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Home , Continued fraction, Fibonacci, Golden angle, Golden ratio, Golden rectangle

THE GOLDEN AND THE SEQUENCE

Todd Cochrane

1 / 24 Everything is Golden

•

• Golden Proportion

• Golden Relation

• Golden

• Golden

• Golden

2 / 24 Geometric Growth, (Exponential Growth):

r = growth rate or common ratio. • Example. Start with 6 and double at each step: r = 2

6, 12, 24, 48, 96, 192, 384, ... Differences:

• Example Start with 2 and triple at each step: r = 3 2, 6, 18, 54, 162,... Differences:

Rule: The differences between consecutive terms of a geometric sequence grow at the same rate as the original sequence.

3 / 24 Differences and of consecutive Fibonacci :

1 1 2 3 5 8 13 21 34 55 89

Is the Fibonacci sequence a geometric sequence?

Lets examine the ratios for the Fibonacci sequence:

1 2 3 5 8 13 21 34 55 89 1 1 2 3 5 8 13 21 34 55 1 2 1.500 1.667 1.600 1.625 1.615 1.619 1.618 1.618

What value is the ratio approaching?

4 / 24 The Golden Ratio

The Golden Ratio, φ = 1.61803398...

• The Golden Ratio is (roughly speaking) the growth rate of the Fibonacci sequence as n gets large.

Euclid (325-265 B.C.) in Elements gives first recorded definition of φ.

n Next try calculating φ . Fn

n 1 2 3 4 5 6 10 12 n φ 1.618 2.618 2.118 2.285 2.218 2.243 2.236 2.236 Fn

φn √ φn ≈ 2.236... = 5, and so Fn ≈ √ . Fn 5

5 / 24 Formula for the n-th Fibonacci

n Rule: The n-th F is the nearest whole number to √φ . n 5 • Example. Find the 6-th and 13-th Fibonacci number.

6 n = 6. √φ = , so F = 5 6

13 n = 13. φ√ = , so F = 5 13 In fact, the exact formula is,

1 n 1 1 Fn = √ φ ± √ , (+ for odd n, − for even n) 5 5 φn

6 / 24 The 100-th Fibonacci Number

Find F100.

100 φ√ = 354224848179261915075.00000000000000000000056 5

F100 = 354224848179261915075

7 / 24 Some calculations with φ

Everyone calculate the following (round to three places):

1 1 = = φ 1.618

φ2 = 1.6182 =

8 / 24 The Amazing Number φ

The Amazing Number φ = 1.61803398... (we’ll round to 3 places):

1/1.618 = .618 that is, 1/φ = φ − 1 1.6182 = 2.618 = 1.618 + 1 that is, φ2 = φ + 1 1.6183 = 4.236 = 2 · 1.618 + 1 that is, φ3 = 2φ + 1 1.6184 = 6.854 = 3 · 1.618 + 2 that is, φ4 = 3φ + 2 1.6185 = 11.089 = 5 · 1.618 + 3 that is, φ5 = 5φ + 3

n φ = Fnφ + Fn−1

9 / 24 Golden Relation

Golden Relation φ2 = φ + 1

The Golden ratio is the unique positive satisfying the Golden Relation.

10 / 24 Exact Value of φ

What is the exact value of φ? From the Golden Relation,

φ2 − φ − 1 = 0

This is a (second degree): ax2 + bx + c = 0. :

√ √ −b + b2 − 4ac 1 + p(−1)2 − 4(−1) 1 + 5 φ = = = . 2a 2 2 √ 1 + 5 φ = = 1.618033988749... irrational 2

11 / 24 Golden Ratio from other sequences

• Example. Next, start with any two numbers and form a recursive sequence by adding consecutive numbers. See what the ratios approach this time. Say we start with 1, 3, 4, 7, 11, 18, 29, 47, 76, 123,...

3 4 7 11 18 29 47 76 123 ratio 1 3 4 7 11 18 29 47 76 value 3 1.33 1.75 1.57 1.64 1.61 1.62 1.617 1.618

Rule: Starting with any two distinct positive numbers, and forming a sequence using the Fibonacci rule, the ratios of consecutive terms will always approach the Golden Ratio!

Recall the Fibonacci Rule: Fn+1 = Fn + Fn−1

12 / 24 Why does the ratio always converge to φ?

WHY? Let An be a sequence satisfying the Fibonacci Rule:

An+1 = An + An−1

13 / 24 Golden Proportion

• Golden Proportion: Divide a into two parts, such that the ratio of the longer part to the shorter part equals the ratio of the whole to the longer part. What is the ratio?

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• Example. Golden Rectangle: Form a rectangle such that when the rectangle is divided into a and another rectangle, the smaller rectangle is similar (proportional) to the original rectangle. What is the ratio of the length to the width?

15 / 24 How to construct a Golden Rectangle

Start with a square ABCD. Mark the midpoint J on a given edge AB. Draw an arc with compass point fixed at J and passing through a vertex C on the opposite edge. Mark the point G where the arc meets the line AB.

• Note: Good to the Golden Rectangle can be obtained using the Fibonacci Ratios.

16 / 24 Partitioning a Golden Rectangle into

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Where is the eye of the spiral located? 18 / 24 • Example. The and .

The ratio of the edge of the inscribed star to the edge of the regular pentagon is φ, the golden ratio.

The ratio of the longer part of an edge of the star to the shorter part is φ.

19 / 24 Construction of a Regular Pentagon

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Divide a into two arcs, so that the ratio of the longer arc to the smaller arc is the golden ratio.

21 / 24 Continued Expansions

• Example. Expand π = 3.141592653....

1 1 3.141592... = 3 + .141592... = 3 + = 3 + 1/.141592... 7.062513... 1 1 1 = 3 + = 3 + = 3 + 7 + .062513... 1 1 7 + 1/.062513... 7 + 15.996594... 1 1 = 3 + = 3 + 1 1 7 + 7 + 15 + .996594... 1 15 + 1+. ..

Every has a unique infinite expansion.

22 / 24 Continued Fraction Expansion of φ

Continued Fraction Expansion of φ. 1 1 1.6180339 = 1 + .6180339 = 1 + = 1 + 1/.6180339 1.6180339 1 1 1 = 1 + = 1 + = 1 + 1 + .6180339 1 1 1 + 1/.6180339 1 + 1.6180339 1 1 = 1 + = 1 + 1 1 1 + 1 + 1 + 1 1 1.6180339 1 + 1+. .. Another way to see this is: From the first line above we see that 1 φ = 1 + φ , Now substitute this expression for φ into the right-hand side and keep repeating: 1 φ = + ,... 1 1 1 + φ

23 / 24 Convergents to the Continued Fraction expansion of φ

1 1 1, 1 + 1 = 2, 1 + 1 = 1+ 1

1 1 + = 1 + 1 1 1 + 1

The convergents to the continued fraction expansion of φ are

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