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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 about it. We will ourselves learn about the 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. Note: Of course, inner beauty is more important

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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+, which is half of 3.23606797749979. That is how the Golden Ratio is calculated to be 1.618033988749895. 1+√5 2

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Phi is the 21 st 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.

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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 Golden Rectangle Note: The Golden Rectangle and the Golden Triangle diagramshave been generated by me using only auto shapes. Golden Triangle

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Size options Size and Position arrow Lock Aspect Ratio Dots to drag and resize to scale Note: All these diagrams have been generated by me usingonly ‘Print Screen’, text boxes, circles and lines. All itemsthat have been ‘Print Screened’ have been created by me.

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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... 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, 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. 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 A diagram showing the Fibonacci numbers. The area ofthe square is the sum of the previous two square areas.

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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!

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Note: This entire diagram has been generated by me using only text boxes and lines

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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 will be adequate. Lets begin the Investigation!

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Ruler Photographs Pen/Pencil/Eraser Laptop Internet PowerPoint MS Paint Sunflower Sea Shells Photoshop

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Note: All the measurements are in centimetres.

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NameBridge Width (mm) Eye Width (mm) Bridge: Eye Ratio Mouth Width (mm) Teeth Width (mm) Mouth: Teeth Ratio Arnav11.51:1.5321:0.67 Ajay N11.51:1.52.50.31:0.12 Jahanvi11.31:1.33.52.31:0.66 Karan0.811:1.252.30.21:0.09 Akshat11.51:1.520.11:0.05 Siddhant0.511:21.5 1:1 Kaevaan0.81.11:1.38221:1 Ajay J0.51.51:31.5 1:1 Prasad1.5 1:121.51:0.75 Nishna1.81.61:0.892.81.51:054 Average3241:1.531:0.57 Note: All measurements are rounded off to thenearest two decimal places.

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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 for the ‘Bridge: Eye’ ratio than the adult, but the adult 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.

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NameBridge Width (cm) Eye Width (cm) Bridge: Eye Ratio Mouth Width (cm) Teeth Width (cm) Mouth: Teeth Ratio Hrithik Roshan 0.850.641.33:11.2 1:1 Megan Fox0.850.641.33:11.48 1:1 Jim Carrey0.850.641.33:11.27 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.

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Bridge Width (cm) Eye Width (cm) Bridge: Eye Ratio Mouth Width (cm) Teeth Width (cm) Mouth: Teeth Ratio 2.332.121.1:16.55.51.2:1 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.

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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.

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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.

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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.

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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.

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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.

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I have learnt a whole load about the Golden Ratio, 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 own speed, I was able to grasp more information 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.

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http://www.intmath.com/Numbers/mathOfBeauty.php [Accessed 5 April 2010 11:39am] http://www.intmath.com/Numbers/mathOfBeauty.php http://en.wikipedia.org/wiki/Golden_ratio [Accessed 5 April 2010 11:54am] http://en.wikipedia.org/wiki/Golden_ratio http://www.insidegraphics.com/textures/images/photoshop_texture_contrast.jpg [Accessed 5 April 2010 1:23pm] http://www.insidegraphics.com/textures/images/photoshop_texture_contrast.jpg http://en.wikipedia.org/wiki/Phi [Accessed 5 April 2010 1:28pm] http://en.wikipedia.org/wiki/Phi http://www.theproblemsite.com/thegoldenratio/nature.asp [Accessed 5 April 2010 2:38pm] http://www.theproblemsite.com/thegoldenratio/nature.asp http://www.poweredtemplates.com/_src_clipart/00001/clipart_b.jpg [Accessed 5 April 2010 3:59pm] http://www.poweredtemplates.com/_src_clipart/00001/clipart_b.jpg http://truthdive.com/wp-content/uploads/2009/06/internet_explorer_logo.jpg [Accessed 5 April 2010 4:07pm] http://truthdive.com/wp-content/uploads/2009/06/internet_explorer_logo.jpg http://www.smccd.edu/accounts/jung/images/powerpoint-logo.jpg [Accessed 5 April 2010 4:13pm] http://www.smccd.edu/accounts/jung/images/powerpoint-logo.jpg http://www.sheeptech.com/images/2008/01/icon_mspaint.png [Accessed 5 April 2010 4:16pm] http://www.sheeptech.com/images/2008/01/icon_mspaint.png http://en.wikipedia.org/wiki/Fibonacci_number [Accessed 6 April 2010 9:30am] http://en.wikipedia.org/wiki/Fibonacci_number http://en.wikipedia.org/wiki/Golden_rectangle [Accessed 6 April 2010 11:17am] http://en.wikipedia.org/wiki/Golden_rectangle http://warrensburg.k12.mo.us/7wonders/artemis/diana2.jpeg [Accessed 6 April 2010 12:16pm] http://warrensburg.k12.mo.us/7wonders/artemis/diana2.jpeg http://en.wikipedia.org/wiki/Temple_of_Artemis [Accessed 6 April 2010 12:16pm] http://en.wikipedia.org/wiki/Temple_of_Artemis

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http://en.wikipedia.org/wiki/Artemis [Accessed 6 April 2010 12:20pm] http://en.wikipedia.org/wiki/Artemis http://en.wikipedia.org/wiki/Golden_triangle_(mathematics) [Accessed 6 April 2010 2:33pm] http://en.wikipedia.org/wiki/Golden_triangle_(mathematics) http://www.frogview.com/uploadimages4/49551a642e5dc5.07604923frogvi ew-gallery.jpg [Accessed 6 April 2010 2:48pm] http://www.frogview.com/uploadimages4/49551a642e5dc5.07604923frogvi ew-gallery.jpg http://jancology.com/blog/archives/da-vinci-vitruvian-man.jpg [Accessed 6 April 2010 3:11pm] http://jancology.com/blog/archives/da-vinci-vitruvian-man.jpg http://en.wikipedia.org/wiki/Salvador_Dal%C3%AD [Accessed 6 April 2010 3:18pm] http://en.wikipedia.org/wiki/Salvador_Dal%C3%AD http://britton.disted.camosun.bc.ca/jblastsupper.jpg [Accessed 6 April 2010 3:26pm] http://britton.disted.camosun.bc.ca/jblastsupper.jpg http://im.rediff.com/movies/2005/jan/10hrithik.jpg [Accessed 6 April 2010 3:31pm] http://im.rediff.com/movies/2005/jan/10hrithik.jpg http://www.rankopedia.com/CandidatePix/2136.gif [Accessed 6 April 2010 3:37pm] http://www.rankopedia.com/CandidatePix/2136.gif http://www.celebritysmackblog.com/wp- content/uploads/2009/08/megan-fox-catwoman.jpg [Accessed 6 April 2010 3:54pm] http://www.celebritysmackblog.com/wp- content/uploads/2009/08/megan-fox-catwoman.jpg http://www.fotosearch.com/bthumb/CSP/CSP164/k1646052.jpg [Accessed 7 April 2010 2:43pm] http://www.fotosearch.com/bthumb/CSP/CSP164/k1646052.jpg http://www.world-mysteries.com/sci_17.htm [Accessed 7 April 2010 4:35pm] http://www.world-mysteries.com/sci_17.htm

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The Golden Ratio What is it? Also known as Phi (rhymes with fly)

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