EXCURSIONS IN MODERN MATHEMATICS SIXTH EDITION Peter Tannenbaum 1

CHAPTER 9 SPIRAL GROWTH IN NATURE Fibonacci Numbers and the Golden Ratio 2

Spiral Growth in Nature Outline/learning Objectives 3  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 4

Fibonacci’s Rabbits 5 “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 Rabbits 6  Let’s say P represents the pairs of rabbits.  P 1 = first month  P 2 = second month  P N = N months  P 12 = a year

Fibonacci’s Rabbits 7  P 0 = 1  P 1 = 1  P 2 = 2  P 3 = 3  P 4 = 5  P 5 = 8  P 6 = 13  P 7 = 21

Fibonacci’s Rabbits 8  P N = P N-1 + P N-2  P 4 = P P 4-2  P 5 = P P 5-2  P 6 = P P 6-2

Fibonacci’s Rabbits 9  P 12 = P P 12-2 = pairs of rabbits!!!

SPIRAL GROWTH IN NATURE 9.2 Fibonacci Numbers 10

Fibonacci Numbers 11 The 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 Numbers 12 The 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, F 1 = 1, F 2 = 1, F 3 = 2, F 4 = 3,...F 10 =55, and so on. A generic Fibonacci number as F N.

Fibonacci Numbers 13 The 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 Numbers 14 Fibonacci Numbers (Recursive Definition) F N = F N-1 + F N-2 (the recursive rule) F N is a generic Fibonacci number. F N-1 is the Fibonacci number right before it. F N-2 is the Fibonacci number two positions before it.

Fibonacci Numbers 15 Fibonacci Numbers (Recursive Definition) F 1 = 1, F 2 = 1 (the seeds) The preceding rule cannot be applied to the first two Fibonacci numbers, F 1 (there are no Fibonacci numbers before it) and F 2 (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 Numbers 16  If you know two Fibonacci numbers then you can derive the numbers before and after. F 97 = 83,621,143,489,848,422,977 F 98 = 135,301,852,344,706,746,049 F 99 = F 97 + F 98 F 96 = F 98 – F 97

Fibonacci Numbers 17  You can use the Fibonacci Notation to solve math problems. F 12-3 = F 9 F 2x8 = F 16 2F 4 = 2x3F 4 x F 3 = 5x2 F 3 / F 1 = 2F 36 / 3 = F 12

Fibonacci Numbers 18 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.

Fibonacci Numbers 19

Fibonacci Numbers 20

Fibonacci Numbers 21

Fibonacci Numbers 22  Bartok’s Sonata for two pianos and percussion.

Fibonacci Numbers 23

Fibonacci Numbers 24

Fibonacci Numbers 25

Fibonacci Numbers 26

Fibonacci Numbers 27 Links:  Fibonacci - World's Most Mysterious Number Fibonacci - World's Most Mysterious Number  Spirals, Fibonacci and being a plant – part 1 Spirals, Fibonacci and being a plant

SPIRAL GROWTH IN NATURE 9.3 The Golden Ratio 28

The Golden Ratio 29  Complete the following Fibonacci equations: F 1 = _____ F 2 = _____ F 3 = _____ F 4 = _____ F 5 = _____ F 6 = _____ F 7 = _____ F 8 = _____ F 9 = _____ F 10 = _____ F 2 /F 1 = _____ F 3 /F 2 = _____ F 4 /F 3 = _____ F 5 /F 4 = _____ F 6 /F 5 = _____ F 7 /F 6 = _____ F 8 /F 7 = _____ F 9 /F 8 = _____ F 10 /F 9 = _____ F 11 /F 10 = _____ 4181/2584 = _____ 6765/4181 = _____ Φ (phi) = “The Golden Ratio” “The Devine Proportion” “The Golden Number” “The Golden Section”

= _______ 1+ = _______ (1+ )/2 = _______ The Golden Ratio 30 The Golden Ratio We 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).

The Golden Ratio 31  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: Φ 2 = Φ +1

The Golden Ratio 32  With that property in mind we can recursively compute higher and higher powers of Φ.  First, we multiply Φ 2 = Φ + 1 times Φ. Φ 3 = Φ 2 + Φ  Replace Φ 2 with Φ +1. Φ 3 = Φ Φ  Φ 3 = 2 Φ +1

The Golden Ratio 33  Recursively multiplying by Φ and substituting you get: Φ 2 = Φ + 1 Φ 3 = 2 Φ + 1 Φ 4 = 3 Φ + 2 Φ 5 = 5 Φ + 3 Φ 6 = 8 Φ + 5 and so on…

The Golden Ratio 34 Powers 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.

The Golden Ratio 35

The Golden Ratio 36

The Golden Ratio 37

The Golden Ratio 38  Nature

The Golden Ratio 39  Music

The Golden Ratio 40  Architecture

The Golden Ratio 41  Art

The Golden Ratio 42  Human Body

The Golden Ratio 43  Paula Zahn

The Golden Ratio 44  And it goes on and on… Where do you think you can find the Golden Ratio?

Fibonacci Numbers 45 Links:  Natures Number: Natures Number:  The Golden Ratio The Golden Ratio  Ted Talk - Golden Ratio Ted Talk - Golden Ratio  Spirals, Fibonacci, and being a plant… part 2 Spirals, Fibonacci, and being a plant

SPIRAL GROWTH IN NATURE 9.4 Gnomons 46

Gnomons 47 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.

Similarity 48 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.

Similarity 49 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.

Similarity 50

Similarity 51 Squares: Squares: Two squares are always similar. Rectangles: Rectangles: Two rectangles are similar if their corresponding sides are proportional.

Similarity 52

Similarity 53 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.

Similarity 54

Gnomons 55 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

56 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´.

Gnomons 57 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.

Gnomons 58 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.

Gnomons 59 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

Gnomons 60

Gnomons 61

Gnomons 62  Based on what you’ve learned in this Unit, where would you expect to see Gnomons?

Fibonacci Numbers 63 Links:  Gnomons Gnomons

SPIRAL GROWTH IN NATURE 9.5 Spiral Growth in Nature 64

Spiral Growth in Nature 65 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 Nature 66 Start with a 1-by-1 square [square 1 in (a).

Spiral Growth in Nature 67 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 Nature 68 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).

Spiral Growth in Nature 69 Next, tack onto it a 3-by-3 square [square 4 in (d)], giving a 5-by-3 Fibonacci rectangle.

Spiral Growth in Nature 70 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.

Spiral Growth in Nature 71 Links:  Spirals, Fibonacci and being a plant Spirals, Fibonacci and being a plant

Spiral Growth in Nature Conclusion 72  Form follows function  Fibonacci numbers  The Golden Ratios and Golden Rectangles  Gnomons