Golden Ratio(1:1.6)

By Sayesha Saigal Grade 7.2

Subject= Mathematics

What is the Golden Ratio?• The golden ratio, also known as divine proportion,

golden mean, or golden section is a number often encountered when taking the ratios of simple geometrical figures such as pentagons, pentagram, decagon and dodecahedron.

• It is often called phi. Its symbol is• Leonardo da Vinci's drawings of the human body

emphasised its proportion. The ratio of the following distances is the Golden Ratio:

• (foot to navel) : (navel to head) Similarly, buildings are more attractive if the proportions used follow the Golden Ratio.

• Scientists believe that we perceive proportional bodies to be more healthy.

Examples of dodecahedrons

Interesting Fact

• Adolf Zeising, who was a qualified mathematician and philosopher, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves.

• He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavours.

Sayesha (Me)

• Bridge width = 12 mm• Eye width = 19 mmBridge/Eye ratio= 1.58• Mouth width = 27 mm• Teeth width = 20 mmMouth/Eye ratio= 1.6

Ratio for 10

Sr. No Name

Bridge width (MM)

Eye width(MM)

Bridge / Eye Ratio

Mouth width (MM)

Teeth width (MM)

Mouth/ Teeth ratio

1 Prasad 20 40 0.5 70 50 1.42 Nana 25 27.5 0.909090909 45 25 1.83 Mummy 35 21 1.666666667 65 40 1.625

4 Parshva 20 30 0.666666667 50 301.666666

6675 Rio 10 10 1 17.5 12.5 1.4

6 Avinash 10 10 1 20 151.333333

333

7 Sachin 10 8 1.25 20 131.538461

538

8 Saira 10 10 1 21 171.235294

118

9 Tehsin 7 8 0.875 13 111.181818

182

10 Barkha 10 9 1.111111111 17 151.133333

333

Ratio for 10

00.20.40.60.8

11.21.41.61.8

2

Bridge / Eye Ratio Mouth/Teeth ratio

Prasad

Nana

Mummy

Parshva

Rio

Avinash

Sachin

Saira

Tehsin

Barkha

Total and Averages for ten• Bridge/Eye Ratio for ten: 1) Total = 9.852) Average = 0.99

• Mouth/Teeth Ratio for ten:1) Total = 14.252) Average = 1.43

Shaheen (Mummy)

• Bridge Width= 35 mm• Eye Width = 21 mmBridge/Eye ratio=

1.666666667 • Mouth Width = 65 mm• Teeth Width = 40 mmMouth/Eye ratio=1.625

Hrithik Roshan

Bridge width = 30 mm• Eye width = 20 mmBridge/Eye ratio = 1:5• Mouth width = 45 mm• Teeth width = 37 mmMouth/Teeth ratio= 1:2

Kareena Kapoor

• Bridge width = 30 mm• Eye width = 20 mmBridge/Eye ratio = 1:5• Mouth width = 65 mm• Teeth width = 47 mmMouth/Teeth ratio=1:4

Aishwarya Rai

• Bridge width = 43 mm• Eye width = 42 mmBridge/Eye ratio = 1:0• Mouth width = 80 mm• Teeth width = 73 mmMouth/Teeth ratio= 1:0

Celebrities vs. Ordinary

00.20.40.60.8

11.21.41.61.8

2

Prasad Nana Mummy HrithikRoshan

KareenaKapoor

AishwaryaRai

Bridge / Eye Ratio

Mouth/Teeth ratio

Poem by Sayesha ‘Perfection’Da Vinci held a brush

When he felt that adrenalin rushFor he knew equal proportion

Wasn’t just a casual notionThe beauty of a face or a building is its grace

Its absolute symmetryThe principle of geometry

You may not be a starBut it’s a different league by far

It’s a beauty with a ratio1:1.6 is that bar.

It could be man or marbleIt may be Stone or girl

If you’re in the golden ratioYou’re as perfect as a pearl !

Comparing…

• According to my information, both ordinary faces and celebrity faces have similar ratios. None of the chosen celebrities fit into the golden ratio mask, while my mum does fit into the ratio (1:1.6) although she is ordinary.

• In my perspective the inside beauty counts more than the face, but it is always an advantage to have mathematical beauty also.

Parthenon, Acropolis, Athens

• Height = 42mm• Width = 25mmRatio of height/width = 1.68

Mathematicians had the contribution of the Greeks in mind when they christened the ratio "phi" in tribute to the great Phidias, who used the proportion frequently in his sculpture.

Porch of Maidens, Acropolis, Athens

• Height = 118 mm• Width = 70 mm Height/width ratio = 1.68 This Greek structure is

also in the golden ratio. We can see how focused these builders were about having each part in equal proportions.

The Egyptians also If we take a cross section

of the Great Pyramid, we get a right triangle, called the Egyptian Triangle. The ratio of the slant height of the pyramid (hypotenuse of the triangle) to the distance from ground centre is 1.61804. This differs from phi by only one unit in the fifth decimal place.

Reflection I have enjoyed this activity. I have now thoroughly understood the different

face proportions that people have and how it can be related to the golden ratio.

I have also learnt how to create my own charts and tables using various people’s measurements.

After measuring ten photographs as also the celebrities, I realized that there isn’t much difference between the ordinary and famous.

I have learnt how to communicate well, be focused and committed to my work.

The AOI in this case is Approaches to Learning since we learnt about the golden ratio and how it can be applied to real life situations and Community and Service since we measured and calculated the faces of our friends and family. I am confident about my work and I know I have done quite well.

Although the one thing that I could have improved was the power point presentation was that instead I could have extended my research on nature but since I had technical difficulties, I was only able to find this much.

I want everyone to enjoy viewing my work and I hope they will be informed about the different faces, calculations and new aspects of the work.

Bibliography • Information on what is golden ratio and examples of dodecahedrons

available from:http://mathworld.wolfram.com/GoldenRatio.html http://www.jimmyr.com/blog/Mask_of_the_Golden_Ratio_203_2007.php

[Accessed 2 April 2010]• Facts on Golden ratio available from: http://en.wikipedia.org/wiki/Golden_ratio#Nature [Accessed 13 April 2010]• Hrithik Roshan’s image available from:http://photogallery.indiatimes.com/celebs/indian-stars/hrithik-roshan/smiling-hrithik/Smiling-Hrithik/photo/3930950/Smiling-Hrithik.jpg

[Accessed 13 April 2010]

Bibliography • Kareena Kapoor’s image available from:http://images.paraorkut.com/img/wallpapers/1024x768/k/kareena_kapoor_smiling-1560_thumb.jpg

[Accessed 13 April 2010]• Aishwarya Rai’s image available from:http://desi-galaxy.sabza.org/wp-content/aishwarya-smiling.jpg

[Accessed 13 April 2010]• Image on Greek buildings available from:http://static.howstuffworks.com/gif/parthenon-and-the-acropolis-landmark-1.jpg

[Accessed 17 April 2010]

Bibliography

• Information on Greek and Egyptian structures available from:

http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm

[Accessed 17 April 2010]