Fibonacci & the golden mean

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2 Fibonacci & the golden mean The merit of painting lies in the exactness of reproduction. Painting is a science and all sciences are based on mathematics. No human inquiry can be a science unless it pursues its path through mathematical exposition and demonstration. Leonardo da Vinci

3 Funny Fibonacci Facts For the definition of the Fibonacci numbers see Fibonacci: The Greatest European Mathematician of the Middle ges (p. 57). Here are some unusual facts about the Fibonacci numbers. If you read Formula for the n th Term of the Fibonacci Sequence (p. 60) or The Golden Mean (p. 5), you will see some of them proved, and you might get ideas about how to prove others.. s n gets larger, the number Fn F gets closer to τ =, the golden mean. n n n+ n+. a) F + F = F for n. b) Fn+ Fn Fn = ( ) for n n+ n n 3n c) F + F F = F n for n. d) If m is a factor of n, then F m is a factor of F n. e) Let G F n = n F. (It follows from part d) that G n is an integer.) Then G = G + G n n n n. f) arctan(/f 4 ) arctan(/f 6 ) + arctan(/f 8 ) arctan(/f 0 ) + = arctan 3. ees have unusual reproductive habits. Each male bee has only one parent (a mother, the queen bee), whereas each female bee has two parents (both a mother and a father). So each male bee has one parent, and you can quickly check that he has two grandparents (the two parents of his mother). How many great-grandparents does he have? How many great-great-grandparents? an you guess the pattern? an you prove it? (rawing a family tree might help.) 54

4 4. a) an you prove this identity, F + F + + Fn = FnF? n+ This diagram may suggest a proof to you This is Mind-boggling! b) an you find a simple formula for F + F + + F n in terms of F n+? It might remind you of the formula for n in terms of n Use a calculator to evaluate /9899. epending on the number of digits on your calculator, you ll find How does this continue? an you guess the next few digits? (The remaining questions will make no sense unless you have already guessed the pattern.) The digits must eventually repeat, as /9899 is a rational number. o you find this surprising? an you find more numbers with a similar property? (Hint: Try /89. Then try to find more.) What if you wanted powers of to appear instead? 6. onsider a circle whose circumference is the golden mean, τ = Start at any point, on the circle and take some number of consecutive steps of arc length one in the clockwise direction. Number the points you step on in the order you encounter them, labelling your first step P, your second step P, and so on. Prove that when you stop, the difference in the subscripts of any two adjacent numbers is a Fibonacci number. What do you think the ratio of two adjacent arc-lengths (e.g. P 6 P 3 to P 3 P 8 in the figure) is? P 4 P P 9 P 6 P 3 difference is F 6 = 8 P 7 P 8 P P 0 P 5 55

5 7. The remarkable interplay between Fibonacci numbers and Pascal s triangle is best shown in a diagram without any words of explanation: There are several ways of proving why this is true. What may be the easiest method is to let P n denote the n th diagonal Pascal sum, and then observe that P = P = and show that P n = P n + P n for n > (which is not obvious). P = P = P 3 = P 4 = 3 P 5 = 5 P 6 = 8 P 7 = nother method is to use the Elvis way of looking at Fibonacci numbers. (See any of the sections dealing with Elvis Numbers.) In how many ways can Elvis get to the n th step? Well, we know (from The Solution to Elvis the Elf's Eccentric Exercise, p. 59) that he can manage this in F n ways. ut let s try an alternate way of looking at it. He can take n single steps and 0 double steps (which he can do in n = way); or he can take n single steps and double step (which he can do 0 n in = n ways, because out of the n- steps he takes, he must choose one to be the double step); or he can take n 4 single steps and double steps (which he n can do in ways, because out of the n- steps, he must choose two of them to be double steps); and so on. The total number of ways in which he can get to the n th step is thus n n n which is just P n. (Just look at the diagram again if you don t believe it!) Thus P n is the n th Elvis number, which is F n! 56

6 8. Elvis the Elf decided to create a landing of marble tiles at the foot of his front hall staircase. He measured off a rectangular area of length 3 feet and width 5 feet and determined that he would need 65 marble tiles measuring one foot square to cover this landing. He purchased 65 tiles but upon arriving home discovered that one tile was damaged beyond repair and only 64 of the tiles could be used. Undaunted by the challenge, he arranged the 64 tiles in an 8 8 square array and then cut the large square into four regions,,, and as shown below left. He then rearranged these four pieces into the 3 5 rectangle shown below right. How did Elvis the Elf create a rectangle of area 65 square feet by rearranging the pieces of a square of area 64 square feet? Take a piece of graph paper, and cut out an 8 8 square. Then cut the square into four pieces as shown above. Rearrange these four pieces to form a 5 3 rectangle. What do you discover? Notice that the numbers (5, 8, 3) are consecutive Fibonacci numbers. Try to create a paradox like the one above by replacing them with another triplet of larger consecutive Fibonacci numbers, such as (8, 3, ). You might notice a similarity to fact # a). I learned of some of the more unusual results in fact # from Georg Gunther of Memorial University of Newfoundland. The bee problem is from H. R. Jacobs, Mathematics, Human Endeavor, pp Fact # 6 was related to me by Greg Kuperberg of the University of alifornia at avis. Fact # 8 is taken from oxeter s, The Golden Section, Phyllotaxis, and Wythoff s Game (see the nnotated References.) 57

7 The Golden Mean The golden mean is the number τ = + 5 ( ), often represented by the Greek letter τ (spelled tau, and pronounced so it rhymes with Yow! ). The ancient Greeks felt that the τ rectangle was the most aesthetically pleasing, and much classical architecture is based on that proportion. (ny rectangle similar to this one is called a golden rectangle.) The Parthenon of ncient Greece, completed in 438.., was constructed so that its front elevation forms a golden rectangle. ut nature also uses the golden mean in its own architecture. The aesthetically-pleasing τ rectangle, 500 years on. τ, sometimes called the golden section or golden ratio, is the first letter of the ancient Greek word τοµη meaning the section. τ has already appeared in our discussions about the Fibonacci sequence, where we saw that: n n τ ( τ) F = n 5 58 In oxeter s The Golden Section, Phyllotaxis, and Wythoff s Game, further unusual occurrences of the golden mean are discussed, including a surprising appearance of Fibonacci numbers in pineapples. (For bibliographic information, see the nnotated References.)

8 Some Interesting Properties of the Golden Mean The formula for Fibonacci numbers (p. 60) is just one of a series of surprising situations in which τ mysteriously appears. Here are some of them, many of which you can try to prove yourself.. a) τ is an irrational number; in other words, there are no positive integers m and n such that τ = m/n. ut τ can be approximated by rational numbers. For example, 3/ =.5 is pretty close; 55/34 (.6764 ) is closer still. In fact, the best rational approximations to the golden mean are given by ratios of consecutive Fibonacci numbers! Fn + b) related fact is that the fraction F converges to τ as n gets large. n c) In fact, if you take any two positive real numbers a and b, and define the sequence G n by G = a, G = b, and G n = G n- +G n-, then the fraction gets closer and closer to τ as n gets large. (This is almost true if you take any two real numbers. The only exception is if b is a certain magic multiple of a. EXPERTS ONLY. a) τ τ = 0. Gn G What is that multiple? Hint: That number appears on p. 60) b) Using this equation, you can show that if you take a golden rectangle (where is the short side), and subtract a square, EF, then the remaining rectangle FE is also golden. E τ - + n = =τ E F This fact is related to fact # 4 a) in Funny Fibonacci Facts (p. 50). 59

9 Some Interesting Properties of the Golden Mean (cont d) 3. Here are a couple of calculator experiments (that can also be done on a computer). a) hoose any positive number. Write it down on a piece of paper, and enter it on your calculator. Then press the following keys: /x + = Record the new number. Repeat by pressing the same sequence of keys, and then once again write down the number you get. Keep doing this. What do you notice? (If you start with F n+ /F n, what are the following terms?) b) Repeat the process in part a), but press this sequence of keys: 4. a) = ( ) + ( + ) + ( + ) + = 5 logτ (This is related to some deep number theory: it is an L-value. similar but easier sum is / + /4 /5 + /7 /8 ) b) sin 8 = τ 4 c) cos = τ 5. The icosahedron is a platonic solid with twenty faces. It will be familiar to some readers as a twenty-sided die. If and are two neighboring vertices, and and are two vertices once removed, then = τ. lso, the four vertices shown in the figure form a golden rectangle. 60 This sequence is not universal to all calculators. I heard this entertaining fact from the analytic number theorist Kannan Soundararajan. n example of an L-function is the zeta function defined on p. 4, which is the subject of the Riemann hypothesis.

10 Some Interesting Properties of the Golden Mean (cont d) 6. y pure coincidence (or some evil conspiracy), mile is approximately equal to τ kilometers. This means that you can use Fibonacci numbers to quickly translate between miles and kilometers. For example, 3 km is about 8 miles; 89 km/h is approximately 55 mph. 7. More experimentation. Try This! a) The greatest-integer function [ ] is defined as follows: [x] is the integral part of x. For example, [9.7] = 9, and [ π ] = 3. Make a table, where in row n we write down [ nτ ] and [ nτ ] (so we multiply τ by n and chop off the decimal, and similarly for τ ). n nτ nτ What patterns can you find? b) (You will only need about four decimal places of accuracy for this one.) Make another list of real numbers: τ/ 5, τ / 5, τ 3 / 5, τ 4 / 5, What patterns can you find? c) o the same thing as in part b) with the sequence: τ, τ, τ 3, τ 4, strange fact about these strange properties: a surprising number of them have in some way to do with the number 5. 5 appears in the definition of τ. It also appears in properties #4 (notice that 8 = 90 /5), #5, and #7 b). Many of these properties are discussed elsewhere (see Funny Fibonacci Facts, p. 50, and The Pythagorean Pentagram, The Golden Mean, & Strange Trigonometry, p.58). If you want to learn more about the golden mean, you should visit these pages. ut if you haven t already done so, you should first check out the very first section on this topic, Elvis Numbers (p. 56). n n τ / n τ n

11 The Pythagorean Pentagram, The Golden Mean, & Strange Trigonometry The golden mean τ (τ = ) is one of those mathematical constants (such as π or e) that pops up in the most unusual places, as we have seen in the previous section. For example, if you use your calculator to compute cos or sin 8 +, you ll get τ. If you compute the reciprocal of sin 8, you ll get τ again. Let s see why this is true. onsider the regular pentagon with all of the diagonals drawn in, and labelled as follows: E H I G F J This figure, called a pentagram, was the symbol of the Pythagorean brotherhood. The Pythagoreans were a secret society dedicated to the pursuit of mathematics and philosophy, established by Pythagoras of Samos (c ) on the southeastern coast of what is now Italy. 6

12 With a little thought, you can fill in all the angles. (Hint: start by showing that the angle between consecutive sides of a pentagon is 08.) E H I G 7 7 F J We choose our unit of length so that the pentagon has side of length.therefore, =. Let t denote the length of. y checking angles, you ll see that I is similar to, so Since = and = t, I =. t I is isosceles, so I =. is, so I =. I =. t = = I + I = t +, so t t = 0. The roots of this quadratic equation in t are + 5 and 5. s t is a length, it is positive, so we can throw out the second solution. Therefore, t = + 5, the golden mean! 63

13 Equipped with this information, we have two basic triangles: and. τ τ E τ You should be able to see lots of triangles similar to both of these in the pentagram. For example, E and are equiangular and therefore similar. From here, we can work out lots of trigonometric values. For example, if we drop an altitude from to E in E, as shown below, E τ K τ we see that cos = EK E = τ You can derive a lot more yourself. The next page provides some ideas to get you started. 64

14 Food for Thought q. pply the cosine law to one of the basic triangles to compute cos 08. sin 7 pply the sine law to compute, and (using sin 7 = sin cos ) sin compute cos in another way. r. Prove that sin 8 = τ. ompute cos 7 and sin 54. s. For the original pentagram, show that I = = = τ. I IJ rea E ompute the ratio rea FGHIJ. t. regular decagon (0-gon) of side is inscribed in a circle. What is the radius of the circle? (Hint: See Some Interesting Properties of the Golden Mean, p. 55.)? u. Using the relationships in exercise 3, and the fact that the ratios of consecutive Fibonacci numbers (especially large ones) are approximately τ, we can fill in lengths of the pentagram that are approximately integers, and Fibonacci numbers to boot. an 34? you determine the missing length? (Warning: these are only approximations!) 3 For an easy way to compute other odd trigonometric values, such as sin 5 = 6 4 flip to Fifteen egress of Separation (p. 88). For the impossibility (in some sense) of computing some values (cos 0, sin ), see Three Impossible Problems of ncient Greece, p. 9., 65