The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)

5. SON OF GOOD NATURE

The nine Indian figures are: 9 8 7 6 5 4 3 2 1.

With these nine figures, and with the sign 0… any number
may be written, as is demonstrated below.

—LEONARDO FIBONACCI (CA. 1170s-1240s)

With the above words, Leonardo of Pisa (in Latin Leonardus Pisanus), also known as Leonardo Fibonacci, began his first and best-known book, Liber abaci (Book of the abacus), published in 1202. At the time the book appeared, only a few privileged European intellectuals who cared to study the translations of the works of al-Khwarizmī and Abu Kamil knew the Hindu-Arabic numerals we use today. Fibonacci, who for a while joined his father, a customs and trading official, in Bugia (in present-day Algeria) and later traveled to other Mediterranean countries (including Greece, Egypt, and Syria), had the opportunity to study and compare different numerical systems and methods for arithmetical operations. Upon concluding that the Hindu-Arabic numerals, which included the place-value principle, were far superior to all other methods, he devoted the first seven chapters of his book to explanations of Hindu-Arabic notation and its use in practical applications.

Leonardo Fibonacci was born in the 1170s to a businessman and government official named Guglielmo. The nickname Fibonacci (from the Latin filius Bonacci, son of the Bonacci family, or “son of good nature”) was most probably introduced by the historian of mathematics Guillaume Libri in a footnote in his 1838 book Histoire des Sciences Mathematique en Italie (History of the mathematical sciences in Italy), although some researchers attribute the first use of Fibonacci to Italian mathematicians at the end of the eighteenth century. In some manuscripts and documents, Leonardo either refers to himself or is referred to as Leonardo Bigollo (or Leonardi Bigolli Pisani), where “Bigollo” means something like “a traveler” or a “man of no importance” in the Tuscan and Venetian dialects respectively. Pisa of the twelfth century was a busy port through which merchandise passed both from inland and from overseas. Spices from the Far East circulated through Pisa on their way to northern Europe, crossing in the port the paths of wine, oil, and salt that were transported between different parts of Italy, Sicily, and Sardinia. The large Pisan leather industry imported goatskins from North Africa, and tanners could be seen processing hides on Pisa's river-banks. The city, on the river Arno, was also proud of its excellent ironwork and shipyards. Pisa is best known today for its famous leaning tower, and the construction of this bell tower began during Fibonacci's youth. Clearly, all of this commercial frenzy required massive records of inventories and prices. Leonardo surely had the opportunity to watch various scribes as they were listing prices in Roman numerals and adding them up using an abacus. Arithmetic operations with Roman numerals are not fun. For example, to obtain the sum of 3,786 and 3,843, you would need to add MMMDCCLXXXVI to MMMDCC-CXLIII; if you think that is cumbersome, try multiplying those numbers. However, for as long as medieval merchants stuck to simple additions and subtractions, they could get by with Roman numerals. The fundamental element that the Roman numerals were lacking was, of course, the place-value system—the fact that a number written as 547 really means (5 × 102) +(4 × 101) +(7 × 10°). The West Europeans overcame the lack of a place-value principle in their number system by the use of the abacus. The name “abacus” may have originated from avaq, the Hebrew word for dust, since the earliest of these calculation devices were simply boards dusted with sand on which numbers could be traced. The abacus during Fibonacci's time had counters sliding along wires. The different wires of the abacus played the role of place value. A typical abacus had four wires, with beads on the bottom wire representing units, those on the one above it tens, those on the third hundreds, and those on the top wire thousands. Thus, while the abacus provided a fairly efficient means for simple arithmetic operations (I was amazed to discover during a visit to Moscow in 1990 that the cafeteria in my hotel was still using an abacus), it clearly presented enormous disadvantages when handling more complex computations. It is impossible to imagine, for example, trying to manipulate the “billions and billions” of astronomy popularizer Carl Sagan using an abacus.

In Bugia (now called Bejaïa), in Algeria, Fibonacci became acquainted with the art of the nine Indian figures, probably with, in his words, the “excellent instruction” of an Arab teacher. Following a tour around the Mediterranean, which he used to expand his mathematical horizons, he decided to publish a book that would introduce the use of Hindu-Arabic numerals more widely into commercial life. In this book, Fibonacci meticulously explains the translation from Roman numerals to the new system and the arithmetic operations with the new numerals. He gives numerous examples that demonstrate the application of his “new math” to a variety of problems ranging from business practices and the filling and emptying of cisterns to the motions of ships. At the beginning of the book, Fibonacci adds the following apology: “If by chance I have omitted anything more or less proper or necessary, I beg forgiveness, since there is no one who is without fault and circumspect in all matters.”

In many cases, Fibonacci gave more than one version of the problem, and he demonstrated an astonishing versatility in the choice of several methods of solution. In addition, his algebra was often rhetorical, explaining in words the desired solution rather than solving explicit equations, as we would do today. Here is a nice example of one of the problems that appear in Liber abaci (as translated in the charming book Leonard of Pisa and the New Mathematics of the Middle Ages by Joseph and Frances Gies):

A man whose end was approaching summoned his sons and said: “Divide my money as I shall prescribe.” To his eldest son, he said, “You are to have 1 bezant [a gold coin first struck at Byzantium] and a seventh of what is left.” To his second son he said, “Take 2 bezants and a seventh of what remains.” To the third son, “You are to take 3 bezants and a seventh of what is left.” Thus he gave each son 1 bezant more than the previous son and a seventh of what remained, and to the last son all that was left. After following their father's instructions with care, the sons found that they had shared their inheritance equally. How many sons were there, and how large was the estate?

For the interested reader, I present both the algebraic (modern) solution and Fibonacci's rhetorical solution to this problem in Appendix 6.

The Liber abaci brought Fibonacci considerable recognition, and his fame reached even the ears of the Roman emperor Frederick II, known as “Stupor Mundi” (“Wonder of the World”) for his patronage of mathematics and the sciences. Fibonacci was invited to appear before the emperor in Pisa in the early 1220s and was presented with a series of what were considered to be very difficult mathematical problems, by Master Johannes of Palermo, one of the court mathematicians. One of the problems read as follows: “Find such a rational number [a whole number or a fraction] that when 5 is either added to or subtracted from its square, the result [in either case] is also the square of a rational number.” Fibonacci solved all the problems using ingenious methods. He later described two of them in a short book called Flos (Flower) and used the one above in the prologue of a book he dedicated to the emperor: Liber quadratorum (Book of squares). Today we have to be impressed by the fact that without relying on computers or calculators of any sort, simply through his virtuosic command of number theory, Fibonacci was able to find out that the solution to the problem above is image. Indeed, image.

Fibonacci's role in the history of the Golden Ratio is truly fascinating. On one hand, in problems in which he consciously used the Golden Ratio, he is responsible for a significant but not spectacular progress. On the other, by simply formulating a problem that on the face of it has no relation whatsoever to the Golden Ratio, he expanded the scope of the Golden Ratio and its applications dramatically.

Fibonacci's direct contributions to the Golden Ratio literature appear in a short book on geometry, Practica Geometriae (Practice of geometry), which was published in 1223. He presented new methods for the calculation of the diagonal and the area of the pentagon, calculations of the sides of the pentagon and the decagon from the diameter of both inscribed and circumscribed circles, and computations of volumes of the dodecahedron and the icosahedron, all of which are intimately related to the Golden Ratio. In the solutions to these problems Fibonacci exhibits a deep understanding of Euclidean geometry. While his mathematical techniques rely to some extent on previous works, in particular on Abu Kamil's On the Pentagon and the Decagon, there is little doubt that Fibonacci brought the use of the Golden Ratio's properties in various geometrical applications to a higher level. However, Fibonacci's main claim to fame and his most exciting contribution to the Golden Ratio derive from an innocent-looking problem in Liber abaci.

ALL THE THOUGHTS OF A RABBIT ARE RABBITS

Many students of mathematics, the sciences, and the arts have heard of Fibonacci only because of the following problem from Chapter XII in the Liber abaci.

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

How can the numbers of the offspring of rabbits have significant mathematical consequences? Indeed, the solution to the problem itself is quite simple. We start with one pair. After the first month, the first pair gives birth to another pair, hence there are two. In Figure 27 I represent a mature pair with a large rabbit symbol and a young pair with a small symbol. After the second month, the mature pair gives birth to another young pair, while the baby pair matures. Hence, there are three pairs, as depicted in the figure. After the third month, each of the two mature pairs gives birth to another pair, and the baby pair matures, so there are five. After the fourth month, each of the three mature pairs gives birth to a pair, and the two baby pairs mature, giving us a total of eight pairs. After five months we have a baby pair from each of the five adult pairs, plus three maturing pairs for a total of thirteen. By now we understand how to proceed to obtain the numbers of mature pairs, of baby pairs, and of pairs in total in successive months. Suppose we examine just the number of adult pairs in any particular month. That number is composed of the number of adult pairs in the previous month, plus the number of baby pairs (which have matured) from that same previous month. However, the number of baby pairs from the previous month is actually equal to the number of adult pairs in the month before that. Therefore, in any given month (starting with the third), the number of adult pairs is simply equal to the sum of the numbers of adult pairs in the two preceding months. The number of adult pairs therefore follows the sequence: 1, 1, 2, 3, 5, 8,… You can easily see from the figure that the numbers of baby pairs follow precisely the same sequence, only displaced by one month. Namely, the numbers of baby pairs are 0, 1, 1, 2, 3, 5, 8,… Of course, the total number of pairs is simply the sum of these, and it gives the same sequence as for the adult pairs, with the first term omitted (1, 2, 3, 5, 8,…). The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…, in which each term (starting with the third) is equal to the sum of the two preceding terms, was appropriately dubbed the Fibonacci sequence in the nineteenth century, by the French mathematician Edouard Lucas (1842–1891). Number sequences in which the relation between successive terms can be expressed by a mathematical expression are known as recursive. The Fibonacci sequence was the first such recursive sequence known in Europe. The general property that each term in the sequence is equal to the sum of the two preceding ones is expressed mathematically as (a notation introduced in 1634 by the mathematician Albert Girard): Fn+2=Fn+1 + Fn Here Fn represents the nthnumber in the sequence (e.g., F5 is the fifth term); Fn+1 is the term following Fn (for n = 5, n+1 = 6), and Fn+2 follows Fn+1

image

Figure 27

The reason that Fibonacci's name is so famous today is that the appearance of the Fibonacci sequence is far from being confined to the breeding of rabbits. Incidentally, the title of this chapter was inspired by Ralph Waldo Emerson's The Natural History of Intellect, which appeared in 1893. Emerson says: “All the thoughts of a turtle are turtles, and of a rabbit, rabbits.” We shall encounter the Fibonacci sequence in an incredible variety of seemingly unrelated phenomena.

To start things off, let us examine a phenomenon that is just about as remote from the topic of rabbit progeny as we could possibly imagine—the optics of light rays. Suppose we have two glass plates made of slightly different types of glass (different light refraction properties, or “indices of refraction”) mounted face to face (as in Figure 28a). If we shine light through the plates, the light rays can (in principle) reflect internally at four reflective surfaces before emerging (Figure 28a). More specifically, they can either pass through without reflecting at all, or they can undergo one internal reflection, two internal reflections, three internal reflections, and so on, potentially an infinite number of internal reflections before reemerging. All of these are paths allowed by the laws of optics. Now count the number of beams that emerge from this two-plate system. There is only one emerging beam in the case of no reflections at all (Figure 28b). There are two emerging beams when all the possibilities for the rays to undergo precisely one internal reflection are considered (Figure 28c), because there are two paths the ray can follow. There are three emerging beams for all the possibilities of two internal reflections (Figure 28d); five beams for three internal reflections (Figure 28e); eight paths if the ray is reflected four times (Figure 28f); thirteen paths for five reflections (Figure 28g); and so on. The numbers of emerging beams—1, 2, 3, 5, 8, 13… —form a Fibonacci sequence.

Now consider the following entirely different problem. A child is trying to climb a staircase. The maximum number of steps he can climb at a time is two; that is, he can climb either one step or two steps at a time. If there are nsteps in total, in how many different ways, Cn, can he climb the staircase? If there is only one step (n = 1), clearly there is only one way to climb it, C1 = 1. If there are two steps, the child can either climb the two steps at once or take them one step at a time; thus, there are two ways, C2 = 2. If there are three steps, there are three ways of climbing: 1+ 1+ 1, 1+2, or 2+ 1; therefore C3 = 3. If there are four steps, the number of ways to climb them increases to C4 = 5: 1+ 1+ 1+ 1, 1+2+ 1, 1+ 1+2, 2+ 1+ 1, 2+ 2. For five steps, there are eight ways, C5 = 8: 1+ 1+ 1+ 1+ 1, 1+ 1+ 1+ 2, 1+ 1+2+ 1, 1+2 + 1+ 1, 2+ 1+ 1+ 1, 2+ 2+ 1, 2+ 1+2, 1+2+2. We find that the numbers of possibilities, 1, 2, 3, 5, 8,…, form a Fibonacci sequence.

image

Figure 28

Finally, let us examine the family tree of a drone, or male bee. Eggs of worker bees that are not fertilized develop into drones. Hence, a drone has no “father” and only a “mother.” The queen's eggs, on the other hand, are fertilized by drones and develop into females (either workers or queens). A female bee has therefore both a “mother” and a “father.” Consequently, one drone has one parent (its mother), two grandparents (its mother's parents), three great-grandparents (two parents of its grand mother and one of its grandfather), five great-great-grandparents (two for each great-grandmother and one for its great-grandfather), and so on. The numbers in the family tree, 1, 1, 2, 3, 5…, form a Fibonacci se quence. The tree is presented graphically in Figure 29.

This all looks very intriguing—the same series of numbers applies to rabbits, to op tics, to stair climbing, and to drone family trees—but how is the Fibonacci sequence related to the Golden Ratio?

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Figure 29

GOLDEN FIBONACCIS

Examine again the Fibonacci sequence; 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,…, and this time let us look at the ratios of successive numbers (calculated here to the sixth decimal place):

image

Do you recognize this last ratio? As we go farther and farther down the Fibonacci sequence, the ratio of two successive Fibonacci numbers oscillates about (being alternately greater or smaller) but comes closer and closer to the Golden Ratio. If we denote the nth Fibonacci number by Fn, and the next one by Fn +1, then we discovered that the ratio Fn + /Fn approaches ö as n becomes larger. This property was discovered in 1611 (although possibly even earlier by an anonymous Italian) by the famous German astronomer Johannes Kepler, but more than a hundred years passed before the relation between Fibonacci numbers and the Golden Ratio was proven (and even then not fully) by the Scottish mathematician Robert Simson (1687–1768). Kepler, by the way, apparently hit upon the Fibonacci sequence on his own and not via reading the Liber abaci.

But why should the terms in a sequence derived from the breeding of rabbits approach a ratio defined through the division of a line? To understand this connection, we have to go back to the astonishing continued fraction we encountered in Chapter 4. Recall that we found that the Golden Ratio can be written as

image

In principle, we could calculate the value of ö by a series of successive approximations, in which we would interrupt the continued fraction farther and farther down. Suppose we attempted to do just that. We would find the series of values (reminder: 1 over a/b is equal to b/a):

image

image

In other words, the successive approximations we find for the Golden Ratio are precisely equal to the ratios of Fibonacci numbers. No wonder then that as we go to higher and higher terms in the sequence the ratio converges to the Golden Ratio. This property is described beautifully in the book On Growth and Form by the famous naturalist Sir D'Arcy Wentworth Thompson (1860–1948). He writes about the Fibonacci numbers: “Of these famous and fascinating numbers a mathematical friend writes to me: ‘All the romance of continued fractions, linear recurrence relations,… lies in them, and they are a source of endless curiosity. How interesting it is to see them striving to attain the unattainable, the Golden Ratio, for instance; and this is only one of hundreds of such relations.’ ” The convergence to the Golden Ratio, by the way, explains the magic trick I described in Chapter 4. If you define a series of numbers by the property that each term (starting with the third) is equal to the sum of the two preceding ones, then irrespective of the two numbers you started with, as long as you go sufficiently far down the sequence, the ratio of two successive terms always approaches the Golden Ratio.

The Fibonacci numbers, like the “aspiration” of their ratios—the Golden Ratio—have some truly amazing properties. The list of mathematical relations involving Fibonacci numbers is literally endless. Here are just a handful of them.

“Squaring” Rectangles

If you sum up an odd number of products of successive Fibonacci numbers, like the three products 1×1+1×2+2×3, then the sum (1+2+6 = 9) is equal to the square of the last Fibonacci number you used in the products (in this case, 32 = 9). To take another example, if we sum up seven products, 1×1+ 1×2+2×3+3×5+5 × 8+ 8 × 13+ 13 × 21 = 441, the sum (441) is equal to the square of the last number used (212 = 441). Similarly, summing up the eleven products 1×1+ 1 ×2+2×3+3×5+5×8+ 8×13+ 13×21+21 × 34+ 34×55+ 55×89+ 89 × 144 = 1442. This property can be represented beautifully by a figure (Figure 30). Any odd number of rectangles with sides equal to successive Fibonacci numbers fits precisely into a square. The figure shows an example with seven such rectangles.

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Figure 30

Eleven Is the Sin

In the drama The Piccohmini by the German playwright and poet Friedrich Schiller, astrologer Seni declares: “Elf ist die Sünde. Elfe über-schreiten die zehn Gebote” (“Eleven is the sin. Eleven transgresses the Ten Commandments”), expressing an opinion that dates back to medieval times. The Fibonacci sequence, on the other hand, has a property related to the number 11, which, far from being sinful, is quite beautiful.

Suppose we sum up the first ten consecutive Fibonacci numbers: 1+ 1+2+ 3+ 5+8+ 13+ 21+ 34+ 55 = 143. This sum is divisible evenly by 11 (143/11 = 13). The same is true for the sum of any ten consecutive Fibonacci numbers. For example, 55+ 89+ 144+ 233+ 377+ 610+ 987+ 1,597+ 2,584+ 4,181 = 10,857, and 10,857 is divisible by 11, 10,857/11 = 987. If you examine these two examples, you discover something else. The sum of any ten consecutive numbers is always equal to 11 times the seventh number. You can use this property to amaze an audience by the speed with which you can add any ten successive Fibonacci numbers.

Revenge of the Sexagesimal?

As you recall, for reasons that are not entirely clear, the ancient Babylonians used base 60 (the sexagesimal base) in their counting system. Although not related to the Babylonian number system, the number 60 happens to play a role in the Fibonacci sequence.

Fibonacci numbers become very large quite rapidly, because you always add two successive Fibonacci numbers to find the next one. In fact, we are quite lucky that rabbits don't live forever, or we would all be inundated with rabbits. While the fifth Fibonacci number is only 5, the 125th is already 59,425,114,757,512,643,212,875,125. Interestingly, the unit digit repeats itself with a periodicity of 60 (namely, after every 60 numbers). For example, the second number is 1, the sixty-second number is 4,052,739,537,881 (also ending in 1); the 122nd number, 14,028,366,653,498,915,298,923,761, also ends in 1; and so does the 182nd; and so on. Similarly, the fourteenth number is 377; the seventy-fourth number (sixty numbers farther along the sequence) 1,304,969,544,928,657 also ends in 7; and so on. This property was discovered in 1774 by the Italian-born French mathematician Joseph Louis Lagrange (1736–1813), who is responsible for many works in number theory and mechanics and who also studied the stability of the solar system. The last two digits (e.g., 01, 01, 02, 03, 05, 08, 13, 21…) repeat in the sequence with a periodicity of 300 and the last three digits repeat with a periodicity of 1,500. In 1963 Stephen P. Geller used an IBM 1620 computer to show that the last four digits repeat every 15,000 times, the last five repeat every 150,000 times, and finally, after the computer ran for nearly three hours, a repetition of the last six digits appeared at the 1,500,000th Fibonacci number. Being unaware of the fact that a general theorem concerning the periodicity of the last digits could be proven, Geller commented: “There does not yet seem to be any way of guessing the next period, but perhaps a new program for the machine which will permit initialization at any point in the sequence for a test will cut down computer time enough so that more data can be gathered.” Shortly thereafter, however, Israeli mathematician Dov Jarden pointed out that one can prove rigorously that for any number of last digits from three and up, the periodicity is simply fifteen times ten to a power that is one less than the number of digits (e.g., for seven digits it is 15 × 10, or 15 million).

Why image?

The properties of our universe, from the sizes of atoms to the sizes of galaxies, are determined by the values of a few numbers known as constants of nature. These constants include a measure of the strengths of all the basic forces—gravitational, electromagnetic, and two nuclear forces. The strength of the familiar electromagnetic force between two electrons, for example, is expressed in physics in terms of a constant known as the fine structure constant. The value of this constant, almost exactly image, has puzzled many generations of physicists. A joke made about the famous English physicist Paul Dirac (1902–1984), one of the founders of quantum mechanics, says that upon arrival to heaven he was allowed to ask God one question. His question was: “Why image

The Fibonacci sequence also contains one absolutely remarkable number—its eleventh number, 89. The value of image in decimal representation is equal to: 0.01123595… Suppose you arrange the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21,…as decimal fractions in the following way:

image

In other words, the units digit in the first Fibonacci number is in the second decimal place, that of the second is in the third decimal place, and so on (the units digit of the nth Fibonacci number is in the (n+1)th decimal place). Now add all of those numbers up. In the preceding list we would obtain 0.01123595…, which is equal to image.

Lightning Addition Trick

Some people can add numbers very quickly in their heads. The Fibonacci sequence allows a person to perform such lightning addition tricks without much effort. The sum of all the Fibonacci numbers from the first to the nth is simply equal to the (n+ 2)th number minus 1. For example, the sum of the first ten numbers, 1+ 1+2+ 3+ 5+ 8+ 13+21+34+ 55 = 143, is equal to the twelfth number (144) minus 1. The sum of the first seventy-eight numbers is equal to the eightieth number minus 1; and so on. Therefore, you can have someone write a long column of numbers starting with 1, 1, and continuing using the definition of the Fibonacci sequence (that each new number be the sum of the two previous ones). Tell this person to draw a line between some arbitrary two numbers in the column and you will be able, at a glance, to give the sum of all the numbers prior to the line. That sum will be equal to the second term after the line minus one.

Pythagorean Fibonaccis

Oddly enough, Fibonacci numbers can even be related to Pythagorean triples. The latter, as you recall, are triples of numbers that can serve as the lengths of the sides of a right-angled triangle (like the numbers 3, 4, 5). Take any four consecutive Fibonacci numbers, such as 1, 2, 3, 5. The product of the outer numbers, 1×5 = 5, twice the product of the inner terms, 2×2×3 = 12, and the sum of the squares of the inner terms, 22+32 = 13, give the three legs in the Pythagorean triple, 5,12,13(52+122=132). But this is not all. Notice also that the third number, 13, is itself a Fibonacci number. This property was discovered by the mathematician Charles Raine.

Given the numerous wonders that the Fibonacci numbers hold in store (we shall soon encounter many more), it should come as no surprise that mathematicians looked for some efficient method for calculating these numbers, Fn'for any value of n While in principle this is not a problem, since if we need the 100th number we simply have to add up the 98th and the 99th numbers, this still means that we first need to calculate all the numbers up to the 99th, which can be quite tedious. As the late comedian George Burns (in his book How to Live to Be 100 or More) once put it: “How do you live to be 100 or more? There are certain things you have to do. The most important one is you have to be sure to make it to 99.”

In the middle of the nineteenth century, the French mathematician Jacques Phillipe Marie Binet (1786–1856) rediscovered a formula that was apparently known already in the eighteenth century to the most prolific mathematician in history, Leonard Euler (1707–1783), and to the French mathematician Abraham de Moivre (1667–1754). The formula allows you to find the value of any Fibonacci number, Fn, if its place in the sequence, n, is known. This Binet formula relies entirely on the Golden Ratio

image

At first glance, this is a formidably disconcerting formula, since it is not even obvious that upon substitution of various values of n it would produce whole numbers (which all the terms in the Fibonacci sequence are). Since we already know that the Fibonacci numbers are intimately related to the Golden Ratio, things start to look a little bit more reassuring when we realize that the first term inside the brackets is, in fact, simply the Golden Ratio raised to the nth power, ö n, and the second is (—1/ ö )n. (Recall from earlier that the negative solution of the quadratic equation defining ö is equal to —1/ ö .) Using a simple scientific pocket calculator you can test for a few values of n that Binet's formula is indeed giving the Fibonacci numbers correctly. For relatively large values of n, the second term in the brackets above becomes very small, and you can simply take Fn to be the closest whole number to image. For example, for n=10, image is equal to 55.0036, and the tenth Fibonacci number is 55.

Just as an amusement, you may wonder if there is a Fibonacci number with precisely 666 digits. Mathematician and author Clifford A. Pickover calls numbers associated with 666 “apocalyptic.” He found that the 3,184thFibonacci number has 666 digits.

Once discovered, Fibonacci numbers seemed to start popping up everywhere in nature. A few fascinating examples are provided by botany

AS THE SUNFLOWER TURNS ON HER GOD

The leaves along a twig of a plant or the stems along a branch tend to grow in positions that would optimize their exposure to sun, rain, and air. As a vertical stem grows, it produces leaves at quite regular spacings. However, the leaves do not grow directly one above the other, because this would shield the lower leaves from the moisture and sunlight they need. Rather, the passage from one leaf to the next (or from one stem to the next along a branch) is characterized by a screw-type displacement around the stem (as in Figure 31). Similar arrangements of re peating units can be found in the scales of a pinecone or the seeds of a sunflower. This phenomenon is called phyllotaxis(“leaf arrangement” in Greek), a word coined in 1754 by the Swiss naturalist Charles Bonnet (1720–1793). For example, in basswoods leaves occur generally on two opposite sides (corresponding to half a turn around the stem), which is known as a ½ phyllotactic ratio. In other plants, such as the hazel, blackberry, and beech, passing from one leaf to the next involves one-third of a turn (⅓ phyllotactic ratio). Similarly, the apple, the coast live oak, and the apricot have leaves every ⅖ of a turn, and the pear and the weeping willow have them every ⅜ of a turn. Figure 31 illustrates a case where it took three complete turns to pass through eight stems (a phyllotactic ratio of ⅜). You'll notice that all the fractions that are observed are ratios of alternate members of the Fibonacci sequence.

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Figure 31

The fact that leaves of plants follow certain patterns was first noted in antiquity by Theophrastus (ca. 372 B.C.—ca. 287 B.C.) in Enquiry into Plants. He remarks: “those that have flat leaves have them in a regular series.” Pliny the Elder (A.D. 23–79) made a similar observation in his monumental Natural History, where he talks about “regular intervals” between leaves “arranged circularly around the branches.” The study of phyllotaxis did not go much beyond these early, qualitative observations until the fifteenth century, when Leonardo da Vinci (1452–1519) added a quantitative element to the description of leaf arrangements by noting that the leaves were arranged in spiral patterns, with cycles of five (corresponding to an angle of ⅖ of a turn). The first person to discover (intuitively) the relation between phyllotaxis and the Fibonacci numbers was the astronomer Johannes Kepler. Kepler wrote: “It is in the likeness of this self-developing series [referring to the recursive property of the Fibonacci sequence] that the faculty of propagation is, in my opinion, formed; and so in a flower the authentic flag of this faculty is shown, the pentagon.”

Charles Bonnet initiated serious studies in observational phyllotaxis. In his 1754 book Recherches sur l'Usage des Feuilles dans les Plantes (Research on the use of leaves in plants) he gives a clear description of ⅖ phyllotaxis. While working with the mathematician G. L. Calandrini, Bonnet may have also discovered that sets of spiral rows (now known as parastichies) appear in some plants, like the scales of a fir cone or a pineapple.

The history of truly mathematical phyllotaxis (as opposed to the purely descriptive approaches) begins in the nineteenth century with the works of botanist Karl Friedric Schimper (published in 1830), his friend Alexander Braun (published in 1835), and the crystallographer Auguste Bravais and his botanist brother Louis (published in 1837). These researchers discovered the general rule that phyllotactic ratios could be expressed by ratios of terms of the Fibonacci series (like ⅖; ⅜) and also noted the appearance of consecutive Fibonacci numbers in the parastichies of pinecones and pineapples.

Pineapples indeed provide a truly beautiful manifestation of a Fibonacci-based phyllotaxis (Figure 32). Each hexagonal scale on the surface of the pineapple is a part of three different spirals. In the figure you can see one of eight parallel rows sloping gently from lower left to upper right, one of thirteen parallel rows that slope more steeply from lower right to upper left, and one of twenty-one parallel rows that are very steep (from lower left to upper right). Most pineapples have five, eight, thirteen, or twenty-one spirals of increasing steepness on their surface. All of these are Fibonacci numbers.

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Figure 32

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Figure 33

How do plants know to arrange their leaves in these Fibonacci patterns? The growth of the plant takes place at the tip of the stem (called the meristem), which has a conical shape (being thinnest at the tip). Leaves that are farther down from the tip (namely, which grew earlier) tend to be radially farther out from the stem's center when viewed from the top (because the stem is thicker there). Figure 33 shows such a view of the stem from the top, where the leaves are numbered according to their order of appearance. The leaf numbered 0, which appeared first, is by now the farthest down from the meristem and the farthest out from the stem's center. Botanist A. H. Church in his 1904 book On the Relation of Phyllotaxis to Mechanical Laws first emphasized the importance of this type of representation for the understanding of phyllotaxis. What we find (by imagining a curve that connects leaves 0 to 5 in Figure 33) is that successive leaves sit along a tightly wound spiral, known as the generative spiral. The important quantity that characterizes the location of the leaves is the angle between the lines connecting the stem's center with successive leaves. One of the discoveries of the Bravais brothers in 1837 was that new leaves advance roughly by the same angle around the circle and that this angle (known as the divergence angle) is usually close to 137.5 °. Are you shocked to hear that this value is determined by the Golden Ratio? The angle that divides a complete turn in a Golden Ratio is 360°/ ö = 222.5 °. Since this is more than half a circle (180 degree), we should measure it going in the opposite direction around the circle. In other words, we should subtract 222.5 from 360, giving us the observed angle of 137.5 degree (sometimes called the Golden Angle).

In a pioneering work in 1907, German mathematician G. van Iterson showed that if you closely pack successive points separated by 137.5 degree on tightly wound spirals, then the eye would pick out one family of spiral patterns winding clockwise and one counterclockwise. The numbers of spirals in the two families tend to be consecutive Fibonacci numbers, since the ratio of such numbers approaches the Golden Ratio.

Such counterwinding spirals are most spectacularly exhibited by the arrangement of the florets in sunflowers. When you look on the head of a sunflower (Figure 34), you will notice both clockwise and counterclockwise spiral patterns formed by the florets. Clearly the florets grow in a way that affords the most efficient sharing of horizontal space. The numbers of these spirals usually depend on the size of the sunflower. Most commonly there are thirty-four spirals going one way and fifty-five the other, but sunflowers with ratios of numbers of spirals of 89/55, 144/89, and even (at least one; reported by a Vermont couple to the Scientific American in 1951) 233/144 have been seen. All of these are, of course, ratios of adjacent Fibonacci numbers. In the largest sunflowers, the structure stretches from one pair of consecutive Fibonacci numbers to the next higher, when we move from the center to the periphery.

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Figure 34

The petal counts and petal arrangements of some flowers also harbor Fibonacci numbers and Golden Ratio connections. Many people have relied (at least symbolically) at some point in their lives on the numbers of petals of daisies to satisfy their curiosity about the intriguing question: “She loves me, she loves me not.” Most field daisies have thirteen, twenty-one, or thirty-four petals, all Fibonacci numbers. (Wouldn't it be nice to know in advance if the daisy has an even or odd number of petals?) The number of petals simply reflects the number of spirals in one family.

The beautifully symmetric arrangement of the petals of roses is also based on the Golden Ratio. If you dissect a rose (petal by petal), you will discover the positions of its tightly packed petals. Figure 35 presents a schematic in which the petals have been numbered. The angles defining the positions (in fractions of a full turn) of the petals are the fractional part of simple multiples of ö. Petal 1 is 0.618 (the fractional part of 1 × ϕ) of a turn from petal 0, petal 2 is 0.236 (the fractional part of 2 × ö) of a turn from petal 1, and so on.

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Figure 35

This description shows that the 2,300-year-old puzzle of the origins of phyllotaxis reduces to the basic question: Why are successive leaves separated by the Golden Angle of 137.5 degree? The attempts to answer this question come in two flavors: theories that concentrate on the geometry of the configuration, and on simple mathematical rules that can generate this geometry; and models that suggest an actual dynamical cause for the observed behavior. Landmark works of the first type (e.g., by mathematicians Harold S. M. Coxeter and I. Adler and by crystallographer N. Rivier) show that buds which are placed along the generative spiral separated by the Golden Angle are close-packed most efficiently. This is easy to understand. If the divergence angle was, let's say, 120 degree (which is 360/3) or any other rational multiple of 360 degree, then the leaves would have aligned radially (along three lines in the case of 120 degree), leaving large spaces in between. On the other hand, a divergence angle like the Golden Angle (which is an irrational multiple of 360 degree) ensures that buds do not line up along any specific radial direction and they fill the spaces efficiently. The Golden Angle proves to be even better than other irrational multiples of 360 degree because the Golden Ratio is the most irrational of all irrational numbers in the following sense. Recall that the Golden Ratio is equal to a continued fraction composed entirely of 1s. That continued fraction converges more slowly than any other continued fraction. In other words, the Golden Ratio is farther away from being expressible as a fraction than any other irrational number.

In a paper that appeared in 1984 in Journal de Physique, a team of scientists led by N. Rivier from the Université de Provence in Marseille, France, used a simple mathematical algorithm to show that when a growth angle equal to the Golden Angle is used, structures that closely resemble real sunflowers are obtained. (See Figure 36.) Rivier and his collaborators suggested that this provided an answer to the question that had been posed in the classical work of biologist Sir D'Arcy Wentworth Thompson. In his monumental book On Growth and Form (first published in 1917 and revised in 1942), Thompson wonders: “… and not the least curious feature of the case [phyllotaxis] is the limited, even the small number of possible arrangements which we observe and recognize.” Rivier's team found that the requirements of homogeneity (that the structure is the same everywhere) and of self-similarity (that when one examines the structure on different scales from small to large, it looks precisely the same) limit drastically the number of possible structures. These two properties may be sufficient to explain the preponderance of Fibonacci numbers and the Golden Ratio in phyllotaxis, but they still do not offer any physical cause.

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Figure 36

The best clues for a possible dynamical cause of phyllotaxis came not from botany but from experiments in physics by L. S. Levitov (in 1991) and by Stephane Douady and Yves Couder (in 1992 to 1996). The experiment by Douady and Couder is particularly fascinating. They held a dish full of silicone oil in a magnetic field that was stronger near the dish's edge than at the center. Drops of a magnetic fluid, which act like tiny bar magnets, were dropped periodically at the center of the dish. The tiny magnets repelled each other and were pushed radially by the magnetic field gradient. Douady and Couder found patterns that oscillated about, but generally converged to, a spiral on which the Golden Angle separated successive drops. Physical systems usually settle into states that minimize the energy. The suggestion is therefore that phyllotaxis simply represents a state of minimal energy for a system of mutually repelling buds. Other models, in which leaves appear at the points of the highest concentration of some nutrient, also tend to produce separations equal to the Golden Angle.

I hope that the next time you eat a pineapple, send a red rose to a loved one, or admire van Gogh's sunflower paintings, you will remember that the growth pattern of these plants embodies this wonderful number we call the Golden Ratio. Realize, however, that plant growth also depends on factors other than optimal spacing. Consequently, the phyllotaxis rules I have described cannot be taken as applying to all circumstances, like a law of nature. Rather, in the words of the famous Canadian mathematician Coxeter, they are “only a fascinatingly prevalent tendency.”

Botany is not the only place in nature where the Golden Ratio and Fibonacci numbers can be found. They appear in phenomena covering a range in sizes from the microscopic to that of giant galaxies. Often that appearance takes the form of a magnificent spiral.

ALTHOUGH CHANGED, I RISE AGAIN THE SAME

No family in the history of mathematics has produced as many celebrated mathematicians (thirteen in total!) as did the Bernoulli family. Disconcerted by the Spanish Fury (the ravaging riot in the Netherlands by Spanish soldiers), the family fled to Basel, Switzerland, from the Catholic Spanish Netherlands. Three members of the family, the brothers Jacques (1654–1705) and Jean (1667–1748), and the latter's second son, Daniel (1700–1782), stood out head and shoulders above the rest. Strangely, the Bernoullis were almost equally famous for their bitter interfamilial rivalries as they were for their numerous mathematical achievements. In one case, the exchanges between Jacques and Jean became particularly acrimonious. The feud was sparked by a dispute over a solution to a famous problem in mechanics. The problem, known as the brachistochrone (from the Greek brachistos, “shortest,” and chronos, “time”), was to find the curve along which a particle acted on by the force of gravity will pass in the shortest time from one point to another. The two brothers proposed the same solution independently, but Jean's derivation was incorrect, and he later attempted to present Jacques' derivation as his own. The sad consequence of this chain of events was that Jeanne became a professor in Groningen and did not set foot in Basel until his brother's death.

Jacques Bernoulli's association with the Golden Ratio comes through another famous curve. He devoted a treatise entitled Spira Mirabilis (Wonderful spiral) to a particular type of spiral shape. Jacques was so impressed with the beauty of the curve known as a logarithmic spiral (Figure 37; the name was derived from the way in which the radius grows as we move around the curve clockwise) that he asked that this shape, and the motto he assigned to it: “Eadem mutato resurgo” (although changed, I rise again the same), be engraved on his tombstone.

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Figure 37

The motto describes a fundamental property unique to the logarithmic spiral—it does not alter its shape as its size increases. This feature is known as self-similarity Fascinated by this property, Jacques wrote that the logarithmic spiral “may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self” f you think about it for a moment, this is precisely the property required for many growth phenomena in nature. For example, as the mollusk inside the shell of the chambered nautilus (Figure 4) grows in size, it constructs larger and larger chambers, sealing off the smaller unused ones. Each increment in the length of the shell is accompanied by a proportional increase in its radius, so that the shape remains unchanged. Consequently, the nautilus sees an identical “home” throughout its lifetime, and it does not need, for example, to adjust its balance as it matures. The latter property applies also to rams, the horns of which are also in the shape of logarithmic spirals (although they do not lie in a plane), and to the curve of elephants' tusks. Increasing by accumulation from within itself, the logarithmic spiral grows wider, with the distance between its “coils” increasing, as it moves away from the source, known as the pole. Specifically, turning by equal angles increases the distance from the pole by equal ratios. If we were, with the aid of a microscope, to enlarge the coils that are invisible to the naked eye to the size of Figure 37, they would fit precisely on the larger spiral. This property distinguishes the logarithmic spiral from another common spiral known as the Archimedean spiral, after the famous Greek mathematician Archimedes (ca. 287–212 B.C.), who described it extensively in his book On Spirals. We can see an Archimedean spiral in the side of a roll of paper towels or a rope coiled on the floor. In this type of spiral, the distance between successive coils remains always the same. As a result of a mistake that surely would have caused Jacques Bernoulli much grief, the mason who prepared Bernoulli's tombstone engraved on it an Archimedean rather than a logarithmic spiral.

Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite “ornament.” The constant shape of the logarithmic spiral on all size scales reveals itself beautifully in nature in the shapes of minuscule fossils or unicellular organisms known as foraminifera. Although the spiral shells in this case are composite structures (and not one continuous tube), X-ray images of the internal structure of these fossils show that the shape of the logarithmic spiral remained essentially unchanged for millions of years.

In his classic book The Curves of Life (1914), English author and editor Theodore Andrea Cook gives numerous examples of the appearance of spirals (not just logarithmic) in nature and art. He discusses spirals in things as diverse as climbing plants, the human body, staircases, and Maori tattoos. In explaining his motivation for writing the book, Cook writes: “for the existence of these chapters upon spiral formations no other apology is needed than the interest and beauty of an investigation.”

Artists have also not failed to see the beauty of logarithmic spirals. In Leonardo da Vinci's study for the mythological subject “Leda and the Swan,” for example, he draws the hair arranged in the shape of a nearly logarithmic spiral (Figure 38). Leonardo repeats this shape many times in his study of spirals in clouds and water in the impressive series of sketches for the “Deluge.” In that work, he combined his scientific observations of frightening floods with the allegorical aspects of destructive forces descended from heaven. Describing the violent flow of water Leonardo wrote: “The sudden waters rush into the pond that contains them, striking the various obstacles with swirling eddies.… The momentum of the circular waves flying from the point of impact hurls them in the way of other circular waves moving in the opposite direction.”

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Figure 38

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Figure 39

Twentieth-century designer and illustrator Edward B. Edwards developed hundreds of decorative designs based on the logarithmic spiral; many can be seen in his book Pattern and Design with Dynamic Symmetry (an example is shown in Figure 39).

The logarithmic spiral and the Golden Ratio go hand in hand. Examine again the series of nested Golden Rectangles obtained when you snip off squares from a Golden Rectangle (Figure 40; we encountered this property already in Chapter 4). If you connect the successive points where these “whirling squares” divide the sides in Golden Ratios, you obtain a logarithmic spiral that coils inward toward the pole (the point given by the intersection of the diagonals in Figure 25, which was called fancifully “the eye of God”).

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Figure 40

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Figure 41

You can also obtain a logarithmic spiral from a Golden Triangle. We saw in Chapter 4 that if you start from a Golden Triangle (an isosceles triangle in which the side is in Golden Ratio to the base) and bisect a base angle, you get a smaller Golden Triangle. If you continue the process of bisecting the base angle ad infinitum, you will generate a series of whirling triangles. Connecting the vertices of the Golden Triangles in the progression will trace a logarithmic spiral (Figure 41).

The logarithmic spiral is also known as the equiangular spiral. This name was coined in 1638 by the French mathematician and philosopher René Descartes (1596–1650), after whom we name the numbers used to locate a point in the plane (with respect to two axes)—Cartesian coordinates. The name “equiangular” reflects another unique property of the logarithmic spiral. If you draw a straight line from the pole to any point on the

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Figure 42

curve, it cuts the curve at precisely the same angle (Figure 42). Falcons use this property when attacking their prey. Peregrine falcons are some of the fastest birds on Earth, plummeting toward their targets at speeds of up to two hundred miles per hour. But they could fly even faster if they would just fly straight instead of following a spiral trajectory to their victims. Biologist Vance A. Tucker of Duke University in North Carolina wondered for years why peregrines don't take the shortest distance to their prey. He then realized that because falcons' eyes are on either side of their heads, to take advantage of their razor-sharp vision, they must cock their heads 40 ° to one side or the other. Tucker found in wind-tunnel experiments that such a head tip would slow them considerably. The results of his research, which were published in the November 2000 issue of the Journal of Experimental Biology, show that falcons keep their head straight and follow a logarithmic spiral. Because of the spiral's equiangular property, this path allows them to keep their target in view while maximizing speeds.

The amazing thing is that the same spiral shape that is found in the unicellular foraminifera and in the sunflower and that guides the flight of a falcon can also be found in those “systems of stars gathered together in a common plane, like those of the Milky Way” which philosopher Immanuel Kant (1724–1804) speculated about long before they were actually observed (Figure 43). These became known as island universes—giant galaxies containing hundreds of billions of stars like our Sun. Observations conducted with the Hubble Space

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Figure 43

Telescope revealed that there are some one hundred billion galaxies in our observable universe, many of which are spiral galaxies. You can hardly think of a better manifestation of the grand vision expressed by English poet, painter, and mystic William Blake (1757–1827), when he wrote:

To see a World in a Grain of Sand,
And a Heaven in a Wild Flower,
Hold Infinity in the Palm of your hand,
And Eternity in an hour.

Why do so many galaxies exhibit a spiral pattern? Spiral galaxies like our own Milky Way have a relatively thin disk (like a pancake) composed of gas, dust (miniature grains), and stars. The entire galactic disk is rotating about the galactic center. In the vicinity of the Sun, for example, the orbital speed around the Milky Way's center is about 140 miles per second, and it takes material about 225 million years to complete one revolution. At other distances from the center the speed is different—higher closer to the center, lower at greater distances—that is, galactic disks do not rotate like a solid compact disk but rather rotate differentially. Seen face on, spiral galaxies show spiral arms originating close to the galactic center and extending outward throughout much of the disk (as in the “Whirlpool Galaxy,” Figure 43). The spiral arms are the part of the galactic disk where many young stars are being born. Since young stars are also the brightest, we can see the spiral structure of other galaxies from afar. The basic question that astrophysicists had to answer is: How do the spiral arms retain their shape over long periods of time? Because the inner parts of the disk rotate faster than the outer parts, any large-scale pattern that is somehow “attached” to the disk material (e.g., the stars) cannot survive for long. A spiral structure tied always to the same collection of stars and gas clouds would inevitably wind up, contrary to observations. The explanation for the longevity of the spiral arms relies on density waves—waves of gas compression sweeping through the galactic disk—squeezing gas clouds along the way and triggering the formation of new stars. The spiral pattern that we observe simply marks the denser-than-average parts of the disk and its newborn stars. The pattern is therefore created repeatedly without winding up. The situation is similar to that observed near a lane closed for repairs by a work crew on a major highway. The density of cars in the vicinity of the closed stretch is higher because cars have to slow down there. If you take a long-exposure photograph of the highway from above, you will record the high-traffic density near the place where repairs are being undertaken. Just as the traffic density wave is not associated with any particular set of cars, the spiral-arms pattern is not tied to any particular piece of disk material. Another similarity is in the fact that the density wave itself moves through the disk more slowly than the motion of the stars and the gas, just as the speed at which the repair work proceeds along the highway is typically much slower than the unperturbed speed of individual cars.

The agent that deflects the motion of the stars and the gas clouds and generates the spiral density wave (analogous to the repair crew that deflects the moving cars to fewer lanes) is the gravitational force resulting from the fact that the distribution of matter in the galaxy is not perfectly symmetric. For example, a set of oval orbits around the center (Figure 44a) in which each orbit is perturbed (rotated) slightly by an amount that changes with distance from the center results in a spiral pattern (Figure 44b).

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Figure 44

Actually, we should be quite happy that the force of gravity behaves in our universe the way it does. According to Newton's universal law of gravitation, every mass attracts every other mass, and the force of attraction decreases as the masses get farther apart. In particular, doubling the distance weakens the force by a factor of four (the force decreases as the inverse of the square of the distance). Newton's laws of motion show that as a result of this dependence on the distance, the orbits of the planets around the Sun are in the shapes of ellipses. Imagine what would have happened had we lived in a universe in which gravity had decreased by a factor of eight (instead of four) upon doubling of the distance (a force decreasing as the inverse of the cube of the distance). In such a universe, Newton's laws predict that one possible orbit of the planets is a logarithmic spiral. In other words, Earth would have spiraled into the Sun or rushed off into space.

Leonardo Fibonacci, who initiated all of this frenzy of mathematical activity, is far from forgotten today. In today's Pisa, a statue of Fibonacci constructed in the nineteenth century stands in the Scotto Garden on the grounds of the Sangallo Fortress, next to a street named after Fibonacci, which runs along the south side of the Arno River.

Since 1963 the Fibonacci Association has published a journal entitled the Fibonacci Quarterly. The group was formed by mathematicians Verner Emil Hoggatt (1921–1981) and Brother Alfred Brousseau (1907–1988) “in order to exchange ideas and stimulate research in Fibonacci numbers and related topics.” Perhaps against the odds, the Fibonacci Quarterly has since grown into a well-recognized journal in number theory. As Brother Brousseau humorously put it: “We got a group of people together in 1963, and just like a bunch of nuts, we started a mathematics magazine.” The Tenth International Conference on Fibonacci Numbers and Their Applications is planned for June 24–28, 2002, at Northern Arizona University in Flagstaff, Arizona.

All of this is but a small tribute to the man who used rabbits to discover a world-embracing mathematical concept. As important as Fibonacci's contribution was, however, the story of the Golden Ratio did not end in the thirteenth century; fascinating developments were still to come in Renaissance Europe.