### Golden Ratios

```GOLDEN RATIOS
By: Arnav Ghosh 7.2
Golden Ratio
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It is also known as ‘Phi’.
It is an irrational mathematical constant and is
usually rounded of to 1.6:1
Scientists usually say that ratio was discovered by
Pythagoras or his followers.
Eulicid was the first person to record the definition
of ‘Phi’ in his textbook ‘Elements’.
The golden ratio, Phi or the divine
proportion was & is used by artists
, architects, scientists etc.
The symbols for ‘Phi’
Golden Ratios in our body
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ratio of 1.6:1. It is therefore set in a golden
rectangle.
The breadth of our front 2 teeth : the length of our
front 2 teeth = the golden ratio
The average human arm has an arm to finger tip :
forearm to fingertip ratio of 1.69:1.
The average body has an shoulder line to
These are only a few, there are many more.
The human head is set in a golden rectangle
My Face
Nose bridge width
(mm)
Eye width (mm)
Nose bridge/Eye
Nose bridge : Eye
35
40
35/40 = 0.875
35:40 = 1:1.40
Mouth (during
smile) width (mm)
Teeth (during
smile) width (mm)
Mouth / Teeth
Mouth : Teeth
75
60
75/60 = 1.25
75:60 = 1.25:1
The ratios that I have found are quite close. The difference could be because of the
fact that my face is a little elongated. Due to this my smile appears to be bigger and
this has therefore cause the measurement to be different. The ratios are also pretty
close to the Golden Ratio.
Friends faces
Name
Bridge
Width
(mm)
Eye width
(mm)
Bridge :
Eye
Mouth
width
(mm)
Teeth
width
(mm)
Mouth :
Teeth
3
3
1:1.00
9
5
1.80:1
Jahnavi
35
40
1:1.14
50
60
1:1.20
Siddhant
1
2
1:2.00
2
1.5
1.33:1
Kaevaan
1
1.5
1:1.50
3.5
2
1.75:1
Um-e-hani
1.3
1.8
1:1.38
3.5
2.5
1.40:1
Jai.T
1.4
1.5
1:1.07
2.5
2
1.25:1
Ahaan
1.5
2
1:1.33
3
2
1.50:1
Sankalp
1.5
1.5
1:1.00
2.5
1.5
1.66:1
Nishna
1.8
2
1:1.11
3
2
1.50:1
Natasha
1.5
2
1:1.33
3.5
2.5
1.40:1
Total
-
-
10:12.86
-
-
14.59:10.20
Average
-
-
1:1.29
-
-
1.43:1
Nose Bridge Width : Eye Width of friend’s faces
2.5
2
Value
1.5
Nose Bridge Width
is to
Eye width
1
0.5
0
Names
The values shown have been
rounded off to the nearest tenth
Mouth Width : Teeth Width of friend’s faces
2
1.8
1.6
Value
1.4
1.2
1
0.8
Mouth width
is to
Teeth Width
0.6
0.4
0.2
0
Names
The values shown have been
rounded off to the nearest tenth
Comparisons
Nose Bridge Width : Eye Width of friend’s faces
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Sankhalp and Aditya have the same ratio here.
Jahnavi, Nishna and Jai.T have the same ratio here.
Natasha and Ahaan have the same ratio here.
The ratios weren’t very different from each other, other than
that of Siddhants’ and the others’.
Mouth Width : Teeth Width of friend’s faces
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Everybody’s mouth is wider than their teeth when they smile
other than Jahnavi.
Natasha and Um – e – hani have the same ratio here.
Siddhant, Ahaan and Nishna have the same ratio here.
Aditya and Kaevaan have the same ratio here.
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I chose my father for this part of the investigation
Nose bridge width (mm)
Eye width (mm)
Nose bridge : Eye
20
40
20:40 = 1:2
Mouth (during smile) width
(mm)
Teeth (during smile) width
(mm)
Mouth : Teeth
81
52
81:52 = 1.56:1
The ratios aren’t that close. This is because my father has a very thin nose and a big
smile causing the difference.
The ratios are similar to those of children’s faces.
Celebrity Faces
Name
Bridge
Width
(mm)
Eye Width
(mm)
Nose
Bridge :
Eye Width
Mouth
Width
(mm)
Teeth
Width
(mm)
Mouth :
Teeth
Hrithik
Roshan
4.5
7
1:1.56
14
7.5
1.87:1
Katrina
Kaif
4
7.5
1:1.88
14.5
8
1.81:1
Amir Khan
4.5
6.5
1:1.44
12.5
8
1.56:1
These ratios aren’t all that different from the face that I have used for my adult
comparison. In fact, Amir Khan’s Mouth : Teeth is the same as my father’s. The Ratios of
all these celebrities are very close to the Phi.
From Left to Right: Hrithik Roshan,
Amir Khan and Katrina Kaif
Nose bridge : Eye Width of Adults
2.5
2
Value
1.5
Nose Bridge Width
is to
Eye Width
1
0.5
0
Hrithik Roshan Katrina Kaif Amir Khan
Name
Father
The values shown have been
rounded off to the nearest tenth
2
1.8
1.6
1.4
1.2
1
Mouth Width
is to
Teeth Width
0.8
0.6
0.4
0.2
0
Hrithik Roshan
Katrina Kaif
Amir Khan
Father
The values shown have been
rounded off to the nearest tenth
Phi – Classical Greek Architecture
After the discovery of ‘Phi’, Greek architects began building buildings that used the
golden ratio in one way or the other. Mostly it was included to make the building look
more attractive but it was also done for other reasons, it was done because then the
pillars could hold the weight of the things above.
Name of building
(H)Height (mm)
(W)Width (mm)
Ratio
Parthenon
59
114
H : W = 1:1.93
Athenian Treasury
85
94
H : W = 1:1.11
Erechtheion
19
29
H : W = 1:1.53
Lion Gate
85
65
H : W = 1.31:1
Simonas Petras Monastery (1st
building)
45
34
H : W = 1.32:1
Temple of Athena Nike
47
60
H : W = 1:1.28
Treasury of Arteus (gate)
115
85
H : W = 1.35:1
Temple of Hephaestus
84
51
H : W = 1.68:1
Classical Greek Buildings
The Parthenon - Height : Width
= 1:1.93
Athenian Treasury Height : Width
= 1:1.11
Erechtheion - Height :
Width = 1:1.53
Lion Gate - Height :
Width = 1.31:1
Classical Greek Buildings
Simonas Petras Monastery (1st
Building) – H : W = 1.32:1
Temple of Athena
Nike – H : W
= 1:1.28
Temple of Hephaestus – H : W
= 1.68:1
Treasury of Arteus –
H : W = 1.35:1
Fibonacci Numbers
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Fibonacci numbers are a sequence of numbers that
follow this pattern: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,
55,89, 144…
Each number is the sum of the previous 2 numbers
The Fibonacci numbers were first by Leonardo
Fibonacci.
After 0,1,1,2 – if the following number is divided
by the number before it, the answer comes out to be
1.6… . This shows the relationship between the
Fibonacci numbers and Phi.
Phi - Nature
I have chosen the relationship between trees, cones,
flowers, fruits and Phi for my case study.
 Nature doesn’t always use Phi for looking nice, it
has other reasons as well…
Trees & Plants
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A Sneezewort
The Golden Ratio application –
8 4 level : 3 level
5 = 5:3
3 =1.67:1
2 5 level : 4 level (Branches)
1 = 8:5
1 =1.60:1
Leaves 6 level : 5 level
= 13:8
=1.63:1
Leafs
5 level : 4 level
= 5:3
=1.67:1
6 level : 5 level
= 8:5
=1.60:1
Phi - Nature
The sneezewort uses the Fibonacci numbers and we get Phi every time. As each branch
progresses the sneezewort looks better. But the sneezewort also uses Phi to save space. If
it didn’t use Fibonacci numbers the flowers on top would either be too cramped or too
spaced out.
If we were to start at the first leaf of the plant and name it ‘0’,
the leaf that will come in the same line will be leaf no.8. If we
were to draw invisible lines to join each leave, we would get 5
revolutions.
An ordinary plant
The number of leaves : The number of revolutions
= 8:5
=1.60:1
Therefore each 1.6 of a leaf there is a full revolution. This makes
the plant look very nice but the plant also uses this pattern to
make sure all the leaves get sunlight.
Phi - Nature
An ordinary tree
When a tree starts growing, only one big
branch or trunk grows. On each level the
number of branches is a Fibonacci number.
4 level : 3 level
6 level : 5 level
= 5:3
= 13:8
=1.67:1
=1.63:1
5 level : 4 level
= 8:5
The tree does so not use Phi only to
=1.60:1
look good but also to make sure
that the branches go high enough
but still use very little space.
A Spiral Leaf Plant
This plant also uses Fibonacci numbers. This plant can be
segregated into levels. From top to bottom: the 1st level has
2 leaves, the 2nd level has 3 leaves, the 3rd level has 5
leaves and the last level has 8 leaves.
Phi - Nature
If we were to look for the golden ratio –
4 level : 3 level
= 8:5
=1.60:1
The plant looks really good but it also uses
Phi to make sure that none of the new
leaves cover the old ones, thus making sure
that each leaf gets enough sunlight. This
configuration also makes sure that water
goes to the roots easily.
Flowers
This is the humble sunflower. A sunflower is a compound
flower or a flower with 2 types of petal: the brown and
yellow. The sunflower is probably one of the best looking
plants. This is because it uses Phi in many more ways than
one. First of all it includes this in the brown petals.
They are arranged in a spiral way. If we
were to count the amount pointing to the
left we would get 21 and the amount
pointing to the right we would get 34…
A Sunflower
The brown petals (shown as yellow)
Phi - Nature
Once we put them in a ratio –
Number pointing to the left : Number pointing to the right
= 34:21
=1.62:1
For every 1.62 of a spiral pointing to the left, there is one spiral pointing to the right. This is
what makes the center of a sunflower look so beautiful.
The next place where we see something close to the golden ratio is in
sunflower petals. The length of the petal on the right is 39mm and it’s width
at the center is 21mm. Once put into ratio –
Length of Petal : Width of Petal
= 39:21
=1.86:1
This number is quite close to Phi and this is what makes sunflower petals look
so good.
A Sunflower Petal
Phi - Nature
Many ordinary flowers also use Phi. For example: if we were to
measure the diameter of the inner petals we would get 8.5mm
and the length of the petals we would get 13.5mm. Once put
into ratio –
Length of Petals : Diameter of inner petals
= 13.5:8.5
=1.59:1
A Flower
Cones
If we were to round this off we would get 1.6:1. This is what
makes the flower look so beautiful.
This is a pinecone. Each ‘fruit’ holds a seed. If we
were to count the number of spirals pointing to the
left we would get 8 and if we were to count the
number of spirals pointing to the right 13. Once
put into ratio A pinecone
Phi - Nature
Number of spirals pointing to the right : Number of spirals pointing to the left
= 13:8
= 1.63:1
For each 1.63 petal pointing to the right there is 1 petal pointing to the left. This makes
the pinecone look very beautiful and it also allows many ‘fruits’ to be stored in a small
space.
Fruits
The pineapple is a very good example of Phi used in fruits. If we look careful at the
pineapple, we can make out Fibonacci
numbers in it’s scales. Once put into ratio –
2nd picture scales : 1st picture scales
= 8:5
=1.6:1
The Fibonacci numbers in the pineapple
3rd picture scales : 2nd picture scales
Phi - Nature
= 13:8
=1.63:1
For every 1.6 of scale going from the left top of the pineapple to the right hand side
bottom of the pineapple, there is 1 scale going horizontally through the pineapple.
For every 1.63 scale going from the right top of the pineapple to the left hand side
bottom of the pineapple, there is 1 scale going from the left top of the pineapple to the
right hand side bottom of the pineapple.
Conclusion
In conclusion, we can see that the golden ratio or Phi is almost like a law of nature. It is
used more for it’s own conveniences than for looking good. It is found in any everyday
object that we pass by. We think they look beautiful but we never find out why. We pluck
ordinary leaves without thinking about the balance we are disrupting. We should learn to
be more careful and should learn to be inquisitive to understand better the world around
us.
Reflection
Through this assessment task I have learnt many things in
many different fields. In math I have learnt more about
the golden ratio and a lot about Fibonacci numbers. I
have also acquired more knowledge in the field of
science. I have learnt many knew things about the items
in nature that I researched about. I have also got more
knowledge in the field of history, I have learnt more
about how the Greek architects used the golden ratio in
one way or another in their buildings.
I used a variety of sources to find the information I
needed: the internet, math text book and worksheets,
Ms. Neha’s explanations, Britannica encyclopedia,
friends, pictures and nature itself.
Reflection
These were very useful sources. The internet and Britannica
encyclopedia gave me the content I needed. The maths textbook
gave an introduction into the topic. Worksheets helped me
understand the concept even better. Ms. Neha’s explanations
helped clear any doubts I had. My friends and their pictures
gave me the measurements I needed. Nature, in a sense gave me
the resources I needed for example flowers.
This unit is connected to the following AOIs:
Health and Social Education: We think how to judge between
beautiful faces and learn that after all, we should only judge the
inner beauty. We learn how this affects our relationships.
Reflection
around us, how it functions, what it takes to do so and
it’s beauties.
The following learner profiles are related to this unit:
Thinkers: We think and calculate to find ratios. We think
of which part of nature would have more of Phi. We
make our own judgments, while comparing faces.
Communicators: We communicate the information we
have using mathematical language and chart, graphs
and tables
Reflection
Inquirers: We inquire on many topics and research on them.
Balanced: We try to balance between information and
diagrams or pictures.
Knowledgeable: We use our previous knowledge on the
topic.
This assignment is linked to every day life because we see
golden ratios very often and we should learn to recognize
them and understand why we think things are so beautiful.
Next time, I will try to include more parts of nature, give
better justifications and use more tables. I will also use
better time management.
Bibliography
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http://goldennumber.net/face.htm
http://www.faqs.org/photodict/photofiles/list/4386/5832grape_leaf.jpg
http://science.howstuffworks.com/evolution/fibonacci-nature1.htm
http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm
http://en.wikipedia.org/wiki/Fibonacci_number
http://www.intmath.com/Numbers/mathOfBeauty.php
http://www.mathsisfun.com/numbers/nature-golden-ratiofibonacci.html
http://www.missouriplants.com/Yellowopp/Helianthus_divaricatus_fl
owers.jpg
http://www.mathsisfun.com/numbers/images/succulent-top.jpg
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/pinecone3.gif
http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
Bibliography
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uc_lc.svg/800px-Phi_uc_lc.svg.png
http://nazmath.net/Online_Classes/HTML2/Wk2/parthenon.jpg
http://images.travelpod.com/users/texaspeyton/1.1248284989.at
henian-treasury.jpg
http://www.macfarlane-web.com/images/ErechtheionMaidens.jpg
http://www.shunya.net/Pictures/Greece/Mycenae/Mycenae-liongate.jpg