THE GOLDEN RATIO AND THE FIBONACCISEQUENCE

Todd Cochrane

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Everything is Golden

• Golden Ratio

• Golden Proportion

• Golden Relation

• Golden Rectangle

• Golden Spiral

• Golden Angle

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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 ageometric sequence grow at the same rate as the originalsequence.

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Differences and ratios of consecutive Fibonacci numbers:

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:

11

21

32

53

85

138

2113

3421

5534

8955

1 2 1.500 1.667 1.600 1.625 1.615 1.619 1.618 1.618

What value is the ratio approaching?

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The Golden Ratio

The Golden Ratio, φ = 1.61803398...

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

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

Next try calculating φn

Fn.

n 1 2 3 4 5 6 10 12φn

Fn1.618 2.618 2.118 2.285 2.218 2.243 2.236 2.236

φn

Fn≈ 2.236... =

√5, and so Fn ≈

φn√

5.

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Formula for the n-th Fibonacci Number

Rule: The n-th Fibonacci Number Fn is the nearest whole number to φn√

5.

• Example. Find the 6-th and 13-th Fibonacci number.

n = 6. φ6√

5= , so F6 =

n = 13. φ13√

5= , so F13 =

In fact, the exact formula is,

Fn =1√5φn ± 1√

51φn , (+ for odd n, − for even n)

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The 100-th Fibonacci Number

Find F100.

φ100√

5= 354224848179261915075.00000000000000000000056

F100 = 354224848179261915075

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Some calculations with φ

Everyone calculate the following (round to three places):

1φ=

11.618

=

φ2 = 1.6182 =

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The Amazing Number φ

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

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

φn = Fnφ+ Fn−1

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Golden Relation

Golden Relation φ2 = φ+ 1

The Golden ratio is the unique positive real number satisfyingthe Golden Relation.

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Exact Value of φ

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

φ2 − φ− 1 = 0

This is a quadratic equation (second degree): ax2 + bx + c = 0.Quadratic Formula:

φ =−b +

√b2 − 4ac

2a=

1 +√(−1)2 − 4(−1)

2=

1 +√

52

.

φ =1 +√

52

= 1.618033988749... irrational

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Golden Ratio from other sequences

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

ratio 31

43

74

117

1811

2918

4729

7647

12376

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, andforming a sequence using the Fibonacci rule, the ratios ofconsecutive terms will always approach the Golden Ratio!

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

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Why does the ratio always converge to φ?

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

An+1 = An + An−1

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Golden Proportion

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

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Golden Rectangle

• Example. Golden Rectangle: Form a rectangle such thatwhen the rectangle is divided into a square and anotherrectangle, the smaller rectangle is similar (proportional) to theoriginal rectangle. What is the ratio of the length to the width?

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How to construct a Golden Rectangle

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

• Note: Good approximations to the Golden Rectangle can beobtained using the Fibonacci Ratios.

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Partitioning a Golden Rectangle into Squares

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Golden Spiral

Where is the eye of the spiral located?18 / 24

• Example. The Pentagon and Pentagram.

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

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

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Construction of a Regular Pentagon

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Golden Angle

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

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Continued Fraction Expansions

• Example. Expand π = 3.141592653....

3.141592... = 3 + .141592... = 3 +1

1/.141592...= 3 +

17.062513...

= 3 +1

7 + .062513...= 3 +

1

7 + 11/.062513...

= 3 +1

7 + 115.996594...

= 3 +1

7 +1

15 + .996594...

= 3 +1

7 +1

15 +1

1+. . .

Every irrational number has a unique infinite continuedfraction expansion.

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Continued Fraction Expansion of φ

Continued Fraction Expansion of φ.

1.6180339 = 1 + .6180339 = 1 +1

1/.6180339= 1 +

11.6180339

= 1 +1

1 + .6180339= 1 +

1

1 + 11/.6180339

= 1 +1

1 + 11.6180339

= 1 +1

1 +1

1 + 11.6180339

= 1 +1

1 +1

1 +1

1+. . .Another way to see this is: From the first line above we see thatφ = 1 + 1

φ , Now substitute this expression for φ into theright-hand side and keep repeating:

φ = 1 +1

1 + 1φ

, . . .

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Convergents to the Continued Fraction expansion of φ

1, 1 + 11 = 2, 1 + 1

1+ 11=

1 +1

1 +1

1 + 11

=

The convergents to the continued fraction expansion of φare

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