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GOLDEN RATIOS By: Arnav Ghosh 7.2

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Golden Ratio 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’

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Golden Ratios in our body The average human head has a length to breadth 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 top of head : length of head ratio of 1.70:1 These are only a few, there are many more. The human head is set in a golden rectangle

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My Face Nose bridge width (mm) Eye width (mm)Nose bridge/EyeNose bridge : Eye 354035/40 = 0.87535:40 = 1:1.40 Mouth (during smile) width (mm) Teeth (during smile) width (mm) Mouth / TeethMouth : Teeth 756075/60 = 1.2575:60 = 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.

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Friends faces NameBridge Width (mm) Eye width (mm) Bridge : Eye Mouth width (mm) Teeth width (mm) Mouth : Teeth Aditya331:1.00951.80:1 Jahnavi35401:1.1450601:1.20 Siddhant121:2.0021.51.33:1 Kaevaan11.51:1.503.521.75:1 Um-e-hani1.31.81:1.383.52.51.40:1 Jai.T1.41.51:1.072.521.25:1 Ahaan1.521:1.33321.50:1 Sankalp1.5 1:1.002.51.51.66:1 Nishna1.821:1.11321.50:1 Natasha1.521:1.333.52.51.40:1 Total--10:12.86--14.59:10.20 Average--1:1.29--1.43:1

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Nose Bridge Width : Eye Width of friend’s faces The values shown have been rounded off to the nearest tenth is to

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Mouth Width : Teeth Width of friend’s faces The values shown have been rounded off to the nearest tenth is to

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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’. 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. Comparisons Nose Bridge Width : Eye Width of friend’s faces Mouth Width : Teeth Width of friend’s faces

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Adult’s faces I chose my father for this part of the investigation Nose bridge width (mm)Eye width (mm)Nose bridge : Eye 204020:40 = 1:2 Mouth (during smile) width (mm) Teeth (during smile) width (mm) Mouth : Teeth 815281:52 = 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.

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Celebrity Faces NameBridge Width (mm) Eye Width (mm) Nose Bridge : Eye Width Mouth Width (mm) Teeth Width (mm) Mouth : Teeth Hrithik Roshan 4.571:1.56147.51.87:1 Katrina Kaif 47.51:1.8814.581.81:1 Amir Khan4.56.51:1.4412.581.56:1 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

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Nose bridge : Eye Width of Adults The values shown have been rounded off to the nearest tenth is to

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Mouth : Teeth of Adults is to The values shown have been rounded off to the nearest tenth

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Phi – Classical Greek Architecture Name of building(H)Height (mm)(W)Width (mm)Ratio Parthenon59114H : W = 1:1.93 Athenian Treasury8594H : W = 1:1.11 Erechtheion1929H : W = 1:1.53 Lion Gate8565H : W = 1.31:1 Simonas Petras Monastery (1 st building) 4534H : W = 1.32:1 Temple of Athena Nike4760H : W = 1:1.28 Treasury of Arteus (gate)11585H : W = 1.35:1 Temple of Hephaestus8451H : W = 1.68:1 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.

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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

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Classical Greek Buildings Simonas Petras Monastery (1 st Building) – H : W = 1.32:1 Temple of Athena Nike – H : W = 1:1.28 Treasury of Arteus – H : W = 1.35:1 Temple of Hephaestus – H : W = 1.68:1

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Fibonacci Numbers 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.

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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 A Sneezewort 853211853211 Leaves The Golden Ratio application – 4 level : 3 level = 5:3 =1.67:1 5 level : 4 level = 8:5 =1.60:1 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 (Branches)

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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. An ordinary plant 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. 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.

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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 = 5:3 =1.67:1 5 level : 4 level = 8:5 =1.60:1 6 level : 5 level = 13:8 =1.63:1 The tree does so not use Phi only to 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 2 nd level has 3 leaves, the 3 rd level has 5 leaves and the last level has 8 leaves.

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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 A Sunflower 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… The brown petals (shown as yellow)

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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. A Sunflower Petal 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.

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Phi - Nature A Flower 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 If we were to round this off we would get 1.6:1. This is what makes the flower look so beautiful. Cones A pinecone 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 -

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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 The Fibonacci numbers in the pineapple pineapple, we can make out Fibonacci numbers in it’s scales. Once put into ratio – 2nd picture scales : 1 st picture scales = 8:5 =1.6:1 3rd picture scales : 2 nd picture scales

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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.

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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.

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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.

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Reflection Environments: We learn more about the environment 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

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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.

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Bibliography http://goldennumber.net/face.htm http://www.faqs.org/photo- dict/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-ratio- fibonacci.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/hosted- sites/R.Knott/Fibonacci/pinecone3.gif http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

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Bibliography http://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Phi_ 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-lion- gate.jpg http://historylink101.net/greek_images/athena-nike.jpg http://upload.wikimedia.org/wikipedia/commons/5/51/Treasury_o f_Atreus_Entrance_Mycenae.jpg http://originallamby.files.wordpress.com/2009/10/sunflower.jpg http://www.usefilm.com/images/5/7/1/571/146186-medium.jpg

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