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Access International Academy NingboRESEARCH PAPERSUBJECT: MathematicsRESEARCH QUESTION : Does the Golden Ratio influence us?NAME: Steven Hsu INSTRUCTOR: Ms. JennieDATE: January 2014WORD COUNT: 3930Abstract A Golden Ratio, or Phi (φ), is a set of numbers that can extend for many pages. Usually it is defined as, or rounded up to, 1.618. This ratio existed in many forms and in many places and fields. It existed in history, in art, in nature and even in our lives. The Golden Ratio, however, is often being recognized as just a ratio. Most people think it is just an ordinary ratio, likes other ratios in mathematics, and do not realize the wide existence of the Golden Ratio.If the Golden Ratio is widely displaced, then does the Golden Ratio have certain effects on us? We, as humans, do not recognize the Golden Ratio consciously, however, we do recognize it unconsciously. If we observed with care then an interesting phenomenon can be discovered. The search of the Golden Ratio will take place in history, in nature and in daily life. Furthermore, there will be precise examples and concise analysis in each field. Different forms of Golden Ratio is discovered and used in a way that is not extremely concealed. In other words, we are constantly and unconsciously expose to the Golden Ratio. Just a side note, the title and the context of this page is in the Golden Ratio. The appearance of the Golden Ratio in all we see, experience and create has unconsciously establishes a sense of harmony, balance, and beauty in our life and nature.Word count: 238Table of ContentsIntroduction 1Golden Ratio and Fibonacci Numbers 2Golden Ratio in History 4Golden Ratio in Nature 9Golden Ratio in Daily Life 14Conclusion 18Bibliography 19 1. Introduction44577003878580Figure SEQ Figure \* ARABIC 1.1Figure SEQ Figure \* ARABIC 1.1A Golden Ratio, or Phi (φ), is a set of numbers that can extend for many pages. Usually it is defined as, or rounded up to, 1.618. With this unbounded irrational number, there is a question that develops. “Does the Golden Ratio influence us?” This question is worth studying because usually people think that the Golden Ratio is just a ratio, however, some other people think it exists in our daily lives and constantly influences us. To determine the impact of the Golden Ratio, I will first prove the existence of Golden Ratio in the field around us. I will find its existence in art, in architectures, in nature, and in our society. After the existence is proven, then the importance of impact can be easily concluded. If we are living in a world where people are constantly exposed to the Golden Ratio, then we are likely to use the Golden Ratio unconsciously in our daily life. “The CN Tower in Toronto…has [incorporated] the golden ratio in its design. The ratio of observation deck at 342 meters to the total height of 553.33 is 0.618 or phi” (Owen). See figure 1.1 for the actual picture for the CN Tower. 2. Golden Ratio and Fibonacci Numbers2.1. Golden Ratio:The Golden Ratio also known as Golden Proportion, Golden Selection, Golden Mean and Divine Proportion. The Golden Ratio, represented as Phi (φ), is a ratio that is round up to 1.618. Phi is a ratio that continues forever and without repeating; which called irrational number. This ratio can be found through different ways. 45434251751965Figure 2.1. SEQ Figure_2.1. \* ARABIC 1Figure 2.1. SEQ Figure_2.1. \* ARABIC 12152650142240Figure 2.1. SEQ Figure_2.1. \* ARABIC 2Figure 2.1. SEQ Figure_2.1. \* ARABIC 2One of the most symbolic ways is through the ratio of the length of a segment. “Golden Ratio…results when a line is divided in one very special and unique way” (Meisner). As shown in Figure 2.1.1, when segment A is separated into segment B and C in a particular way, the Golden Ratio is created. Sometime 0.618 and 0.382 can also be recognized as Golden Ratio. This is because they are components of forming actual Golden Ratio or Phi.Besides from segments, Golden Ratio can also be generated through the concept of Golden Rectangle. In Figure 2.1.2, the concept of Golden Rectangle and Golden Ratio can be visualized. If a square is cut out from the Golden Rectangle, then another Golden Rectangle is formed. This procedure can be repeated and received same result. As the square is removed, the ratio of the square and the new Golden Rectangle is Phi.4762528575Figure 2.1. SEQ Figure_2.1. \* ARABIC 3Figure 2.1. SEQ Figure_2.1. \* ARABIC 3Another form of the Golden Ratio is the Golden Angles. As shown in Figure 2.1.3, a circle is being divided into two sections. The 222.5 section is also known as 0.618 turns of a circle and the 137.5 section is also known as 0.382 turns of a circle. Both sections can be considered as Golden Angle because the Golden Ratio can be formed when the ratio of these two sections is formulated.2.2. Fibonacci Numbers: The Fibonacci Numbers is, sometime known as Fibonacci Series and Fibonacci Sequence, “A sequence of numbers in which each number is the sum of the two preceding numbers, e.g. 0,1,1,2,3,5,8,…”( Daintith). In other words, the next number in the series can be found by adding the two previous numbers before it or through the equation of F0=0，F1=1，Fn=F(n-1)+F(n-2); (n>=2). Another way to find the sequence is through Pascal’s Triangle. As shown in Figure 2.2.1, Fibonacci Numbers can be obtained by adding the diagonal numbers. 7334255715Figure 2.2. SEQ Figure_2.2. \* ARABIC 10Figure 2.2. SEQ Figure_2.2. \* ARABIC 12.3. Their Relationship:The Golden Ratio and Fibonacci Numbers are closely related. Even though they looked different and have usage, but they can end up with same result. “The [G]olden [S]ection number is closely connected with the Fibonacci series and has a value of (5 + 1)/2” (Knott). In other words, the Golden Ratio is ± 1.618 and Fibonacci Numbers can also result 1.618. The way Fibonacci Numbers result in 1.618 is through the average ratio between successive Fibonacci Numbers. -666753232150Figure 2.3. SEQ Figure_2.3. \* ARABIC 1Figure 2.3. SEQ Figure_2.3. \* ARABIC 2Figure 2.3. SEQ Figure_2.3. \* ARABIC 1Figure 2.3. SEQ Figure_2.3. \* ARABIC 2Another way to show their relationship is through rebuilding Golden Rectangle. As shown in Figure 2.3.1, using Fibonacci Numbers allow us to reconstruct the Golden Rectangle. The reason Golden Rectangle can be formed by Fibonacci Numbers is because they have same ratio. In Figure 2.3.1, the concept of reconstructing Golden Rectangle can be visualized. Moreover, if the sequence in Figure 2.3.1 is reversed, the concept of Golden Rectangle reducing can be visualized; it will be the same concept as shown in Figure 2.3 and Figure 2.1.2. “Spiral shells also exhibit patterns related to the Fibonacci sequence” (Smoller). This concept of reconstructing Golden Rectangle with Fibonacci Numbers can also be known as the formation of Golden Spiral. The concept and the formation of Golden Spiral are shown in Figure 2.3.2. 3. Golden Ratio in HistoryGolden Ratio is not something just discovered recently. It actually existed in various famous artworks and architectures in our history. 3.1 In Famous ArtworksOne of the earliest forms of Golden Ratio existed in famous artworks. The existences of Golden Ratio can be observed and discovered in different famous artworks. 34556703776980Figure 3.1.20Figure 3.1.2-15240029845Figure 3.1.1 Figure 3.1.1 One of the famous artwork that contains Golden Ratio is Leonardo da Vinci (1452-1519)’s Mona Lisa. The Golden Ratio in Mona Lisa existed in the form of Golden Rectangle. As shown in Figure 3.1.1, Mona Lisa included Golden Rectangles. In addition, this Golden Rectangles created significant purpose for the painting. “… [T]he edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth…” ("The Fibonacci Series."). The Focal points help to bring attentions to particular part of the painting. However, Mona Lisa included more than one set of Golden Rectangles. Another set of Golden Rectangles can be discovered from the close view of the woman’s face. As shown in Figure 3.1.2, the woman’s face can be divided into many Golden Rectangles. Similarly as the purpose in Figure 3.1.1, the Golden Rectangles existed in the face created numerous amounts of focal points. “[These focal points helped] to create a sense of beauty and balance…” ("Could You Explain the Most Basic Types of Balance Used in Compositions?").36245803399155Figure 3.1.4Figure 3.1.4952589535Figure 3.1.3Figure 3.1.3Besides from Leonardo da Vinci’s Mona Lisa, Golden Rectangles also existed in other artists’ artworks. Joseph Mallord William Turner (1775-1851)’s Norham Castle at Sunrise also contained Golden Rectangle. At the first glimpse, this artwork may not indicate any sign or clue of Golden Rectangle. However, the Golden Rectangle can be discovered as the viewer’s attentions gradually draw to the brownish creature. As shown in Figure 3.1.3, the brownish creature actually marks the borderline of two Golden Rectangles. “Joseph Mallord William Turner is admired for his use of color and light… [and these] particular interests are the geometric similarities in his various canvases…” (Britton). In other words, even though Joseph Mallord William Turner is well-known for “his use of color and light,” but his usage of Golden Rectangle do existed in most of his artworks and have significant influences.French neo-impressionist Seurat’s (1859-1891) “Bathers” is another famous artwork that contain a Golden Rectangle. As shown in Figure 3.1.4, the Bathers includes many Golden Rectangles. The three main people in the painting are formed in the way of Golden Rectangles. From the head to the waist, each person created an unconscious Golden Rectangle. Moreover, the whole painting can be divided into four Golden Rectangles. The boundaries are set by the horizon and the head of the person in the middle. Golden Rectangles exist in different famous paintings that we normally think of as beautiful. The importance of Golden Rectangle can be observed in various famous artworks and it has dramatic influence on the paintings. Perhaps, just like Luca Pacioli said, “without mathematics there is no art” (Meisner).3.2 In Ancient Architectures 2525395508000Figure 3.2. SEQ Figure_3.2. \* ARABIC 1Figure 3.2. SEQ Figure_3.2. \* ARABIC 1Besides from famous artworks, Golden Ratio can also be discovered in numerous amounts of ancient architectures. -3511552921000Figure 3.2. SEQ Figure_3.2. \* ARABIC 200Figure 3.2. SEQ Figure_3.2. \* ARABIC 2The most classic example is the Parthenon in Acropolis, Athens. The Parthenon has different form of Golden Ratio existed in different section or part of the Parthenon. The first existence of Golden Ratio is at its main entrance. Moreover, its main entrance can show two forms of Golden Ratio. As shown in Figure 3.2.1, the Golden Ratio discovered in the main entrance of Parthenon are Golden Rectangle and Golden Spiral. Another part of Parthenon that contains Golden Ratio is the columns. The columns of Parthenon have Golden Ratio in the form of Golden Rectangles. “…The width of the columns is in a golden ratio proportion formed by the distance from the center line of the columns to the outside of the columns…” (Meisner). This existence can be visualized in Figure 3.2.2. 7620053975Figure 3.2. SEQ Figure_3.2. \* ARABIC 3Figure 3.2. SEQ Figure_3.2. \* ARABIC 3If the top of the columns is magnified, then another example of Golden Ratio will appeared. The top of column can be divided into three arrangements of Golden Ratio. -2047875802640Figure 3.2. SEQ Figure_3.2. \* ARABIC 4Figure 3.2. SEQ Figure_3.2. \* ARABIC 4The arrangements are created by the design and the boundary of the Parthenon. One is formed by the section that is above the columns, shown in Figure 3.2.3 as vertical rectangle. Another one is formed by the carving that is between the boundaries, shown in Figure 3.2.3 as horizontal rectangle.Another arrangement of Golden Ratio in this section is shown in Figure 3.2.4. The Golden Ratio existed in the form of Golden Spiral and it is similar to the horizontal rectangle shown in Figure 3.2.3.294259016510Figure 3.2. SEQ Figure_3.2. \* ARABIC 5Figure 3.2. SEQ Figure_3.2. \* ARABIC 5The Golden ratio in the Parthenon did not just exist in the outer form. In other words, the Parthenon includes some Golden Ratios that cannot be discovered from outside views. The way Parthenon is arranged contain many Golden Ratios; in the form of both Golden Rectangles and Golden Spirals. In Figure 3.2.5, a floor plan of the Parthenon is shown. From the floor plan, many Golden Rectangles and Golden Spirals can be identified.The Parthenon contain dramatic amounts of the Golden Ratio. These dramatic existences may seem as the architects designed it on purpose. However, this conjecture is invalid. “If… the golden ratio was intended to be included among the many numbers and proportions included, then one can find some rather compelling evidence that they applied it… with the deeper knowledge recorded by Euclid…150 years later” (Meisner). 4. Golden Ratio in Nature Golden Ratio also existed in the field of Nature. The existence of Golden Ratio in Nature is created naturally and constantly expose to human beings. The reason of existence is unknown and remains uncanny. However, this ratio existed in an extremely conceal way; if the observations are not carried carefully and precisely, then its existence cannot be discovered easily. 4.1 In Plants:The most obvious example among all the plant examples is the flower petals. The flower petals include Fibonacci Numbers and Golden Ratio. Many species of flower have the same number of petals as one of the numbers in the Fibonacci Numbers. Some famous examples are clovers- even though they are not flowers and they have leaves instead of petals, but they correspond with the Fibonacci Numbers- buttercups, chicory, and daisy; the clovers have three petals, buttercups have five petals, chicories have 21 petals, and daisies have 34 petals (Dvorsky). As shown in Figure 4.1.1 and mentioned above, petals, and leaves, do correspond with the Fibonacci Numbers. 168275-472440Figure 4.1. SEQ Figure_4.1. \* ARABIC 1Figure 4.1. SEQ Figure_4.1. \* ARABIC 1More importantly, Golden Ratio also appeared in the petals; it appeared in the form of Golden Angle. Even though it required very precise and chary analysis, but we still discovered its existences. “[Golden Ratio] appears in petals on account of …each petal is placed at 0.618034 per turn (out of a 360° circle)” (Dvorsky).43180003687445Figure 4.2. SEQ Figure_4.2. \* ARABIC 2Figure 4.2. SEQ Figure_4.2. \* ARABIC 22889250102870Figure 4.1. SEQ Figure_4.1. \* ARABIC 2Figure 4.1. SEQ Figure_4.1. \* ARABIC 2Besides from flowers, trees also have Fibonacci Numbers in them. The Fibonacci Numbers can be seen in the formation or the arrangement of the tree branches. “This pattern of branching is repeated for each of the new stems. A good example is the sneezewort. [The] [r]oot systems and [the] algae exhibit this [kind of] pattern” (Dvorsky). As shown in Figure 4.1.2, the way trees are branching follow the pattern of the Fibonacci Numbers. This pattern is extremely mysterious. What is the chance of branching pattern follows the pattern of the Fibonacci Numbers? Is it mere coincidence or something deeper?4.2 In Animals:The Golden Ratio and Fibonacci Numbers also existed in animals. Among all the examples of Golden Ratio and Fibonacci Numbers in animals, the most miraculous example is the animal body. The Golden Ratio existed in animals in the form of proportion. The Golden Ratio of a penguin is the most obvious example. The proportions existed in a penguin is marked by the key body parts of penguins. Each section can be clearly separated by the body features of penguin and formed the Golden Ratio. “The eyes, beak, wing and key body markings of [a] penguin all fall at golden sections [proportion to] its height” (Meisner). As shown in Figure 4.2.1, a penguin can be divided into six sections that are significant examples of Golden Ratio. 3838575180975Figure 4.2. SEQ Figure_4.2. \* ARABIC 10Figure 4.2. SEQ Figure_4.2. \* ARABIC 1These kind of patterns also existed in other animals, such as, moths and ants. The pattern on a particular kind of moth has marked the boundaries of the Golden Ratio. The boundaries created by this particular kind of moth have created the Golden Ratio of its width and length. The sections are formed by the eye-like pattern on the moth and this phenomenon can be visualized in Figure 4.2.2. 3542665393065Figure 4.2. SEQ Figure_4.2. \* ARABIC 3Figure 4.2. SEQ Figure_4.2. \* ARABIC 3In comparison, ants have more Golden Ratio in them. Ants can have two Golden Ratios in them. “The body sections of an ant are defined by the golden sections of its length. [In addition,] [i]ts leg sections are also golden sections of its length” (Meisner). In other words, the first Golden Ratio can be spotted through the body sections, similar to the Golden Ratio in a penguin. The second Golden Ratio can be spotted through the leg sections; this Golden Ratio is relatively rare, since most Golden Ratios in animals are in the way or form of body section or pattern sections. This rare phenomenon can be seen in Figure 4.2.3. Other than animal bodies or anatomies, Golden Ratio can also be discovered in animals’ reproductive dynamics. Two most famous examples are the idea rabbit reproductive pattern, representation of Fibonacci Numbers, and the gender ratio of honey bees, representation of Golden Ratio. 01500505Figure 4.2. SEQ Figure_4.2. \* ARABIC 4Figure 4.2. SEQ Figure_4.2. \* ARABIC 4The idea rabbit reproductive pattern is actually a mathematic puzzle about hypothetical rabbits. Even though this example is a hypothetical one, but it follow the Fibonacci Numbers and mimic the exponential growth of a population. The question states “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” (Smoller). The solution and Fibonacci Numbers can be seen in the Figure 4.2.4. The honey bees reproduce in an interesting way. The gender ratio of honey bees is a representative example of Golden Ratio in reproductive dynamic. This example is not hypothetical. “[Ratio of] the number of females in a colony by the number of males (females always outnumber males)… is typically something very close to 1.618” (Dvorsky). 4.3 Other Natural Phenomena:Nature includes more than animals and plants. This aspect is same with the Golden Ratio. The Golden Ratios existed beyond animals and plants; the Golden Ratio also exited in non-living things in the Nature, in the natural disasters, and in even outer spaces.037465Figure 4.3. SEQ Figure_4.3. \* ARABIC 1Figure 4.3. SEQ Figure_4.3. \* ARABIC 1The most famous example among non-living things that contain Golden Ratio is the shell. The Golden Ratio existed in shell is in the form of Golden Rectangle and Golden Spiral. As shown in Figure 4.3.1, a typical shell can have the existence of both Golden Rectangle and Golden Spiral. The width and length of a shell can form a Golden Rectangle easily and the curving boundaries of the shell can form a Golden Spiral. Similar phenomenon can also be discovered in the natural disasters. 347662557150Figure 4.3. SEQ Figure_4.3. \* ARABIC 2Figure 4.3. SEQ Figure_4.3. \* ARABIC 2In disasters, the most significant example is hurricane or the typhoon. In hurricane and typhoon both Golden Rectangle and Golden Spiral can be witnessed in a single existence. Even though hurricane and typhoon are dangerous and disastrous, but they contain Golden Rectangle and Golden Spiral in them. As shown in Figure 4.3.2, the hurricane looks a typical shell in the Nature and includes both types of Golden Ratio. -257810139065Figure 4.3. SEQ Figure_4.3. \* ARABIC 300Figure 4.3. SEQ Figure_4.3. \* ARABIC 3A more surprising fact of the Golden Ratio is that it existed beyond the Earth. In other words, it also can be seen in the Universe. Many examples of Golden Ratio can be found in the Universe, but the clear ones are in the Milky Way or the Galaxy. “[S]piral galaxies also follow the familiar Fibonacci [Numbers and the Golden Ratios]” (Dvorsky) Even though the shape of a spiral galaxy looks like an ordinary shell, but it is a significant example of the Golden Ratio that is beyond the Earth. 5. Golden Ratio in Daily LifeNot surprisingly, the Golden Ratio is also hidden in our daily life. It has existed in our life for a long time and not everyone knows the existence of Golden Ratio in daily routine. The Golden Ratio, however, surprisingly existed in many famous products and many interesting phenomena in finances and music.5.1 In Finance:Even though there is no direct example of Golden Ratio in finance phenomena, but Fibonacci Numbers do existed in the finance field. One of the most famous aberrations in the finance field is the Fibonacci Retracement. The Fibonacci Retracement is a method of analysis in finance. The Fibonacci Retracement is derived from Fibonacci Numbers. “Fibonacci numbers are important to [investors] because [they] take note of the key ratios 0.382, 0.50, 0.618, 0.786, 1.00, 1.27, 1.618 [and] 2.618. They expect retracements to find support when the price drops 38.2%, 50%, 61.8%, 78.6%, 100%, 127%, 161.8% [and] 261.8%. Similarly, when a stock price has dropped it may retrace to an extent related to these Fibonacci ratios” ("FIFTI? Education Using Fibonacci."). In other words, Fibonacci Retracement is widely used in finance and has relatively high reputation. 2190750944245Figure 5.1. SEQ Figure_5.1. \* ARABIC 1Figure 5.1. SEQ Figure_5.1. \* ARABIC 1As mentioned above, the Fibonacci Retracement is used in finance and is quite reliable. However, this method may sounds too ideal to be true for some experts. Even though there are experts who do not believe in this method, but the Fibonacci Retracement does existed in reality. The Figure 5.1.1 shows the price movement of USD and CAD currency pair between September 23, 2013 and September 24, 2013. During this time period, the price retraced approximately 38.2%. As shown in Figure 5.1.1, the Fibonacci Retracement does exist. Even though finance only contain examples of Fibonacci Numbers, but the Golden ratio does exist in a tool that many people use; the credit card. The credit card has a very good proportion of the Golden Ratio. As the author of this research paper measures a typical credit card, he finds that the length is approximately 86 millimeters and width is approximately 54 millimeters. When the ratio of length and width is taken, the ratio is around 1.6. The ratio means that a typical credit card does not just have credits in it, but also the Golden Ratio.5.2 In Music:3804285481965Figure 5.2. SEQ Figure_5.2. \* ARABIC 1Figure 5.2. SEQ Figure_5.2. \* ARABIC 1This may sounds ridiculous, but Golden Ratios also exists in music! The existence of Golden Ratio can be discovered in two main fields; the instruments and the tools. 939801407795Figure 5.2. SEQ Figure_5.2. \* ARABIC 2Figure 5.2. SEQ Figure_5.2. \* ARABIC 2The representative example of Golden Ratio in the instruments is the violin. In comparison, the existence of Golden Ratio in violin is the simplest and most understandable. As shown in Figure 5.2.1, the violin can be divided into two sections and the ratio of the two sections is Golden Ratio. As the author of this research paper measures his own violin, he finds out that the longer section is approximately 36 centimeters and the shorter section is approximately 22.5 centimeters. Then the ratio of the violin is around 1.6. Even though the ratio is not a perfect Golden Ratio, but it is close enough to call it a Golden Ratio.Other than instruments, the Golden Ratio also existed in musical related tools. One of the most famous examples is the Cardas Audio. “George Cardas founded the [Cardas Audio] to perfect audio cables using ultra-pure materials, innovative Golden ratio resonance control techniques and uniquely insightful solutions to transmission line problems” 38227001177290Figure 5.2. SEQ Figure_5.2. \* ARABIC 3Figure 5.2. SEQ Figure_5.2. \* ARABIC 3( Cardas). In other words, the Cardas Audio used the concept of Golden Ratio to improve the speakers or the audios. Moreover, the Casrdas Audio does not only follow the Golden Ratio when the company constructs the speakers, but also provides setup plans that are based on the concept of Golden Ratio. One of the official setup plans can be seen in Figure 5.2.2.The Golden Ratio existed in the speakers also helped the company to win an award, as shown in Figure 5.2.3. Perhaps, the Golden Ratio is not just a ratio but something deeper. 5.3 In Famous Modern Designs:2538095987425Figure 5.3. SEQ Figure_5.3. \* ARABIC 1Figure 5.3. SEQ Figure_5.3. \* ARABIC 1Among all the Golden Ratio examples, the existences of the Golden Ratio in famous modern designs are the aberrations. Many famous modern designs do have Golden Ratio in them and more importantly, they contained relatively more Golden Ratio examples then other fields. The most famous example of the Golden Ratio in modern designs is the logo of the Apple Inc. “[The] Apple [Inc.] ha[s] used the [Golden Ratio] in designing their Logo” (Kditz, Malte). The Apple Inc.’s logo contains numerous amounts of Golden Ratio in it; in form of Golden Rectangle, Golden Ratio and Golden Spiral. This fabulous phenomenon can be visualized in Figure 5.3.1. In Figure 5.3.1, the right part shows the actual logo of the Apple Inc. and the left part shows the proportion of the curves used in the logo. In other words, the curves used in the logo are proportioned to the sections in the Golden Spiral and corresponded to the numbers in the Fibonacci Numbers. Besides the tremendous numbers of Golden Ratios in Apple Inc.’s logo, the Golden Ratio also existed in the iCloud logo. “[T]he new iCloud logo is heavily based on the Golden Ratio” (Kditz, Malte). In comparison, the iCloud logo has less Golden Ratios than the Apple Inc. logo. As shown in Figure 5.3.2, the iClound logo only has two main sets of Golden Ratio. 141922573025Figure 5.3. SEQ Figure_5.3. \* ARABIC 2Figure 5.3. SEQ Figure_5.3. \* ARABIC 26. ConclusionThe Golden Ratio is not just a ratio; it is something deeper and more mysterious. The Golden Ratio appears in history, in nature and in our daily life. The wide dissemination of the Golden Ratio in many fields may not be just a coincidence, but something that exists intentionally. Is it the signature of Natural forces? Or it is the result of something more superior; like God? The reason of the formation of Golden Ratio does not matter. Most importantly the Golden Ratio has existed on the Earth for a long time and the Golden Ratio does influence the human population unconsciously. As a human, we tend to use and prefer the existence of Golden Ratio in our creations and environments; it is something we had for an extremely long period of time. The appearance of the Golden Ratio in all we see, experience and create has unconsciously established a sense of harmony, balance, and beauty in our life and nature.BibliographyARSA. "Lady Blunt, a Rare Stradivarius Violin Sets $15.9 Million Auction Record to Help Japan Quake Relief."?EXtravaganzi. N.p., 21 June 2011. Web. 18 Dec. 2013. <, Jill. "TITLE." N.p., 6 May 2012. Web. 18 Dec. 2013. <, George. "Cardas Audio."?Cardas Audio. N.p., n.d. Web. 18 Dec. 2013. < John. "Fibonacci series." A Dictionary of Computing. 2004. . 8 Dec. 2013 <;. Dvorsky, George. "15 Uncanny Examples of the Golden Ratio in Nature."?. N.p., 20 Feb. 2013. Web. 18 Dec. 2013. <, Michele Anna. 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