Excursions in Modern Mathematics, 7e: 9.1 - 1Copyright © 2010 Pearson Education, Inc. 9 The...

Excursions in Modern Mathematics, 7e: 9.1 - 1Copyright © 2010 Pearson Education, Inc.

9 The Mathematics of Spiral Growth

9.1 Fibonacci’s Rabbits

9.2 Fibonacci Numbers

9.3 The Golden Ratio

9.4 Gnomons

9.5 Spiral Growth in Nature


In 1202 a young Italian named Leonardo Fibonacci published a book titled Liber Abaci (which roughly translated from Latin means “The Book of Calculation”). Although not an immediate success, Liber Abaci turned out to be one of the most important books in the history of Western civilization.

Leonardo Fibonacci


Liber Abaci was a remarkable book full of wonderful ideas and problems, but our story in this chapter focuses on just one of those problems–a purely hypothetical question about the growth of a very special family of rabbits. Here is the question, presented in Fibonacci’s own (translated) words:

“The Book of Calculation”


A man puts one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if the nature of these rabbits is such that every month each pair bears a new pair which from the second month on becomes productive?

Fibonacci’s Rabbits


We will call P1 the number of pairs of rabbits in the first month, P2 the number of pairs of rabbits in the second month, P3 the number of pairs of rabbits in the third month, and so on. With this notation the question asked by Fibonacci (...how many pairs of rabbits can be produced from [the original] pair in a year?) is answered by the value P12 (the number of pairs of rabbits in month 12). For good measure we will add one more value, P0, representing the original pair of rabbits introduced by “the man” at the start.



Let’s see now how the number of pairs of rabbits grows month by month. We start with the original pair, which we will assume is a pair of young rabbits. In the first month we still have just the original pair (for convenience, let’s call them Pair A), soP1 = 1. By the second month the original pair matures, becomes “productive,” and generates a new pair of young rabbits. Thus, by the second month we have the original mature Pair A plus the new young pair we will call Pair B, so P2 = 2.



By the third month Pair B is still too young to breed, but Pair A generates another new young pair, Pair C, so P3 = 3.By the fourth month Pair C is still young, but both Pair A and Pair B are mature and generate a new pair each (Pairs D and E). It follows that P4 = 5.We could continue this way, but our analysis can be greatly simplified by the following two observations:



1. In any given month (call it month N) the number of pairs of rabbits equals the total number of pairs in the previous month (i.e., in month N – 1 ) plus the number of mature pairs of rabbits in month N (these are the pairs that produce offspring–one new pair for each mature pair).

2. The number of mature rabbits in month N equals the total number of rabbits in month N – 2 (it takes two months for newborn rabbits to become mature).



Observations 1 and 2 can be combined and simplified into a single mathematical formula:


PN = PN – 1 + PN – 2

The above formula reads as follows: The number of pairs of rabbits in any given month (PN) equals the number of pairs of rabbits the previous month (PN – 1) plus the number of pairs of rabbits two months back (PN – 2).


It follows, in order, that


P5 = P4 + P3 = 5 + 3 = 8

P6 = P5 + P4 = 8 + 5 = 13

P7 = P6 + P5 = 13 + 8 = 21

P8 = P7 + P6 = 21 + 13 = 34

P9 = P8 + P7 = 34 + 21 = 55

P10 = P9 + P8 = 55 + 34 = 89

P11 = P10 + P9 = 89 + 55 = 144

P12 = P11 + P10 = 144 + 89 = 233


So there is the answer to Fibonacci’s question: In one year the man will have raised 233 pairs of rabbits.

This is the end of the story about Fibonacci’s rabbits and also the beginning of a much more interesting story about a truly remarkable sequence of numbers called Fibonacci numbers.







9.4 Gnomons



The sequence of numbers shown above is called the Fibonacci sequence, and the individual numbers in the sequence are known as the Fibonacci numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

THE FIBONACCI SEQUENCE


You should recognize these numbers as the number of pairs of rabbits in Fibonacci’s rabbit problem as we counted them from one month to the next.

The Fibonacci sequence is infinite, and except for the first two 1s, each number in the sequence is the sum of the two numbers before it.

Fibonacci Sequence


We will denote each Fibonacci number by using the letter F (for Fibonacci) and a subscript that indicates the position of the number in the sequence. In other words, the first Fibonacci number is F1 = 1, the second Fibonacci number is F2 = 1, the third Fibonacci number is F3 = 2, the tenth Fibonacci number is F10 = 55. We may not know (yet) the numerical value of the 100th Fibonacci number, but at least we can describe it as F100.

Fibonacci Number


A generic Fibonacci number is usually written as FN (where N represents a generic position). If we want to describe the Fibonacci number that comes before FN we write FN – 1 ; the Fibonacci number two places before FN is FN – 2, and so on. Clearly, this notation allows us to describe relations among the Fibonacci numbers in a clear and concise way that would be hard to match by just using words.

Fibonacci Number


The rule that generates Fibonacci numbers–a Fibonacci number equals the sum of the two preceding Fibonacci numbers–is called a recursive rule because it defines a number in the sequence using earlier numbers in the sequence. Using subscript notation, the above recursive rule can be expressed by the simple and concise formulaFN = FN – 1 + FN – 2 .

Fibonacci Number


There is one thing still missing. The formula FN = FN – 1 + FN – 2 requires two consecutive Fibonacci numbers before it can be used and therefore cannot be applied to generate the first two Fibonacci numbers, F1 and F2. For a complete definition we must also explicitly give the values of the first two Fibonacci numbers, namely F1 = 1 andF2 = 1. These first two values serve as “anchors” for the recursive rule and are called the seeds of the Fibonacci sequence.

Fibonacci Number


■ F1 = 1, F2 = 1 (the seeds)

■ FN = FN – 1 + FN – 2 (the recursive rule)

FIBONACCI NUMBERS (RECURSIVE DEFINITION)


How could one find the value of F100? With a little patience (and a calculator) we could use the recursive definition as a “crank” that we repeatedly turn to ratchet our way up the sequence: From the seeds F1 and F2 we compute F3, then use F3 and F4 to compute F5, and so on. If all goes well, after many turns of the crank (we will skip the details) you will eventually get to

F97 = 83,621,143,489,848,422,977

Example 9.1 Cranking Out Large Fibonacci Numbers


and then to

F98 = 135,301,852,344,706,746,049

one more turn of the crank gives

F99 = 218,922,995,834,555,169,026

and the last turn gives

F100 = 354,224,848,179,261,915,075

converting to dollars yields

$3,542,248,481,792,619,150.75



$3,542,248,481,792,619,150.75

How much money is that? If you take $100 billion for yourself and then divide what’s left evenly among every man, woman, and child on Earth (about 6.7 billion people), each person would get more than $500 million!



In 1736 Leonhard Euler discovered a formula for the Fibonacci numbers that does not rely on previous Fibonacci numbers. The formula was lost and rediscovered 100 years later by French mathematician and astronomer Jacques Binet, who somehow ended up getting all the credit, as the formula is now known as Binet’s formula.

Leonard Euler


BINET’S FORMULA

FN

1

5

1 5

2

N

1 5

2

N


You can use the following shortcut of Binet’s formula to quickly find the Nth Fibonacci number for large values of N:

Using a Programmable Calculator

Step 1 Store in the calculator’s memory.

Step 2 Compute AN.

Step 3 Divide the result in step 2 by

Step 4 Round the result in Step 3 to the nearest whole number. This will give you FN.

1 5 / 2A

5.


Use the shortcut to Binet’s formula with a programmable calculator to compute F100.

Example 9.2 Computing Large Fibonacci Numbers: Part 2

Step 1 Compute The calculator should give something like: 1.6180339887498948482.

Step 2 Using the power key, raise the previous number to the power 100. The calculator should show 792,070,839,848,372,253,127.

1 5 / 2.


Step 3 Divide the previous number by The calculator should show 354,224,848,179,261,915,075.

Step 4 The last step would be to round the number in Step 3 to the nearest whole number. In this case the decimal part is so tiny that the calculator will not show it, so the number already shows up as a whole number and we are done.

Example 9.2 Computing Large Fibonacci Numbers: Part 2

5.


We find Fibonacci numbers when we count the number of petals in certain varieties of flowers: lilies and irises have 3 petals; buttercups and columbines have 5 petals; cosmos and rue anemones have 8 petals; yellow daisies and marigolds have 13 petals; English daisies and asters have 21 petals; oxeye daisies have 34 petals, and there are other daisies with 55 and 89 petals

Why Fibonacci Numbers Are Special


Fibonacci numbers also appear consistently in conifers, seeds, and fruits. The bracts in a pinecone, for example, spiral in two different directions in 8 and 13 rows; the scales in a pineapple spiral in three

Why Fibonacci Number Are Special

different directions in 8, 13, and 21 rows; the seeds in the center of a sunflower spiral in 55 and 89 rows.Is it all a coincidence?Hardly.






9.4 Gnomons



This number is one of the most famous and most studied numbers in all mathematics. The ancient Greeks gave it mystical properties and called it the divine proportion, and over the years, the number has taken many different names: the golden number, the golden section, and in modern times the golden ratio, the name that we will use from here on. The customary notation is to use the Greek lowercase letter (phi) to denote the golden ratio.

Golden Ratio


The golden ratio is an irrational number–it cannot be simplified into a fraction, and if you want to write it as a decimal, you can only approximate it to so many decimal places.For most practical purposes, a good enough approximation is 1.618.

Golden Ratio


THE GOLDEN RATIO

1 5

2 1.618


Find a positive number such that when you add 1 to it you get the square of the number.To solve this problem we let x be the desired number. The problem then translates into solving the quadratic equation x2 = x + 1. To solve this equation we first rewrite it in the form x2 – x – 1 = 0 and then use the quadratic formula. In this case the quadratic formula gives the solutions

The Golden Property

1 1 2 4 1 1 2

1 5

2


Of the two solutions, one is negative

The Golden Property

1 5 / 2 0.618

1 5 / 2.

is the only positive number with the property that when you add one to the number you get the square of the number, that is,2 = + 1. We will call this property the golden property. As we will soon see, the golden property has really important algebraic and geometric implications.

golden ratio It follows that

and the other is the


We will use the golden property 2 = + 1 to recursively compute higher and higher powers of . Here is how:

Fibonacci Numbers - Golden Property

If we multiply both sides of 2 = + 1 by , we get

3 = 2 + Replacing 2 by + 1 on the RHS gives

3 = ( + 1) + = 2 + 1


If we multiply both sides of 3 = 2 + 1 by , we get

4 = 22 +

Replacing 2 by + 1 on the RHS gives

4 = 2( + 1) + = 3 + 2



If we multiply both sides of 4 = 3 + 2 by , we get

5 = 32 + 2

Replacing 2 by + 1 on the RHS gives

5 = 3( + 1) + 2 = 5 + 3



If we continue this way, we can express every power of in terms of :

6 = 8 + 5 7 = 13 + 8 8 = 21 + 13 and so on.

Notice that on the right-hand side we always get an expression involving two consecutive Fibonacci numbers. The general formula that expresses higher powers of in terms of and Fibonacci numbers is as follows.



N = FN+ FN –

1

POWERS OF THE GOLDEN RATIO


We will now explore what is probably the most surprising connection between the Fibonacci numbers and the golden ratio. Take a look at what happens when we take the ratio of consecutive Fibonacci numbers. The table that appears on the following two slides shows the first 16 values of the ratio FN / FN – 1.

Ratio: Consecutive Fibonacci Numbers


The table shows an interesting pattern:

As N gets bigger, the ratio of consecutive Fibonacci numbers appears to settle down to a fixed value, and that fixed value turns out to be the golden ratio!



FN/ FN – 1 ≈

RATIO OF CONSECUTIVE FIBONACCI NUMBERS

and the larger the value of N, the better the approximation.






9.4 Gnomons



The most common usage of the word gnomon is to describe the pin of a sundial– the part that casts the shadow that shows the time of day. The original Greek meaning of the word gnomon is “one who knows,” so it’s not surprising that the word should find its way into the vocabulary of mathematics.

Gnomons


In this section we will discuss a different meaning for the word gnomon. Before we do so, we will take a brief detour to review a fundamental concept of high school geometry–similarity.

Gnomons


We know from geometry that two objects are said to be similar if one is a scaled version of the other. (When a slide projector takes the image in a slide and blows it up onto a screen, it creates a similar but larger image. When a photocopy machine reduces the image on a sheet of paper, it creates a similar but smaller image.) The following important facts about similarity of basic two-dimensional figures will come in handy later in the chapter:

Similar Figures


Two triangles are similar if and only if the measures of their respective angles are the same. Alternatively, two triangles are similar if and only if their sides are proportional. In other words, if Triangle 1 has sides of length a, b, and c, then Triangle 2 is similar to Triangle 1 if and only if its sides have length ka, kb, and kc for some positive constant k.

Similar Figures - Triangles


Two squares are always similar.

Two rectangles are similar if their corresponding sides are proportional.

Similar Figures - Squares - Rectangles


Two circles are always similar.Any circular disk (a circle plus all its interior) is similar to any other circular disk. Two circular rings are similar if and only if their inner and outer radii are proportional

Similar Figures - Circles - Disks - Rings


In geometry, a gnomon G to a figure A is a connected figure that, when suitably attached to A, produces a new figure similar to A.

By “attached,” we mean that the two figures are coupled into one figure without any overlap.

Gnomon


Informally, we will describe it this way: G is a gnomon to A if G & A is similar to A. Here the symbol “&” should be taken to mean “attached in some suitable way.”

Gnomon


Consider the square S. The L-shaped figure G is a gnomon to the square–when G is attached to S as shown, we get the square S’.

Example 9.3 Gnomons to Squares


Consider the circular disk C with radius r. The O-ring G with inner radius r is a gnomon to C. Clearly, G & C form the circular disk. Since all circular disks are similar, C’ is similar to C.

Example 9.4 Gnomons to Circular Disks


Consider a rectangle R of height h and base b. The L-shaped figure G can clearly be attached to R to form the larger rectangle. This does not, in and of itself, guarantee that G is a gnomon to R.

Example 9.5 Gnomons to Rectangles


The rectangle R’ [with height (h + x) and base (b + y)] is similar to R if and only if their corresponding sides are proportional, which


requires that

b

h

b y h x .

This can be simplified to

b

h

y

x.


There is a simple geometric way to determine if the L-shaped G is a gnomon to R–just extend the diagonal of R in G & R. If the extended diagonal passes through the outside corner of G, then G is a gnomon; if it doesn’t, then it isn’t.



Let’s start with an isosceles triangle T, with vertices B, C, and D whose angles measure 72º, 72º, and 36º, respectively. On side CD we mark the point A so that BA is congruent

Example 9.6 A Golden Triangle

to BC. (A is the point of intersection of side CD and the circle of radius BC and center B.)


Since T’ is an isosceles triangle, angle BAC measures 72º and it follows that angle ABC measures 36º. This implies that triangle T’ has equal angles as triangle T and thus


they are similar triangles.


“So what?” you may ask. Where is the gnomon to triangle T? We don’t have one yet! But we do have a gnomon to triangle T’ – it is triangle BAD, labeled G’. After all, G’ & T’


is a triangle similar to T’. Note that gnomon G’ is an isosceles triangle with angles that measure 36º, 36º, and 108º.


We now know how to find a gnomon not only to triangle T’ but also to any 72-72-36 triangle, including the original triangle T: Attach a 36-36-108 triangle, G, to one of the longer sides of T.

72-72-36 and 36-36-108 Triangles


If we repeat this process indefinitely, we get a spiraling series of ever increasing 72-72-36 triangles.

72-72-36 and 36-36-108 Triangles


It’s not too far-fetched to use a family analogy: Triangles T and G are the “parents,” with T having the “dominant genes;” the “offspring” of their union looks just like T (but bigger). The offspring then has offspring of its own (looking exactly like its grand-parent T), and so on ad infinitum.

72-72-36 and 36-36-108 Triangles


Example 9.6 is of special interest to us for two reasons. First, this is the first time we have an example in which the figure and its gnomon are of the same type (isosceles triangles). Second, the isosceles triangles in this story (72-72-36 and 36-36-108) have a property that makes them unique: In both cases, the ratio of their sides (longer side over shorter side) is the golden ratio.These are the only two isosceles triangles with this property, and for this reason they are called golden triangles.

Golden Triangles


Consider a rectangle R with sides of length l (long side) and s (short side), and suppose that the square G withsides of length l is a gnomon to R.

Example 9.7 Square Gnomons to Rectangles


If so, then the rectangle R´ must be similar to R, which implies that their corresponding sides must be proportional (long side of R´ / short side of R´ = long side of R / short side of R):


l sl

ls


After some algebraic manipulation the preceding equation can be rewritten in the form


ls

2

l

s1


Since (1) l/s is positive (l and s are the lengths of the sides of a rectangle), (2) this last equation essentially says l/s that satisfies the golden property, and (3) the only positive number that satisfies the golden property is , we can conclude that


l

s


We can summarize all the above with the following conclusion:

A rectangle with sides of length l and s (long side and short side, respectively) has a square gnomon if and only if

Rectangles-Squares and Gnomons

l

s.


A rectangle whose sides are in the proportion of the golden ratio is called a golden rectangle. In other words, a golden rectangle is a rectangle with sides l (long side) and s (short side) satisfying l/s = . A close relative to a golden rectangle is a Fibonacci rectangle–a rectangle whose sides are consecutive Fibonacci numbers.

Golden and Fibonacci Rectangles l

s.


This rectangle has l = 1

and s = 1/.

Since l/s = 1/(1/) = ,

this is a golden rectangle.

Example 9.8 Golden and Almost Golden Rectangles


This rectangle has l = + 1

and s = .

Here l/s = ( + 1)/ .

Since + 1 = 2,

this is a golden rectangle.



This rectangle has l = 8 and s = 5. This is a Fibonacci rectangle, since 5 and 8


are consecutive Fibonacci numbers. The ratio of the sides is l/s = 8/5 = 1.6 so this is not a golden rectangle. On the other hand, the ratio 1.6 is reasonably close to so we will think of this rectangle as “almost golden.”


This rectangle has l = 89 and s = 55 and is a Fibonacci rectangle. The ratio of the sides is l/s = 89/55 = 1.61818…, in theory this is not a golden rectangle. In practice, this


rectangle is as good as golden–the ratio of the sides is the same as the golden ratio up to three decimal places.


This rectangle is neither a golden nor a Fibonacci rectangle. On the other hand, the ratio of the sides (12/7.44 ≈ 1.613) is very close to the golden ratio.It is safe to say that, sitting on a supermarket shelf, that box of Corn Pops looks temptingly golden.



From a design perspective, golden (and almost golden) rectangles have a special appeal, and they show up in many everyday objects, from posters to cereal boxes. In some sense, golden rectangles strike the perfect middle ground between being too “skinny” and being too “squarish.”

Golden Rectangles


A prevalent theory, known as the golden ratio hypothesis, is that human beings have an innate aesthetic bias in favor of golden rectangles, which, so the theory goes, appeal to our natural sense of beauty and proportion.

Golden Ratio Hypothesis






9.4 Gnomons



In nature, where form usually follows function, the perfect balance of a golden rectangle shows up in spiral-growing organisms, often in the form of consecutive Fibonacci numbers. To see how this connection works, consider the following example, which serves as a model for certain natural growth processes.

Spiral Growth in Nature


Start with a 1 by 1 square. Attach to it a 1 by 1 square. Squares 1 and 2 together form a 1 by 2 Fibonacci rectangle. We will call this the “second generation” shape.

Example 9.9 Stacking Squares on Fibonacci Rectangles


For the third generation, tack on a 2 by 2 square (3). The “third-generation” shape is the 3 by 2 Fibonacci rectangle.



Next, tack on a 3 by 3 square, giving a 3 by 5 Fibonacci rectangle.



Tacking on a 5 by 5 square results in an 8 by 5 Fibonacci rectangle. You get the picture–we can keep doing this as long as we want.



We might imagine these growing Fibonacci rectangles as a living organism. At each step, the organism grows by adding a square (a very simple, basic shape). The interesting feature of this growth is that as the Fibonacci rectangles grow larger, they become very close to golden rectangles, and become essentially similar to one another. This kind of growth–getting bigger while maintaining the same overall shape–is characteristic of the way many natural organisms grow.



Let’s revisit the growth process of the previous example, except now let’s create within each of the squares being added an interior “chamber” in the form of a quarter-circle. We need to be a little more careful about how we attach the chambered square in each successive generation, but other than that, we can repeat the sequence of steps in Example 9.9 to get the sequence of shapes shown on the next two slides.

Example 9.10 Growth of a “Chambered” Fibonacci Rectangle


These figures depict the consecutive generations in the evolution of the chambered Fibonacci rectangle.



The outer spiral formed by the circular arcs is often called a Fibonacci spiral.



Natural organisms grow in essentially two different ways. Humans, most animals, and many plants grow following what can informally be described as an all-around growth rule. In this type of growth, all living parts of the organism grow simultaneously–but not necessarily at the same rate. One characteristic of this type of growth is that there is no obvious way to distinguish between the newer and the older parts of the organism. In fact, the distinction between new and old parts does not make much sense.

Gnomon Growth


Contrast this with the kind of growth exemplified by the shell of the chambered nautilus, a ram’s horn, or the trunk of a redwood tree. These organisms grow following a one-sided or asymmetric growth rule, meaning that the organism has a part added to it (either by its own or outside forces) in such a way that the old organism together with the added part form the new organism. At any stage of the growth process, we can see not only the present form of the organism but also the organism’s entire past.

Gnomon Growth


All the previous stages of growth are the building blocks that make up the present structure. The other important aspect of natural growth is the principle of self-similarity: Organisms like to maintain their overall shape as they grow. This is where gnomons come into the picture. For the organism to retain its shape as it grows, the new growth must be a gnomon of the entire organism. We will call this kind of growth process gnomonic growth.

Gnomon Growth


We know from Example 9.4 that the gnomon to a circular disk is an O-ring with an inner radius equal to the radius of the circle.

Example 9.11 Circular Gnomonic Growth

We can thus have circular gnomonic growth by the regular addition of O-rings.


O-rings added one layer at a time to a starting circular structure preserve the circular shape through-out the structure’s growth. When carried to three dimensions, this is a good model for the way the trunk of a redwood tree grows. And this is why we can “read” the history of a felled redwood tree by studying its rings.

Example 9.11 Circular Gnomonic Growth


The figure shows a diagram of a cross section of the chambered nautilus. The

Example 9.12 Spiral Gnomonic Growth

chambered nautilus builds its shell in stages, each time adding another chamber to the already existing shell.


At every stage of its growth, the shape of the


chambered nautilus shell remains the same–the beautiful and distinctive spiral.


This is a classic example of gnomonic growth–each new chamber added to the shell is a gnomon of the entire shell. The gnomonic growth of the shell proceeds, in essence, as follows: Starting with its initial shell (a tiny spiral similar in all respects to the adult spiral shape), the animal builds a chamber (by producing a special secretion around its body that calcifies and hardens). The resulting, slightly enlarged spiral shell is similar to the original one.



The process then repeats itself over many stages, each one a season in the growth of the animal. Each new chamber adds a gnomon to the shell, so the shell grows and yet remains similar to itself. This process is a real-life variation of the mathematical spiral-building process discussed in Example 9.10. The curve generated by the outer edge of a nautilus shell–a cross section is called a logarithmic spiral.



More complex examples of gnomonic growth occur in sunflowers, daisies, pineapples, pinecones, and so on. Here, the rules that govern growth are somewhat more involved, but Fibonacci numbers and the golden ratio once again play a prominent role.

Complex Gnomonic Growth

Excursions in Modern Mathematics, 7e: 9.1 - 1Copyright © 2010 Pearson Education, Inc. 9 The...

Documents

Transcript of Excursions in Modern Mathematics, 7e: 9.1 - 1Copyright © 2010 Pearson Education, Inc. 9 The...