibit the Fibonacci sequence in the following way: Count the spirals formed by the scales of the cone, first counting from lower left to upper right Then count the spirals from lower right to upper left You should find that in one direction you get 8 spirals, and in the other you get 3 spirals, once again successive terms of the Fibonacci sequence Many pinecones exhibit 5 and 8 spirals, and the cone of the giant sequoia has 3 and 5 spirals
55 The Fibonacci Sequence and the Golden Ratio 35 Continuing the Discussion of the Golden Ratio A fraction such as is called a continued fraction This continued fraction can be evaluated as follows Let x Then By the quadratic formula from algebra, x x 5 Notice that the positive solution is the golden ratio x x x x x x 0 4 5 The Golden Ratio If we consider the quotients of successive Fibonacci numbers, a pattern emerges It seems as if these quotients are approaching some limiting value close to 68 This is indeed the case As we go farther and farther out into the sequence, these quotients approach the number 5, known as the golden ratio, and often symbolized by, the Greek letter phi The golden ratio appears over and over in art, architecture, music, and nature Its origins go back to the days of the ancient Greeks, who thought that a golden rectangle exhibited the most aesthetically pleasing proportion A golden rectangle is defined as a rectangle whose dimensions satisfy the equation If we let L represent length and W represent width, we have Since LW, LL, and WL, this equation can be written Multiply both sides by 3 5 5 3 666 8 5 6 Length Width to get 3 8 65 3 6538465 34 6904769 55 34 67647059 89 55 68888 Length Width Length L W L W L L W L L W L Subtract and subtract from both sides to get the quadratic equation 0 Using the quadratic formula from algebra, the positive solution of this equation is found to be 5 68033989, the golden ratio
36 CHAPTER 5 Number Theory An example of a golden rectangle is shown in Figure 7 The Parthenon (see the photo), built on the Acropolis in ancient Athens during the fifth century BC, is an example of architecture exhibiting many distinct golden rectangles A Golden Rectangle in Art The rectangle outlining the figure in St Jerome by Leonardo da Vinci is an example of a golden rectangle FIGURE 7 To see how the terms of the Fibonacci sequence relate geometrically to the golden ratio, a rectangle that measures 89 by 55 units is constructed (See Figure 8 on the next page) This is a very close approximation to a golden rectangle Within this rectangle a square is then constructed, 55 units on a side The remaining rectangle is also approximately a golden rectangle, measuring 55 units by 34 units Each time this process is repeated, a square and an approximate golden rectangle are formed As indicated in the figure, vertices of the square may be joined by a smooth curve known as a spiral This spiral resembles the outline of a cross section of the shell of the chambered nautilus, as shown in the photograph next to Figure 8 FOR FURTHER THOUGHT Or, should we say, For Further Viewing? The 959 animated film Donald in Mathmagic Land has endured for more than 40 years as a classic It provides a 30-minute trip with Donald Duck, led by the Spirit of Mathematics, through the world of mathematics Several minutes of the film are devoted to the golden ratio (or, as it is termed there, the golden section) The Walt Disney Company Disney provides animation to explain the golden ratio in a way that the printed word simply cannot do The golden ratio is seen in architecture, nature, and the human body The film is available on video and can easily be purchased or rented For Group Discussion Verify the following Fibonacci pattern in the conifer family Obtain a pineapple, and count spirals formed by the scales of the cone, first counting from lower left to upper right Then count the spirals from lower right to upper left What do you find? Two popular sizes of index cards are 3 by 5 and 5 by 8 Why do you think that these are industry-standard sizes? 3 Divide your height by the height of your navel Find a class average What value does this come close to?
55 The Fibonacci Sequence and the Golden Ratio 37 55 34 55 8 8 34 3 FIGURE 8