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Topics in Mathematical Modeling
By K. K. Tung PRINCETON UNIVERSITY PRESS
Copyright © 2007 Ka-Kit Tung
All rights reserved.
ISBN: 978-1-4008-8405-6
CHAPTER 1
Fibonacci Numbers, the Golden Ratio, and Laws of Nature?
Mathematics required:
high school algebra, geometry, and trigonometry; concept of limits from precalculus
Mathematics introduced:
difference equations with constant coefficients and their solution; rational approximation to irrational numbers; continued fractions
1.1 Leonardo Fibonacci
Leonardo of Pisa (1175–1250), better known to later Italian mathematicians as Fibonacci (Figure 1.1), was born in Pisa, Italy, and in 1192 went to North Africa (Bugia, Algeria) to live with his father, a customs officer for the Pisan trading colony. His father arranged for the son's instruction in calculational techniques, intending for Leonardo to become a merchant. Leonardo learned the Hindu-Arabic numerals (Figure 1.2) from one of his "excellent" Arab instructors. He further broadened his mathematical horizons on business trips to Egypt, Syria, Greece, Sicily, and Provence. Fibonacci returned to Pisa in 1200 and published a book in 1202 entitled Liber Abaci (Book of the Abacus), which contains a compilation of mathematics known since the Greeks. The book begins with the first introduction to the Western business world of the decimal number system:
These are the nine figures of the Indians: 9, 8, 7, 6, 5, 4, 3, 2, 1. With these nine figures, and with the sign 0, which in Arabic is called zephirum, any number can be written, as will be demonstrated.
Since we have ten fingers and ten toes, one may think that there should be nothing more natural than to count in tens, but that was not the case in Europe at the time. Fibonacci himself was doing calculations using the Babylonian system of base 60! (It is not as strange as it seems; the remnant of the sexagesimal system can still be found in our measures of angles and time.)
The third section of Liber Abaci contains a puzzle:
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 each month each pair begets a new pair which from the second month on becomes productive?
In solving this problem, a sequence of numbers, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..., emerges, as we will show in a moment. This sequence is now known as the Fibonacci sequence.
The above problem involving incestuous rabbits is admittedly unrealistic, but similar problems can be phrased in more plausible contexts: A plant (tree) has to grow two months before it branches, and then it branches every month. The new shoot also has to grow for two months before it branches (see Figure 1.3). The number of branches, including the original trunk, is, if one counts from the bottom in intervals of one month's growth: 1, 1, 2, 3, 5, 8, 13,. ... The plant Achillea ptarmica, the "sneezewort," is observed to grow in this pattern.
The Fibonacci sequence also appears in the family tree of honey bees. The male bee, called the drone, develops from the unfertilized egg of the queen bee. Other than the queen, female bees do not reproduce. They are the worker bees. Female bees are produced when the queen mates with the drones. The queen bee develops when a female bee is fed the royal jelly, a special form of honey. So a male bee has only one parent, a mother, while a female bee, be it the queen or a worker bee, has both a mother and a father. If we count the number of parents and grandparents and great grandparents, etc., of a male bee, we will get 1, 1, 2, 3, 5, 8, ..., a Fibonacci sequence.
Let's return to the original mathematical problem posed by Fibonacci, which we haven't yet quite solved. We actually want to solve it more generally, to find the number of pairs of rabbits n months after the first pair was introduced. Let this quantity be denoted by Fn. We assume that the initial pair of rabbits is one month old and that we count rabbits just before newborns arrive.
One way to proceed is simply to enumerate, thus generating a sequence of numbers. Once we have a sufficiently long sequence, we would hopefully be able to see the now famous Fibonacci pattern (Figure 1.4).
After one month, the first pair becomes two months old and is ready to reproduce, but the census is taken before the birth. So F1 = 1, but F2 = 2; by the time they are counted, the newborns are already one month old. The parents are ready to give birth again, but the one-month-old offspring are too young to reproduce. Thus F3 = 3. At the end of three months, both the original pair and its offspring are productive, although the births are counted in the next period. Thus F4 = 5. A month later, an additional pair becomes productive. The three productive pairs add three new pairs of offspring to the population. Thus F5 = 8. At five months, there are five productive pairs: the first-generation parents, four second-generation adults, and one third-generation pair born in the second month. Thus F6 = 13. It now gets more difficult to keep track of all the rabbits, but one can use the aid of a table to keep account of the ages of the offspring. With some difficulty, we obtain the following sequence for the number of rabbit pairs after n months, for n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,....
This is the sequence first generated by Fibonacci. The answer to his original question is F12 = 233.
If we had decided to count rabbits after the newborns arrive instead of before, we would have to deal with three types of rabbits: newborns, one-month-olds, and mature (two-month-old or older) rabbits. In this case, the Fibonacci sequence would have shifted by one, to: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,. ... The initial 1 is missing, which, however, can be added back if we assume that the first pair introduced is newborn. It then takes two months for them to become productive. The discussion below works with either convention.
To find Fn for a general positive integer n, we hope that we can see a pattern in the sequence of numbers already found. A sharp eye can now detect that any number in the sequence is always the sum of the two numbers preceding it. That is,
Fn+2 = Fn+1 + Fn, for n = 0, 1, 2, 3,. ... (1.1)
A second way of arriving at the same recurrence relationship is more preferable, because it does not depend on our ability to detect a pattern from a partial list of answers:
Let Fn (k) be the number of k-month-old rabbit pairs at time n. These will become (k + 1)-month-olds at time n + 1. So,
Fn+1(k + 1) = Fn(k).
The total number of pairs at time n + 2 is equal to the number at n + 1 plus the newborn pairs at n + 2:
Fn+2 = Fn+1 + new births at time n + 2.
The number of new births at n + 2 is equal to the number of pairs that are at least one month old at n + 1, and so:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Therefore,
Fn+2 = Fn+1 + Fn.
which is the same as Eq. (1.1). This recurrence equation is also called the renewal equation. It uses present and past information to predict the future. Mathematically it is a second-order difference equation.
To solve Eq. (1.1), we try, as we generally do for linear difference equations whose coefficients do not depend on n,
Fn = λn,
for some as yet undetermined constant λ. When we substitute the trial solution into Eq. (1.1), we get
λn+2 = λn+1 + λn.
Canceling out λn, we obtain a quadratic equation,
λ2 = λ + 1. (1.2)
which has two roots (solutions):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus λn1 is a solution, and so is λn2. By the principle of linear superposition, the general solution is
Fn = aλn1 + bλn2. (1.3)
where a and b are arbitrary constants. If you have doubts on the validity of the superposition principle used, I encourage you to plug this general solution back into Eq. (1.1) and see that it satisfies that equation no matter what values of a and b you use. Of course these constants need to be determined by the initial conditions. We need two such auxiliary conditions since we have two unknown constants. They are F0 = 1 and F1 = 1. The first requires that a + b = 1, and the second implies that λ1a + λ2b = 1. Together, they uniquely determine the two constants. Finally, we find:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
With the irrational number √5 in the expression, it is surprising that Eq. (1.4) would always yield whole numbers, 1, 1, 2, 3, 5, 8, 13, ..., when n goes from 0, 1, 2, 3, 4, 5, ..., but you can verify that amazingly it does.
1.2 The Golden Ratio
The number λ1 = ½ (1 + [square root of (5)])is known as the Golden Ratio. It has also been called the Golden Section (in an 1835 book by Martin Ohm) and, since the 16th century, the Divine Proportion. It is thought to reflect the ideal proportions of nature and to even possess some mystical powers. It is an irrational number, now denoted by the Greek symbol Φ:
Φ = 1.6180339887....
It does have some very special, though not so mysterious, properties. For example, its square,
Φ2 = 2.6180339887 ...,
is obtainable by adding 1 to Φ. Its reciprocal,
1/Φ = 0.6180339887 ...,
is the same as subtracting 1 from Φ. These properties are not mysterious at all, if we recall that Φ is a solution of Eq. (1.2).
In terms of Φ, the general solution (1.3) can be written as
Fn = aΦn + b (-1/Φ)n.
Since Φ > 1, the second term diminishes in importance as n increases, so that for n >> 1,
Fn [congruent to] aΦn.
Therefore the ratio of successive terms in the Fibonacci sequence approaches the Golden Ratio:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
(In fact, since this property about the ratio converging to the Golden Ratio is independent of a and b, as long as a is not zero, it is satisfied by all solutions to the difference equation (1.1), including the Lucas sequence, which is the sequence of numbers starting with F0 = 2 and F1 = 1:2, 1, 3, 4, 7, 11, 18, 29, ...).
For our later use, we also list the result
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)
As you may recall, an irrational number is a number that cannot be expressed as the ratio m/n of two integers, m and n. Mathematicians sometimes are interested in the rational approximation of an irrational number; that is, finding two integers, m and n, whose ratio, m/n, gives a good approximation of the irrational number with an error that is as small as possible under some constraints. For example, the irrational number π = 3.14159265 ... can be approximated by the ratio 22/7 = 3.142857 ..., with error 0.00126. This is the best rational approximation if n is to be less than 10. When we make m and n larger, the error goes down rapidly. For example, 355/113 is a rational approximation of π (with n less than 200) with an error of 0.000000266. We measure the degree of irrationality of an irrational number by how slowly the error of its best rational approximation approaches zero when we allow m and n to get bigger and bigger. In this sense π is "not too irrational."
From Eq. (1.5) we see that the value of Φ can thus be approximated by the rational ratio: 8/5 = 1.6, or 13/8 = 1.625, or 21/13 = 1.615385 ..., or 34/21 = 1.619048 ..., or 55/34 = 1.617647 ..., or 89/55 = 1.618182 ..., or 144/89 = 1.617978. ... The ratios of successive terms in the Fibonacci sequence will eventually converge to the Golden Ratio. One therefore can use the ratio of successive Fibonacci numbers as the rational approximation to the Golden Ratio. Such rational ratios, however, converge to the Golden Ratio extremely slowly. Thus we might say that the Golden Ratio is the most irrational of the irrational numbers. (How do we know it is the most irrational of the irrational numbers? A proof requires the use of continued fractions. See exercise 2 for some examples.)
More importantly, the Golden Ratio has its own geometrical significance, first recognized by the Greek mathematicians Pythagoras (560– 480 BC), and Euclid (365–325 BC). The Golden Ratio is the only positive number that, when 1 is subtracted from it, equals its reciprocal. Euclid in fact defined it, without using the name Golden Ratio, when he studied the division of a line into what he called the "extreme and mean ratio":
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
(Does this sound like Greek to you? If so, you may find Figure 1.5 helpful. Consider the straight line abc, cut into two segments ab and bc, in such a way that the "extreme ratio" [bar.abc]/[bar.ab] is equal to its "mean ratio" [bar.ab]/[bar.bc]. Without loss of generality, let the length of small segment [bar.bc] be 1, and [bar.ab] be x, so the whole line [bar.abc] is 1 + x. The line is said to be cut in extreme and mean ratio when (1 + x)/x = x/1; this is the same as x2 = x + 1, which is Eq. (1.2). Φ is the only positive root of that equation.)
Many authors reported that the ancient Egyptians possessed the knowledge of the Golden Ratio even earlier and incorporated it in the geometry of the Great Pyramid of Khufu at Giza, which dates to 2480 BC. Midhat Gazale, who was the president of AT&T-France, wrote in his popular 1999 book, Gnomon: From Pharaohs to Fractals:
It was reported that the Greek historian Herodotus learned from the Egyptian priests that the square of the Great Pyramid's height is equal to the area of its triangular lateral side.
Referring to Figure 1.6, we consider the upright right triangle formed by the height of the pyramid (from its base to its apex), the slanted height of the triangle on its lateral side (the length from the base to the apex of the pyramid along the slanted lateral triangle), and a horizontal line joining these two lines inside the base. We see that if the above statement is true, then the ratio of the hypotenuse to the base of that triangle is equal to the Golden Ratio. (Show this!) However, as pointed out by Mario Livio in his wonderful 2002 book, The Golden Ratio, Gazale was repeating an earlier misinterpretation by the English author John Taylor in his 1859 book, The Great Pyramid: Why Was It Built and Who Built It, in which Taylor was trying to argue that the construction of the Great Pyramid was through divine intervention. What the Greek historian Herodotus (ca. 485–425 BC) actually said was: "Its base is square, each side is eight plethra long and its height the same." One plethron was 100 Greek feet, approximately 101 English feet (see Fischler, 1979; Markowsky, 1992).
Nevertheless, there is no denying that the physical dimensions of the Great Pyramid as it stands now do give a ratio of hypotenuse to base rather close to the Golden Ratio. The base of the pyramid is approximately a square with sides measuring 756 feet each, and its height is 481 feet. So the base of the upright right triangle is 756/2 = 378 feet, while the hypotenuse is, by the Pythagorean Theorem, 612 feet. Their ratio is then 612/378 = 1.62, which is in fact quite close to the Golden Ratio. The debate continues. All we can say is that, casting aside the claims of some religious cults, there is no historical or archeological evidence that the ancient Egyptians knew about the Golden Ratio.
(Continues...)
Excerpted from Topics in Mathematical Modeling by K. K. Tung. Copyright © 2007 Ka-Kit Tung. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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