Excursions in Modern MathematicsSixth Edition

Peter Tannenbaum

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Chapter 9Spiral Growth in NatureFibonacci Numbers and the Golden Ratio

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Spiral Growth in NatureOutline/learning

Objectives3

To generate the Fibonacci sequence and identify some of its properties.

To identify relationships between the Fibonacci sequence and the golden ratio.

To define a gnomon and understand the concept of similarity.

To recognize gnomonic growth in nature.

Spiral Growth in Nature

9.1 Fibonacci’s Rabbits

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Fibonacci’s Rabbits5

“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?”

- Leonardo Fibonacci, 1202

Fibonacci’s Rabbits6

Let’s say P represents the pairs of rabbits. P1 = first month P2 = second month PN = N months P12 = a year

Fibonacci’s Rabbits7

P0 = 1 P1 = 1 P2 = 2 P3 = 3 P4 = 5 P5 = 8 P6 = 13 P7 = 21

Fibonacci’s Rabbits8

PN = PN-1 + PN-2

P4 = P4-1 + P4-2

P5 = P5-1 + P5-2

P6 = P6-1 + P6-2

Fibonacci’s Rabbits9

P12 = P12-1 + P12-2 = 144 + 89

233 pairs of rabbits!!!

Spiral Growth in Nature9.2 Fibonacci Numbers

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Fibonacci Numbers11

The Fibonacci SequenceThe Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…

The Fibonacci numbers form what mathematician call an infinite sequence– an ordered list of numbers that goes on forever. As with any other sequence, the terms are ordered from left to right.

Fibonacci Numbers12

The Fibonacci SequenceThe Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…

In mathematical notation we express this by using the letter F (for Fibonacci) followed by a subscript that indicates the position of the term in the sequence. In other words, F1 = 1, F2 = 1, F3 = 2, F4 = 3, ...F10 =55, and so on. A generic Fibonacci number as FN.

Fibonacci Numbers13

The Fibonacci SequenceThe Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…

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 other (earlier) numbers in the sequence.

Fibonacci Numbers14

Fibonacci Numbers (Recursive Definition)

FN = FN-1 + FN-2 (the recursive rule)

FN is a generic Fibonacci number. FN-1 is the Fibonacci number right before it. FN-2 is the Fibonacci number two positions before it.

Fibonacci Numbers15

Fibonacci Numbers (Recursive Definition)

F1 = 1, F2 = 1 (the seeds)

The preceding rule cannot be applied to the first two Fibonacci numbers, F1 (there are no Fibonacci numbers before it) and F2 (there is only one Fibonacci number before it– the rule requires two), so for a complete description, we must “anchor” the rule by giving the values of the first Fibonacci numbers as named above.

Fibonacci Numbers16

If you know two Fibonacci numbers then you can derive the numbers before and after.

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

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

F99 = F97 + F98 F96 = F98 – F97

Fibonacci Numbers17

You can use the Fibonacci Notation to solve math problems.

F12-3 = F9 F2x8 = F16

2F4 = 2x3 F4 x F3 = 5x2

F3 / F1 = 2 F36 / 3 = F12

Fibonacci Numbers18

Is there an explicit (direct) formula for computing Fibonacci numbers?

Binet’s Formula

Binet’s formula is an example of an explicit formula– it allows us to calculate a Fibonacci number without needing to calculate all the preceding Fibonacci numbers.

1 1 5 1 5

2 25

N N

NF

Fibonacci Numbers19

Fibonacci Numbers20

Fibonacci Numbers21

Fibonacci Numbers22

Bartok’s Sonata for two pianos and percussion.

Fibonacci Numbers23

Fibonacci Numbers24

Fibonacci Numbers25

Fibonacci Numbers26

Fibonacci Numbers27

Links:Fibonacci - World's Most Mysterious NumberSpirals, Fibonacci and being a plant – part 1

Spiral Growth in Nature9.3 The Golden Ratio

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The Golden Ratio29

Complete the following Fibonacci equations:F1 = _____F2 = _____F3 = _____F4 = _____F5 = _____F6 = _____F7 = _____F8 = _____F9 = _____F10 = _____

F2/F1 = _____F3/F2 = _____F4/F3 = _____F5/F4 = _____F6/F5 = _____F7/F6 = _____F8/F7 = _____F9/F8 = _____F10/F9 = _____F11/F10 = _____

4181/2584 = _____6765/4181 = _____

Φ (phi) = 1.618“The Golden Ratio”“The Devine Proportion”“The Golden Number”“The Golden Section”

= _______1+ = _______(1+ )/2 = _______

The Golden Ratio30

The Golden RatioThe Golden RatioWe will now focus our attention on the number

one of the most remarkable and famous numbers in all of mathematics. The modern tradition is to denote this number by the Greek letter (phi).

1 5 / 2

The Golden Ratio31

The Golden Property: When adding one to the number you get the square of the number.

Φ turns out to be the only positive number with that property: Φ2 = Φ +1

The Golden Ratio32

With that property in mind we can recursively compute higher and higher powers of Φ.

First, we multiply Φ2 = Φ + 1 times Φ.

Φ3 = Φ2 + Φ

Replace Φ2 with Φ +1.Φ3 = Φ + 1 + Φ Φ3 = 2Φ +1

The Golden Ratio33

Recursively multiplying by Φ and substituting you get:

Φ2 = Φ + 1 Φ3 = 2Φ + 1Φ4 = 3Φ + 2 Φ5 = 5Φ + 3Φ6 = 8Φ + 5 and so on…

The Golden Ratio34

Powers of the Golden RatioPowers of the Golden Ratio

In some ways you may think of the preceding formula as the opposite of Binet’s formula. Whereas Binet’s formula uses powers of the golden ratio to calculate Fibonacci numbers, this formula uses Fibonacci numbers to calculate powers of the golden ratio.

1N

NNF F

The Golden Ratio35

The Golden Ratio36

The Golden Ratio37

The Golden Ratio38

Nature

The Golden Ratio39

Music

The Golden Ratio40

Architecture

The Golden Ratio41

Art

The Golden Ratio42

Human Body

The Golden Ratio43

Paula Zahn

The Golden Ratio44

And it goes on and on…

Where do you think you can find the Golden Ratio?

Fibonacci Numbers45

Links:Natures Number: 1.618033988...The Golden RatioTed Talk - Golden RatioSpirals, Fibonacci, and being a plant… part 2

Spiral Growth in Nature

9.4 Gnomons

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Gnomons47

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.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.

Similarity48

We know from geometry that two objects are said to be similar if one is a scaled version of the other.

The following important facts about similarity of basic two-dimensional figures will come in handy later in the chapter.

Similarity49

Triangles: Triangles: 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.

Similarity50

Similarity51

Squares: Squares: Two squares are always similar.

Rectangles: Rectangles: Two rectangles are similar if their corresponding sides are proportional.

Similarity52

Similarity53

Circles and disks: Circles and disks: Two circles are always similar. Any circular disk (a circle plus all of its interior) is similar to any other circular disk.

Circular rings: Circular rings: Two circular rings are similar if and only if their inner and outer radii are proportional.

Similarity54

Gnomons55

We will now return to the main topic of this section– gnomon. In geometry, a gnomon G to a figure A is a connected figure which, when suitably attached to A, produces a new figure similar to A. Informally, we will describe it this way: G is a gnomon to A if G&A is similar to A.

Gnomons

Gnomons56

Consider the square S in (a). The L-shaped figure G in (b) is a gnomon to the square– when G is attached to S as shown in (c), we get the square S´.

Gnomons57

Consider the circular disk C with radius r in (a). The O-ring G in (b) with inner radius r is a gnomon to C. Clearly, G&C form the circular disk C´ shown in (c). Since all circular disks are similar, C´ is similar to C.

Gnomons58

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

Gnomons59

The rectangle R´ is similar to R if and only if their corresponding sides are proportional. With a little algebraic manipulation, this can be simplified to

b y

h x

Gnomons60

Gnomons61

Gnomons62

Based on what you’ve learned in this Unit, where would you expect to see Gnomons?

Fibonacci Numbers63

Links:Gnomons

Spiral Growth in Nature9.5 Spiral Growth in Nature

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Spiral Growth in Nature65

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.

Spiral Growth in Nature66

Start with a 1-by-1 square [square 1 in (a).

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Attach to it a 1-by-1 square [square 2 in (b)]. Squares 1 and 2 together form a 2-by-1 Fibonacci rectangle. We will call this the “second-generation” shape.

Spiral Growth in Nature68

For the third generation, tack on a 2-by-2 square [square 3 in (c)]. The “third-generation” shape (1, 2, and 3 together) is the 2-by3 Fibonacci rectangle in (c).

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Next, tack onto it a 3-by-3 square [square 4 in (d)], giving a 5-by-3 Fibonacci rectangle.

Spiral Growth in Nature70

Then tack o a 5-by-5 square [square 5 in (e)}, resulting in an 8-by-5 Fibonacci rectangle. We can keep doing this as long as we want.

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Links:Spirals, Fibonacci and being a plant

Spiral Growth in Nature Conclusion

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Form follows functionForm follows function Fibonacci numbersFibonacci numbers The Golden Ratios and The Golden Ratios and Golden RectanglesGolden Rectangles

GnomonsGnomons