Golden Ratio (1:1.6)

1. Golden Ratio(1:1.6) By Sayesha Saigal Grade 7.2 Subject= Mathematics

2. 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.

3. Examples of dodecahedrons

4. 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.

5. Bridge width = 12 mm Eye width = 19 mm Bridge/Eye ratio= 1.58 Mouth width = 27 mm Teeth width = 20 mm Mouth/Eye ratio= 1.6 Sayesha (Me)

8. Total and Averages for ten • Bridge/Eye Ratio for ten: • Total = 9.85 • Average = 0.99 • Mouth/Teeth Ratio for ten: • Total = 14.25 • Average = 1.43

9. Bridge Width= 35 mm Eye Width = 21 mm Bridge/Eye ratio= 1.666666667 Mouth Width = 65 mm Teeth Width = 40 mm Mouth/Eye ratio=1.625 Shaheen (Mummy)

10. Bridge width = 30 mm Eye width = 20 mm Bridge/Eye ratio = 1:5 Mouth width = 45 mm Teeth width = 37 mm Mouth/Teeth ratio= 1:2 Hrithik Roshan

11. Bridge width = 30 mm Eye width = 20 mm Bridge/Eye ratio = 1:5 Mouth width = 65 mm Teeth width = 47 mm Mouth/Teeth ratio=1:4 Kareena Kapoor

12. Bridge width = 43 mm Eye width = 42 mm Bridge/Eye ratio = 1:0 Mouth width = 80 mm Teeth width = 73 mm Mouth/Teeth ratio= 1:0 Aishwarya Rai

15. Poem by Sayesha ‘Perfection’ Da Vinci held a brush When he felt that adrenalin rush For he knew equal proportion Wasn’t just a casual notion The beauty of a face or a building is its grace Its absolute symmetry The principle of geometry You may not be a star But it’s a different league by far It’s a beauty with a ratio 1:1.6 is that bar. It could be man or marble It may be Stone or girl If you’re in the golden ratio You’re as perfect as a pearl !

16. 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.

17. Height = 42mm Width = 25mm Ratio 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. Parthenon, Acropolis, Athens

18. 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. Porch of Maidens, Acropolis, Athens

19. 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. The Egyptians also

20. 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.

21. 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]

22. 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]

23. Bibliography • Information on Greek and Egyptian structures available from: http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm [Accessed 17 April 2010]