t golden ratio and the fibonacci sequencecjbalm/quest/day7_notes.pdfthe golden ratio the golden...

54
T HE GOLDEN RATIO AND THE F IBONACCI S EQUENCE Todd Cochrane 1/54

Upload: dangdieu

Post on 23-May-2018

229 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

THE GOLDEN RATIO AND THE FIBONACCISEQUENCE

Todd Cochrane

1 / 54

Page 2: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Everything is Golden

• Golden Ratio

• Golden Proportion

• Golden Relation

• Golden Rectangle

• Golden Spiral

• Golden Angle

2 / 54

Page 3: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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.

3 / 54

Page 4: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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?

4 / 54

Page 5: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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?

5 / 54

Page 6: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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.

6 / 54

Page 7: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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.

7 / 54

Page 8: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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)

8 / 54

Page 9: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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)

9 / 54

Page 10: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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)

10 / 54

Page 11: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

The 100-th Fibonacci Number

Find F100.

φ100√

5= 354224848179261915075.00000000000000000000056

F100 = 354224848179261915075

11 / 54

Page 12: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Some calculations with φ

Everyone calculate the following (round to three places):

1φ=

11.618

=

φ2 = 1.6182 =

12 / 54

Page 13: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

13 / 54

Page 14: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

14 / 54

Page 15: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

15 / 54

Page 16: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

16 / 54

Page 17: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

17 / 54

Page 18: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

18 / 54

Page 19: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Golden Relation

Golden Relation φ2 = φ+ 1

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

19 / 54

Page 20: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

20 / 54

Page 21: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

21 / 54

Page 22: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

22 / 54

Page 23: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

23 / 54

Page 24: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

24 / 54

Page 25: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

25 / 54

Page 26: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

26 / 54

Page 27: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

27 / 54

Page 28: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

28 / 54

Page 29: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

29 / 54

Page 30: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

30 / 54

Page 31: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

31 / 54

Page 32: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

32 / 54

Page 33: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

33 / 54

Page 34: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

34 / 54

Page 35: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

35 / 54

Page 36: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Why does the ratio always converge to φ?

WHY?

Let An be a sequence satisfying the Fibonacci Rule:

An+1 = An + An−1

36 / 54

Page 37: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Why does the ratio always converge to φ?

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

An+1 = An + An−1

37 / 54

Page 38: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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?

38 / 54

Page 39: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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?

39 / 54

Page 40: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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.

40 / 54

Page 41: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Partitioning a Golden Rectangle into Squares

41 / 54

Page 42: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Golden Spiral

Where is the eye of the spiral located?42 / 54

Page 43: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

• 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.

43 / 54

Page 44: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

• 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.

44 / 54

Page 45: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

• 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.

45 / 54

Page 46: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Construction of a Regular Pentagon

46 / 54

Page 47: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

Golden Angle

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

47 / 54

Page 48: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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.

48 / 54

Page 49: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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.

49 / 54

Page 50: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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.

50 / 54

Page 51: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

, . . .

51 / 54

Page 52: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

, . . .

52 / 54

Page 53: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

, . . .

53 / 54

Page 54: T GOLDEN RATIO AND THE FIBONACCI SEQUENCEcjbalm/Quest/Day7_notes.pdfThe Golden Ratio The Golden Ratio, ˚= 1:61803398::: The Golden Ratio is (roughly speaking) the growth rate of the

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

54 / 54