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Excursions in Modern MathematicsSixth Edition
Peter Tannenbaum
1
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
10
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).
Spiral Growth in Nature67
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).
Spiral Growth in Nature69
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.
Spiral Growth in Nature71
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
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