### The Golden Ratio

```Note: Of course, inner beauty is more important
Currently in Math, our unit is Ratio. We have
learnt about proportion, scaling an object to a
size that can be drawn, interpreting the size of
a diagram drawn to scale and the Golden
Ratio. This activity is to give us a deeper
understanding about the Golden Ratio by
allowing us to experiment with various
resources to collect and analyze information
places the Golden Ratio is found, as well as
how it affects the way someone looks. This
exciting activity will teach us that physical
beauty can be mathematical.
How can the study of the Golden Ratio aid us in
designing more aesthetically pleasing art and
architecture?
Community
and Service
Human Ingenuity
The reflections on the Unit Question and the Areas
of Interaction are at the end of the project.
Also called the ‘Golden Section’ or the ‘Golden
Proportion’, the Golden Ratio is approximately
equivalent to 1.618033988749895. This is
represented by the Greek letter φ (phi), like how
3.14… is represented by π (pi). The Golden Ratio
is irrational, meaning it goes on forever. Even
before the Renaissance, architects and artists
have scaled their designs using the Golden Ratio.
They used the Golden Rectangle the most often,
which they felt looked pleasing to the eye. The
way to calculate the Golden Ratio is 1+ 1+√5
,
2
which is half of 3.23606797749979. That is how
the Golden Ratio is calculated to be
1.618033988749895.
Phi is the 21st letter of the Greek alphabet. The
uppercase version of Phi is Φ, while the more
commonly used one is the lowercase, which is
φ. There are many mathematical uses for this
alphabet. A few of the things it represents are:
The Golden Ratio (1.618)
A ring or group homomorphism
The second angle after θ (Theta)
The polar angle from the z-axis
The Azimuthal angle from the x-axis
It was the idea of Mark Barr, to use the letter φ
for representing the golden ratio.
Golden
Rectangle
Golden
Triangle
Note: The Golden Rectangle and the Golden Triangle diagrams
have been generated by me using only auto shapes.
The Golden Rectangle is a rectangle in which the ratio of the longer side to the
shorter side is 1.6:1, which is the Golden Ratio. It can be created on the
computer very easily:
1.
Draw a rectangle
2.
Change the size to 1.6cm and 1 cm using the size options
3.
Click on the ‘Size and Position’ arrow
4.
Tick ‘Lock Aspect Ratio’
5.
To change the size, drag the rectangle by the four dots on the corners.
(Pictures on the next page)
The Golden Triangle is an isosceles triangle in which one of the two equal sides
(a) and the base (b) form the Golden Ratio when put in the ratio of a:b.
Below are pictures of a Golden Rectangle and a Golden Triangle
Size and
Position arrow
Dots to drag
and resize to
scale
Lock Aspect
Ratio
Note: All these diagrams have been generated by me using
only ‘Print Screen’, text boxes, circles and lines. All items
that have been ‘Print Screened’ have been created by me.
Size options
Fibonacci numbers are directly related to the Golden Ratio and φ (Phi).
When a Fibonacci number is in a ratio with the Fibonacci number before it,
and that ratio is simplified, you will end up with a number extremely close to
the Golden Ratio. For example,
•8:5 :: 1.6:1
•13:8 :: 1.625:1
•21:13 :: 1.6154:1
•34:21 :: 1.619:1
•55:34 :: 1.6176:1
But the Fibonacci number ratio closest to the Golden Ratio (Approximately
1.618) is:
___
89:55 :: 1.618:1
Fibonacci numbers are found in everyday life. Your body has one nose, two
eyes, three parts in the leg and five fingers on each hand. A pine cone has five
spirals facing one way and eight spirals facing the other way. A pineapple has
eight spirals going one way and thirteen facing the other way. These are
everyday examples of the Golden Ratio being present in nature.
A diagram showing the Fibonacci numbers. The area of
the square is the sum of the previous two square areas.
Fibonacci numbers are a sequence in which a number is equal to the sum of
the previous two numbers of the sequence. The first few Fibonacci numbers
are 1, 1, 2, 3, 5, 8,13, 21, 34, 55, 89...
A honeybee colony is made up of Drones, Queens and Workers. The
females (queens and workers) have a drone and a queen as parents.
However, a drone is hatched from unfertilised eggs, laid by a queen but
not fertilized by a male drone. Therefore, they only have one parent.
The family tree on the next page illustrates the fact that a drone has:
1 parent
2 grandparents
3 great-grandparents
5 great-great-grandparents
8 great-great-great-grandparents
13 great-great-great-great-grandparents
21 great-great-great-great-great-grandparents
34 great-great-great-great-great-great-grandparents
55 great-great-great-great-great-great-great-grandparents
89 great-great-great-great-great-great-great-great-grandparents
144 great-great-great-great-great-great-great-great-great-grandparents
Etc.
All these bold and underlined numbers are Fibonacci Numbers.
The family of a Drone happens to relate to the Golden Ratio!
Note: This entire diagram has been generated by me using only text boxes and lines
After researching about the Golden Ratio, I will
now start finding out about it in nature and the
human body. However, this time I won’t get the
figures off the Internet or books, but I will go out
and actually measure it. I will use a 30 cm ruler, as
I will be working only with plants and parts of the
body. The ruler will be convenient to travel around
with and it will not accidentally hit someone while
being transported, like a metre scale might. I will
work using centimetres, as they fit the ruler I will
be using. I am not measuring things that are too
big, so I will not need to use metres; centimetres
Lets begin the Investigation!
Ruler
Photographs
Pen/Pencil/Eraser
Laptop
Internet
PowerPoint
MS
Paint
Sunflower
Sea Shells
Photoshop
Note: All the measurements are in centimetres.
=19.5:13
(Top of head to pupil):(Pupil to edge of lip)
=10.5:6
(Nose tip to chin):(Lips to chin)
= 9:5
(Nose tip to chin):(Pupil to nose tip)
=9:5
(Width of nose at base):(Nose tip to lips)
=3.5:3
(Outside distance between eyes):(Hairline to pupil)
=5:5
Note: All the measurements are in centimetres.
(Length of Lips):(Width of nose at base)
=5:3
(Hairline to chin):(Hairline to bottom of nose)
=16:10
(Width of bridge of nose):(Width of eye)
=1.5:1
(Width of mouth):(Width of teeth)
=0.9:0.5
Bridge
Width
(mm)
Eye
Width
(mm)
Bridge:
Eye
Ratio
Mouth
Width
(mm)
Teeth
Width
(mm)
Mouth:
Teeth
Ratio
Arnav
1
1.5
1:1.5
3
2
1:0.67
Ajay N
1
1.5
1:1.5
2.5
0.3
1:0.12
Jahanvi
1
1.3
1:1.3
3.5
2.3
1:0.66
Karan
0.8
1
1:1.25
2.3
0.2
1:0.09
Akshat
1
1.5
1:1.5
2
0.1
1:0.05
Siddhant
0.5
1
1:2
1.5
1.5
1:1
Kaevaan
0.8
1.1
1:1.38
2
2
1:1
Ajay J
0.5
1.5
1:3
1.5
1.5
1:1
1.5
1.5
1:1
2
1.5
1:0.75
Nishna
1.8
1.6
1:0.89
2.8
1.5
1:054
Average
324
1:1.53
1:0.57
Note: All measurements are rounded off to the
nearest two decimal places.
Name
The average result in the previous page for the ‘Bridge:
Eye’ ratio shows that the Golden Ratio is found in
children as well, not just adults. This is because as we
grow, our body grows in proportion, and that ratio is set
from childhood. Compared to the adult, it is different
because the classmates are closer to the Golden Ratio
is closer for the ‘Mouth: Teeth’. This is because
nowadays, most people don’t like to show their teeth
while smiling, and so smile with their mouth mostly
closed. This way, only a small percentage of their teeth
is visible. In the pictures accessible to me, some of my
classmates were not even ready for the picture to be
taken, and so weren’t smiling the way they naturally
smile in school, and were smiling in a very artificial way.
So, their result got affected slightly.
Name
Bridge
Width
(cm)
Eye
Width
(cm)
Bridge:
Eye
Ratio
Mouth
Width
(cm)
Teeth
Width
(cm)
Mouth:
Teeth
Ratio
Hrithik
Roshan
0.85
0.64
1.33:1
1.2
1.2
1:1
Megan Fox
0.85
0.64
1.33:1
1.48
1.48
1:1
Jim Carrey
0.85
0.64
1.33:1
1.27
1.27
1:1
As seen in the above results, ALL the ratios of the actors are the same. All
the actors were measured at the same width, although the heights
differed. This was just for me to have a uniformed layout. There is a
possibility that many actors have the above ratios, and like the Golden
Ratio making someone look good, the above ratio is another proportion
that makes someone’s face look nice.
There is also a possibility that the Mouth: Teeth ratio is 1:1 because their
smile may be slightly artificial, and therefore, teeth is visible throughout
the smile.
Bridge
Width
(cm)
2.33
Eye
Width
(cm)
2.12
Bridge:
Eye
Ratio
1.1:1
Mouth
Width
(cm)
6.5
Teeth
Width
(cm)
5.5
Mouth:
Teeth
Ratio
1.2:1
The two ratios for an adult are not the exactly
same, but they are extremely close to each
other; there is just a 0.1:1 difference in them.
The Mouth: Teeth ratio is more than the Bridge:
Eye ratio by a small margin, so one could say
that the two ratios are similar. They are both not
very close to the Golden Ratio. The second ratio,
Mouth: Teeth, is quite close to that of the actors.
There is just a 0.1:1 difference in that as well.
The face of the adult compared to that of the
film stars is not very different, but not very
close either. For the Bridge: Eye ratio, the
actors are closer to the golden ratio than the
adult. This may make them look more
attractive. However, the adult is closer to the
golden ratio for the Mouth: Teeth ratio. This
may be because the adult’s smile is more
natural, and therefore their teeth are not
covered by their smile. The actor’s smiles may
be slightly unnatural, which may be why there
is no part of their inner-mouth shown apart
from their teeth.
The Parthenon in Acropolis, Athens was one of the first temples in
Ancient World to be made perfectly like the Golden Rectangle. It
was a temple dedicated to Athena, the goddess of knowledge.
Below is a picture of the Parthenon and how the different parts of
it fit perfectly into a Golden Rectangle. The grid that has been
fitted above the picture is like the grid seen in the slide about the
Fibonacci numbers; each box is equal to the sum of the previous
two. The whole rectangle is a Golden Rectangle, as are the smaller
rectangles within it.
One of the greatest artists who used the Golden Ratio for painting
human figures was Leonardo da Vinci. He presented the Golden
Ratio in his famous sketch, ‘The Vitruvian Man’ (Picture bottom
right). This sketch was drawn in 1492, and is named after a Roman
architect Vitruvius who agreed with Leonardo da Vinci about using
certain proportion of the human body (i.e. The golden ratio) in
architecture. It is also believed that the Golden Ratio has been
used in the Mona Lisa.
Salvador Dalí, a Spanish painter, used the Golden Rectangle as the
size of his canvases. A study conducted in 1999 proved that out of
the 350 artworks checked, over a 100 of them were using the
Golden Ratio. In Salvador’s work ‘The Sacrament of the Last
Supper’ (Picture bottom left), he drew a dodecahedron that had
sides proportional to each other in the Golden Ratio.
The temple of Artemis was a Greek temple which was
in Ephesus, Turkey around 550 BC. It was dedicated to
Artemis, the goddess of hunting and wild animals. She
is the twin sister of the sun, Apollo and the daughter of
Zeus and Leto. She is represented by bow and arrows,
as she is a Hunter. Her Roman equivalent is Diana.
The temple was one of the first to be made entirely of
marble and was one of the largest in its time. Built on
a platform measuring 131.1 x 78.9m, it was even larger
than the Parthenon.
Length: Width= 5.9:2.5 :: 2.36:1 (Red)
Length: Height = 5.9:4.45 :: 1.32:1 (Yellow)
Base : Triangle Length= 6.56:2.96:: 2.21:1 (Orange)
Length: Width= 7.2: 2.6 :: 2.7:1 (Red)
Length: Height= 7.2: 2.81 :: 2.56 (Yellow)
There are no ratios found that are close to the Golden Ratio. This
may be because the monument was built and designed a long time
ago, before the Greeks started to utilise the Golden Ratio in their
architecture.
In addition to being found in the body, pine-cones and
pineapples, the golden ratio can be found in various
other aspects of nature as well, such as the shell. A the
distance between the spirals of a shell follow the
Fibonacci numbers and the Golden Ratio. The diagram
at the bottom shows the distance between the areas of
the shell growing. This is easy to understand because of
blocks that are on the diagram. These boxes are like
those in the page about Fibonacci Numbers; each box’s
area is the area of the previous two boxes. This shows
that the shell grows in proportion to the Golden Ratio.
A very famous and known example of the Golden Ratio in
nature is a sunflower. Sunflowers have spirals winding in two
directions: clockwise and anticlockwise. But when they are
counted, they are not equal. Sometimes it is 144: 89, 89:55 or
55:34. One can clearly see that they are all Fibonacci
numbers. It is extremely interesting that these Fibonacci
numbers are in sequential order. As we know, when we divide
two consecutive Fibonacci numbers, we will get a number
close to the Golden Ratio. Here are the results rounded off to
three decimal places:
144:89 :: 1.618:1
89:55 :: 1.618:1
55:34 :: 1.618:1
All three of these ratios, when simplified, are extremely close
to the Golden Ratio when rounded off. The Golden Ratio is
also 1.618 when rounded off to three decimal places! This
shows that the sunflower is an example of the Golden Ratio in
nature.
As seen in the page about the Parthenon, the Golden Ratio
can be utilised to create more aesthetically pleasing buildings
and monuments for the community. Also having learnt about
the family of the Drone relating to Fibonacci numbers, we can
use exciting examples like that to interest students about
Fibonacci numbers. If you give a student the family diagram
of drone, and ask them to find out how many parents,
grandparents, great-grandparents, etc. the Drone has, then
they will explore and learn the Fibonacci number themselves.
This will teach them about it in an interactive way, which is a
way in which I learn very fast.
It is human intelligence that has discovered that the Golden
Ratio is aesthetically pleasing. This enables humans to use the
Golden Ratio to make nice architecture for everyone to
appreciate. Humans can also recognise the people who have
the Golden Ratio and are good looking. However, the bad
consequence is that some people might make fun of people
who do no have the Golden Ratio in their body.
We can use the Golden Ratio for geometrical
shapes, such as triangles, rectangles and other
polygons. This will make that shape look good, as
the shape will have the Golden Ratio. We can then
use these shapes in our architectural designs. The
windows, doors, height and width of the building,
roof, false ceiling heights, geometrical designs and
carvings. This will make the building look very
attractive and it will be appreciated more. For art,
the same can be drawn. The canvas size of the
artwork can follow the Golden Ratio and if a human
body is being drawn, it can be extremely accurate
using the Golden Ratio. This is why the Greek
carvings are said to be very accurate in size.
Fibonacci numbers, φ (Phi), where we find them in
everyday life and how we can improve art and
architecture with the knowledge of it. Since I was
allowed to research and make the project in my
and understand more about the topic. I knew what
the Fibonacci numbers were, and how they were
formed, but I didn’t know that they were called
Fibonacci numbers. I also utilized various modes of
research, such as actually testing whatever I could
to ensure that the information I was receiving was
accurate, such as the seashells or the sunflower.
Since I couldn’t actually go and measure the Greek
monuments, I cross checked the information in
various websites, to ensure quality.
http://www.intmath.com/Numbers/mathOfBeauty.php
[Accessed 5 April 2010
11:39am]
http://en.wikipedia.org/wiki/Golden_ratio
[Accessed 5 April 2010 11:54am]
http://www.insidegraphics.com/textures/images/photoshop_texture_contrast.jpg
[Accessed 5 April 2010 1:23pm]
http://en.wikipedia.org/wiki/Phi
[Accessed 5 April 2010 1:28pm]
http://www.theproblemsite.com/thegoldenratio/nature.asp
[Accessed 5 April
2010 2:38pm]
http://www.poweredtemplates.com/_src_clipart/00001/clipart_b.jpg
[Accessed 5
April 2010 3:59pm]
[Accessed 5 April 2010 4:07pm]
http://www.smccd.edu/accounts/jung/images/powerpoint-logo.jpg
[Accessed 5
April 2010 4:13pm]
http://www.sheeptech.com/images/2008/01/icon_mspaint.png
[Accessed 5 April
2010 4:16pm]
http://en.wikipedia.org/wiki/Fibonacci_number
[Accessed 6 April 2010 9:30am]
http://en.wikipedia.org/wiki/Golden_rectangle
[Accessed 6 April 2010 11:17am]
http://warrensburg.k12.mo.us/7wonders/artemis/diana2.jpeg
[Accessed 6 April
2010 12:16pm]
http://en.wikipedia.org/wiki/Temple_of_Artemis
[Accessed 6 April 2010 12:16pm]
http://en.wikipedia.org/wiki/Artemis
[Accessed 6 April 2010 12:20pm]
http://en.wikipedia.org/wiki/Golden_triangle_(mathematics) [Accessed 6
April 2010 2:33pm]
ew-gallery.jpg [Accessed 6 April 2010 2:48pm]
http://jancology.com/blog/archives/da-vinci-vitruvian-man.jpg [Accessed
6 April 2010 3:11pm]
3:18pm]
http://britton.disted.camosun.bc.ca/jblastsupper.jpg [Accessed 6 April
2010 3:26pm]
http://im.rediff.com/movies/2005/jan/10hrithik.jpg [Accessed 6 April
2010 3:31pm]
http://www.rankopedia.com/CandidatePix/2136.gif [Accessed 6 April 2010
3:37pm]