& The Fibonacci numbers

1, 1, 2, 3, 5, ..

1:1.618...

Lalith Varaprasad National Institute of Design

Student Manual

The Golden Ratio

The Golden Ratio & The Fibonacci numbers

1, 1, 2, 3, 5, ..

Designed as a part of Diploma Project at National Institute of Design, Ahmedabad. Project Sponsor: Ford Foundation Project Guide: Immanuel Suresh Designer: Lalith Varaprasad Special thanks to: Immanuel Suresh, Tarun Deep Girdher, Manu Gajjar and Praveen Nahar.

Creative Commons Attribution-Non Commercial 4.0 International Public License You are free to: Share – copy and redistribute the material in any medium or format. Adapt – remix, transform, and build upon the material. Under the following terms: NonCommercial – you may not use the material for commercial purposes.

THE GOLDEN RATIO AND THE FIBONACCI NUMBERS 1, 1, 2, 3, 5, ...

LALITH VARAPRASAD

NATIONAL INSTITUTE OF DESIGN, INDIA

The Golden Ratio is also known as

the golden section, golden proportion, golden mean, golden number, divine section and divine proportion.

It is represented with the Greek letter â&#x20AC;&#x2DC;phiâ&#x20AC;&#x2122;

IF YOU WANT TO UNDERSTAND NATURE &

BECOME AN ARTIST, DESIGNER, ARCHITECT, ETC.,. GOLDEN RATIO WILL HELP YOU TO CREATE EYE- PLEASING WORK.

CONTENTS GOLDEN RATIO > Activity 1 to 3: Understanding your body proportions

1

> Activity 4 to 5: Repeating the experiment with more body measurements

9

> Golden ratio explanation

14

> Drawing Golden rectangle

16

> Golden ratio examples and exercise sheet

18

FIBONACCI NUMBERS > Activity 1: Solving Fibonacciâ&#x20AC;&#x2122;s rabbits problem

28

> The Fibonacci numbers explanation

32

> Drawing Fibonacci Golden rectangle and spiral

33

> Fibonacci numbers examples

36

TOOLS > Tools to measure Golden ratio

38

> Creating tools to measure Golden ratio

40

> Using tools to measure Golden ratio with exercise sheets

44

Using tools, draw your observations based on your understanding

50

Resources & Colophon

52

B

A

Activity 1

Understanding your body proportions Stand against a wall and mark your total height and your navel height. Using a measuring tape or a scale, Measure your height from feet to navel (A) =

cm

Measure your height from navel to head top (B) =

cm

The ratio A to B (A /B) = A ratio is a relationship between two numbers of the same kind, expressed as "a /b" or a:b. A ratio is a number and it does not have any unit.

1

2

B

B

A

A

Activity 2

Understanding your friend’s body proportions Ask a few friends to take their measurements like you did in Activity 1 1. Your friend’s name: Measure your friend’s height from feet to navel (A) =

cm

Measure your friend’s height from navel to head top (B) =

cm

The ratio A to B (A /B) = 2. Your friend’s name: Measure your friend’s height from feet to navel (A) =

cm

Measure your friend’s height from navel to head top (B) =

cm

The ratio A to B (A /B) =

3

Activity 3

If you are more curious, try this exercise with many friends

B

B

B

A

A

A

S.no Your friendâ&#x20AC;&#x2DC;s name

4

His body proportion A is equal to

His body proportion B is equal to

Ratio A /B is equal to

From Activity 1, 2 and 3

WHAT DID YOU OBSERVE?

The ratio A to B (A /B) is approximately =

5

B

A

6

The ratio of the total height from the feet to the navel (A) to the height from the navel to the head top (B) in any human body is approximately equal to 1.618

7

C

A

8

Activity 4

Repeating the experiment with more body measurements Stand against a wall and mark your total height and your navel height. Using a measuring tape or a scale, Measure your total height from feet to top of your head (C) =

cm

Measure your height from feet to navel (A) =

cm

The ratio C to A (C/A) =

9

Activity 5

Understanding your friendâ&#x20AC;&#x2122;s body proportions

C

A

S.no Your friendâ&#x20AC;&#x2DC;s name

10

C

A

C

A

His body proportion C is equal to

His body proportion A is equal to

Ratio C /A is equal to

From Activity 4 and Activity 5

WHAT DID YOU OBSERVE?

The ratio C to A (C /A) is approximately =

11

C

A

12

The ratio of the total height from the feet to the head (C) to the height from the feet to the navel (A) in any human body is approximately equal to 1.618

13

The Golden ratio and the idea behind it B C A

When we divide a line into two parts so that: the longer part divided by the smaller part

=

the whole length divided by the longer part

then we have the golden ratio. A B A B

14

=

C A

= 1.618 =

sometimes expressed as A:B

The golden ratio (symbol is the Greek letter ‘phi’) is a special number approximately equal to 1.618. It appears many times in geometry, architecture, art and other areas. ‘Phi’ came from Phidias was a Greek sculptor, painter and architect, who lived in the 5th century BC, and is commonly regarded as one of the greatest of all sculptors of classical Greece. Phidias used golden ratio when he designed the Parthenon. The Parthenon is a temple on the Athenian Acropolis, Greece, dedicated to the goddess Athena, who the people of Athens considered their patron.

15

Drawing Golden rectangle Here is one way to draw a rectangle with the Golden Ratio: Draw a square (of size 1 inch) 1”

Place a dot half way along one side

1”

1/2 ” 1/2 ”

Draw a line from that point to an opposite corner (it will be √5/2 in length)

Using compass draw an arc from the point so that it runs along the square’s side

1” 5/2 ” 1”

5/2 ”

1/2 ” 1/2 ”

1”

Then you can extend the square to be a rectangle

5/2 ” 5/2 ” 1/2 ” 1”

16

The side length of this rectangle is equal to 5 2

1 5 1+ 5 + = = 2 2 2

1 2

1

In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : (1+ 5 )/2, which is 1: (the Greek letter phi), where is approximately 1.618.

17

Examples of the Golden ratio

1

2

3

Golden ratio in architecture (Taj Mahal in India)

18

4

1

2 3

4

19

8.37 cm Golden ratio in Painting ( Michelangeloâ&#x20AC;&#x2122;s Creation of Adam ) (13.55cm / 8.37cm ! 1.618... =

20

)

13.55 cm You can measure above rectangles with a scale

21

1 2

3

4

Golden ratio in human hand (The skeleton gives a more accurate measurement)

22

1 3

2

4

23

1

3

Golden ratio in Honey Bees

24

2

4

2

1

4

DRONE

3

WORKER

QUEEN

25

Golden rectangle exercise sheet Look around and try to ďŹ nd rectangles whose dimensions are in the golden ratio Object

26

Width

Length

Ratio of length to width

Draw anything you like within the golden rectangle boxes below

27

Fibonacci numbers

How many spirals go in the clockwise direction (green lines)? How many spirals go in a counter-clockwise direction (yellow lines)? Isn’t that strange? Wouldn’t you expect that they would be the same? To understand the spirals in pinecones, pineapples, sunflowers and lots of other things in nature, we have to meet an Italian mathematician named Leonardo de Pisa. Most people call him Fibonacci. He ﬁrst posed this problem in 1202.

28

Fibonacciâ&#x20AC;&#x2122;s rabbits problem:

If 2 newborn rabbits are put in a pen, how many rabbits will be in the pen after 1 year? Assume that rabbits... > Always produce one male and one female offspring > Can reproduce once every month > Can start reporducing once they are two months old > Never die!

29

Complete the following chart

Month 1

2

3

1

2

3

1

4

1

2

5

5

1

2

3

1

6

6

1

2

3

4

4

7 8

30

Pairs of rabbits

1

1

1 represents this pair of rabbits are 1 month old.

This arrow represents given birth to a new pair

This arrow represents age transformation

Total pairs 1

1

2 3

5

1

2

8

31

Fibonacci’s work on this problem led him to this sequence of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... 0+1=1

Take any two Fibonacci numbers next to each other and try to ﬁnd out the ratio of those two numbers, you will then have golden ratio. (Note: for best results take bigger Fibonacci numbers)

1+2=3

3+5=8

Can you ﬁgure out what the next number in the sequence will be? We call this the Fibonacci sequence, and the numbers are called Fibonacci numbers. To get the next number in the sequence, you add the previous two numbers together. Now go back and look at those pinecone spirals. What do you notice about the number of pinecone spirals in each direction?

32

Fibonacci and the Golden rectangle An easy one to do using graph paper. 1. Start by colouring in one square. This is a 1 x 1 square because it is one length on each side. Now add another 1 x 1 square next to it using a different coloured pencil. 2. Now add a 2 x 2 square and then add a 3 X 3 square. 4. Continue drawing squares in a clockwise direction until your paper runs out of space. 5. Now look at the Fibonacci sequence, what size square should you add next? 6. What do you notice about the ratio between the length and width of each rectangle as you measure larger and larger rectangles?

33

Fibonacci and Golden spiral

Now, with your compass, make an arc in the squares with a radius the size of the edge of the square. Donâ&#x20AC;&#x2122;t get nervous about these big words; they just mean that the arc will be one-quarter of a circle. Look at the picture below to see what I mean. The arcs in the ďŹ rst squares will be really, really tiny. But look how they grow!

34

Look at this picture of a nautilus shell. What do you notice?

35

Examples of the Fibonacci numbers in nature

The number of petals on the above flowers is a Fibonacci number

Sunflower

36

Pinecone

13 8 5 3 2 1 Branches

Growing trees

37

Tools You can use following tools to measure golden ratio in nature

1.618

1

Golden ratio Caliper

Patent: Jack Nestor, William A. Shoemaker, Jr. in 1986 http://www.google.com/ patents/US4768953

38

Ruler

1.618

1

Golden rectangle

39

Creating tools Tool 1: Creating Golden ratio Caliper A

ﬁg 1

AE = AG = 158.5mm

B C

BF = 98mm AB = AC = BD = CD = 60.5mm

D E

F

G

DF = 37.5mm

Material required:

Process of making Caliper

> Pencil, Ruler, Cutter and Tracing paper > Used carton or cardboard or any used shoe box > 4 small bolts and nuts or 4 rivets for joints

1. Trace the individual parts of ﬁg 2 on a tracing paper with a pencil and transfer the drawings on a carton or cardboard sheet 2. Cut the cardboard along the drawing and make holes at proper distance as shown in drawing ﬁg 1

Measuring golden ratio in palm

3. Fix the joints with rivets or nut and bolts as shown in drawing ﬁg 1 4. Now your tool is ready. Go and explore to understand the golden ratio in nature

40

ďŹ g 2

1

1.618

1.618

1

You can create your own bigger or smaller calipers with different measurements but the ratio should always be 1:1.618 as shown in ďŹ g 2.

41

Tool 2: Creating Golden rectangle

1.618

1

Material required:

Making Golden rectangle

> Pencil, Ruler, Cutter

1. Take a two coloured transparent sheets and create equal golden rectangles in the size that you have learnt in page number 20 (ďŹ g 2)

> Coloured transparent sheets > Cello tape (transparent sticking tape)

2. Cut two coloured transparent sheets along the drawing ( you will get 2 squares, 2 rectangles ) Measuring golden ratio in palm

3. Join opposite color sheets together to form golden rectangle with a cello tape 4. Now your tool is ready. Go and explore to understand the golden ratio in nature

42

ďŹ g 2

5 2

1 2

1 You can create your own bigger or smaller golden rectangle with different measurements but the ratio should always be 1:1.618 as shown in ďŹ g 2.

43

Golden Ratio exercise sheets 1. Measure the blue and red colour line distances on the face drawing using a (Ruler) 2. Find the ratio of the blue line to the red line of exercise sheet 1 Example: 4 cm

/

2.47 cm

Bigger line distance/ Smaller line distance = 4/ 2.47 = 1.619 Exercise sheet 1 1

line ratio =

/

=

/

=

2

line ratio =

/

=

/

=

3

line ratio =

/

=

/

=

4

line ratio =

/

=

/

=

5

line ratio =

/

=

/

=

3. In exercise sheet 1, the ratio of blue line to the red line is approximately = 4. Do you agree that the human face follow (Yes / No) golden ratio proportions ?

44

Exercise sheet 1

1

5

2

3

4

45

You can also use Golden Caliper and the Golden rectangle for quick understanding of Golden ratio of human face in exercise sheet 2

Golden ratio Caliper

Golden rectangle

46

Exercise sheet 2

47

Fibonacci numbers exercise sheet 1. Write Fibonacci numbers starting from 0 to 89 (look at page no.30 if you need help)

2. How many number of petals are there in the

sunflower (exercise sheet 3)? Is that a Fibonacci number?

(Yes / No)

Here you can notice that the head of the sunflower has two series of seed curves, one winding in one direction and one in another. 3. In spiral 1, total number of seeds =

Is that a Fibonacci number?

(Yes / No)

4. In spiral 2, total number of seeds =

Is that a Fibonacci number?

(Yes / No)

5. Do you agree that the sunflower follows

Fibonacci numbers?

48

(Yes / No)

Exercise sheet 3

137.508

2 1

222.492

222.492 137.508

= 1.618...

Sunflower

49

Draw & Write your observations of Golden ratio in nature

Title

Title

Title

50

Draw & Write your observations of Fibonacci numbers in nature

Title

Title

Title

51

RESOURCES Books Elam Kimberly (2001). Gemoetry of Design, New York: Princeton Architectural Press Images p35: Nautilus Shell, p36: Flower, http://io9.com/5985588/15-uncanny-exam ples-of-the-golden-ratio-in-nature p28: Pinecones: http://www.3villagecsd.k12.ny.us/wmhs/ Departments/Math/OBrien/ďŹ bonacci2.html p15: The Parthenon in Athens by Steve Swayne, Wikipedia Web references http://illuminations.nctm.org/Lesson.aspx?id=2202 http://www.mensaforkids.org/lessons/Fibonacci Videos Youtube: Golden Ratio Fibonacci Sequence TEDxEast: Matthew Cross Youtube: Arthur Benjamin: The Magic of Fibonacci Numbers by TED Youtube: The Golden Ratio & Fibonacci Numbers: Fact versus Fiction, Professor Keith Devlin from Stanford

Set in Roboto (designed by Christian Robertson) and Free Serif (designed by PrimoĹž Peterlin, Steve White) Designed in Adobe Illustrator CC Document size: 5 X 8.09 inches, Inside pages Natural Buff 100 gsm, Cover: Cartridge 140 gsm

52