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2857500-11430000-114300-11430000The Golden RatioChristopher ShenThe Wheatley School12/17/12AbstractThe following research was done to investigate the derivations, properties, and applications of the golden ratio. It is claimed that the golden ratio coincides with paintings such as those of Leonardo da Vinci and great architecture feats such as the Parthenon. In the paper you will find a few of many ways to formulate the golden ratio, as well as its applications in the real world, and its meaning in the branches of mathematics. It was found that the golden ratio appears in nature in the form of rectangles, spirals, and pentagons, pentagrams, dodecahedrons, and icosahedrons. IntroductionThe golden ratio is an irrational number, which is defined as 1+52. It has fascinated and has proved to be an interest to mathematicians, philosophers, architects, artists, and naturalists over many years. From the shells of mollusks, such as the chambered nautilus, to the Parthenon in Greece, the golden ratio finds its way into architecture and nature. Some have gone to claim that paintings such as those of Leonardo da Vinci are based upon the golden ratio in golden rectangles and spirals. As time progressed, the golden ratio was given alternate names such as the golden mean, the golden number, the golden section, and the divine proportion. The quantity is denoted by the Greek letter ?, phi, which comes from the Greek sculptor Phidias. In some cases, it will be denoted by τ. Although the golden ratio has existed since the beginning of the development of mathematics, it is still unsure when it was first discovered and applied to the real world. However, Euclid of Alexandria proposed the first clear definition for the golden ratio. He derived the golden ratio from a simple line segment. Divide that line segment into two unequal line segments so that the ratio of the whole segment to the larger segment equals the ratio of the larger segment to the smaller segment. When this is the case, the two equal ratios simplify to the golden ratio. This was the first and simplest of many derivations to be formulated later in time. Interesting and unique properties emerged with new derivations. Derivations from rectangles, Fibonacci numbers, pentagons, continued fractions, and continued radicals have also been derived through studies of the golden ratio. Equivalent to the nonrepeating decimal 1.61803398875… the golden ratio is a unique irrational number with applications in both the real world and other branches in mathematics such as number theory, search algorithms, 2D and 3D geometry, and spirals related to the Fibonacci numbers.DerivationsDuring the evolution of mathematics, the golden ratio was discovered by various mathematicians in diverse areas of mathematics. The first and most primitive derivation is the one by Euclid, as mentioned above. In the figure 1, the golden ratio would be derived from the ratio of ABAC=ACCB. 11430005842000914400203263500Descending deeper into geometry, the golden ratio has also surfaced in the 2 and 3 dimensional aspects of geometry. In two-dimensional geometry, the golden ratio was derived mainly from golden rectangles. In figure 2, a rectangle is drawn so that the ratio of the length, l, to the width, w is 1:x. A rectangle is said to be a golden rectangle if dividing the original rectangle into a square and a smaller rectangle results in a rectangle with sides in an equivalent ratio of 1:x. Figure 2From this definition, it can be noted that1x=x-11cross-multiplying and rearranging terms yields the quadraticx2-x-1=0which has the two roots being1+52 and 1-521028065171640500After solving for the two roots, one must reject the negative value since the ratios were relating lengths, which are of positive value. From the golden rectangle we have derived the exact value of the golden mean and have opened the door to some unique and interesting properties of the golden ratio. But, first, let’s construct a golden rectangle and see why this construction works. In figure 3, the construction is demonstrated for ease of explaining. First, unit square ABCD is constructed. The midpoint of CD, E, is then drawn. Next, EB is drawn so that an arc can be constructed from it. Then, extend AB and CD. When the arc intersects extended CD at new point F, construct a perpendicular to CDF. The large, newly created rectangle is a golden rectangle. The construction above works as follows. After the construction on radius EB, a right triangle is formed. Substituting the values of the legs in the Pythagorean theorem gives 12+122=c2which simplifies to54=c2giving the length of radius EB to be equal to52Given the definition of a radius, EB and EF must be equal. So the length of the whole rectangle is CE+EF. Substituting in values gives 12+52 which arrives at the golden ratio once again.One property of the golden ratio that can be drawn from the golden rectangle by rearranging the earlier quadratic is that the reciprocal of the golden ratio plus one is equivalent to the golden ratio itself. Recalling the earlier quadratic and rearranging it arrives at,?2=?+1and dividing both sides by phi gives,?=1+1?As a result of this property, phi’s reciprocal is just ? – 1. All the numerical places succeeding the decimal point remain the same. In addition ?2 is equivalent to the conjugate, ?+1. Furthermore, in the same quadratic, ?2=?+1, it is seen that ? squared is the sum of phi raised to the first power and phi raised to the zero power. This property can be expanded as a generality for higher powers. Phi to some power is equivalent to the sum of itself to smaller powers. For example, ?9=?8+?7. Because of this property, the left side of the above equation can be reduced several times further so that for any power of phi will be equivalent to the sum of an integer and a multiple of phi. For example, ?9=34?+21. Take note that the coefficients are consecutive Fibonacci numbers. In fact, every power of phi can be written this way. 102870080010000Next, if we extended the sequence of dividing a golden rectangle into more and more squares and golden rectangles similar to a fractal, we could form a spiral such as the one below.Figure 42857500260604000The spiral turns out to be a type of logarithmic spiral and is closely related to the Fibonacci numbers. If you do not know so already, Fibonacci numbers are integers in a sequence where the next term in the sequence in the sum of the two previous terms. The sequence starts off as 1, 1, 2, 3, 5, 8, 13, 21, 34, and goes on forever. If you look closely at the figure, the squares left behind from constructing golden rectangles have Fibonacci numbers in them, or so it seems. It is very closely related to the Fibonacci spiral, which approximates the golden ratio and is constructed from quarter arcs in squares with dimensions of Fibonacci sequence numbers. Furthermore, if you take any two consecutive terms in the Fibonacci sequence, you will get an approximation of the golden ratio. The larger the terms from the sequence are, the more accurate the approximation will be. For example, 53=1.6 while 8955=1.618, which is more accurate. But, golden spirals do not have to be constructed from rectangles. For example, golden triangles are isosceles triangles where the ratio of its sides is equivalent to phi. These can be drawn as a fractal similar to Figure 4 and the same logarithmic spiral could be constructed. (See Figure). Going back to derivations, the golden ratio also appears in regular pentagons. The ratio between a side of a regular pentagon to one of its diagonals is 1: ?. A regular pentagon with side lengths of 1 unit would have a diagonal of ? units. To examine this take a look at figure six.-2286002486025Figure 600Figure 6-114300260032500-114300000In the figure it is seen that when a diagonal intersects another diagonal, the “golden line segment” is made and the corresponding ratios are the golden ratio. As an alternative, one might have set up a ratio corresponding to the lengths of the sides in the similar triangles. That would yield the ratio d1=11 ? = ?.Simlifying gets the quadratic d=? 2- ?-1 which has been previously mentioned as a derivation of the golden ratio. -228600129794000The golden ratio is seen in three-dimensional figures as well. A polyhedron is a platonic solid if all the sides are congruent regular polygons where the same amount of edges meet at every vertex. There are exactly five platonic solids known in three dimensions; they are the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosohedron. The golden ratio appears in both the dodecahedron, a polyhedron with twelve pentagonal sides, and the icosohedron, a twenty triangular sided polyhedron. Both of the solids can be constructed from the the intersection of three perpendicular rectangles in three dimensional space. In the figure above, three rectangles are intersected along the mid-segment of the other rectangles. Connecting adjacent vertices formed by the intersecting rectangles forms triangular faces, which if constructed forms the icosahedron. The same can be said about dodecahedron, however the intersecting rectangles figure exists in a dodecahedron so that each of the vertices of the rectangles meets the midpoint of one of the dodecahedrons pentagonal faces. It happens that the three rectangles used for this construction are golden rectangles, where the ratio of its length to its width is the golden ratio. Continuing our search for derivations, interesting repeating fractals as well as radicals conceive the golden ratio. Given the property that 1+1? = ?, which was mentioned earlier, you can substitute an equivalent of phi, which is the left side of this equation, in for the ? dividing the 1. This gives the complex fraction1+11+1?= ?This can be done an infinite amount of times to yield the continued fraction 1+11+11+11+…= ?.Similar to continued fractions, the golden ratio also appears in repeated radicals. In a calculator based investigation, the golden ratio was found to be equivalent to the nested radical 1+1+1+1+1+…. This radical can be calculated to the golden ratio in just a few simple steps. First, set the nested radical equal to x. Then, square both sides. x2=1+1+1+1+1+…2After simplifying and reordering the terms, the result isx2=1+1+1+1+1+1+…Then, you can substitute x in for the nested radical on the right side of the equation and to finally obtain x2=1+xx2-x-1=0which is a familiar quadratic, whose root are phi itself, and its negative reciprocal, -1?. The negative root is rejected in this case, because the square root is composed of positive integers and radicals, so the square root must be a positive, real number.Applications0203454000The golden ratio itself is fascinating and branches out to many areas of mathematics. What makes it even more fascinating is its ubiquitous nature. It has appeared in art, architecture, and nature in the real world. The golden rectangle reveals itself very often in architecture, and there is just something about its unique qualities, which makes it pleasing to the eye. First, the golden ratio was widely found in various architectures such as those of the Greeks, because of their vast knowledge of mathematics and precision. On of the most famous monuments left standing today from classical Greece is the Parthenon. It is not surprising that Phidias was the designer for this complex architectural feat. He must have taken into account the golden ratio in the designs of the Parthenon. The spaces between the columns form golden rectangles as well as many other aspects of the Parthenon as illustrated to the left. Even the shape of the room in the Parthenon is a golden rectangle. Travelling back further in time into Egypt, the golden ratio is present in the Great Pyramids of Giza. The slant height, the height, and the line segment connecting the two on the base form a right triangle. If we were to call the leg of the right triangle on the base of the pyramid one unit, the slant height would be evaluated as phi, while the height would be ?. Going to more modern architecture, a set of ten floors of the UN building forms a golden ratio. In addition, the CN tower, which was once the tallest building, has the golden ratio in its design. The skyscraper’s observation deck divides the tower so that the ratio of the tower’s full height to its height up to the observation deck is equal to the ratio of the tower’s height up to the observation deck to its height from the observation deck to the top. This familiar ratio once again yields the golden ratio. 062801500The golden ratio also appears everywhere in nature. First, it is commonly seen in golden spirals, in which the shells of mollusks such as the chambered nautilus closely resemble. Furthermore, it is also seen in pinecones, pineapples, and sunflowers. In most pinecones, there are five spirals going towards one direction and eight pointing in the other. These are Fibonacci numbers! In pineapples, the spirals are in a ratio of 13:8 and in sunflowers the spirals can be in ratios of 55:34, 89:55, or even 144:89. The ratios of the spirals are approximations for the golden ratio. The growth of such plants’ leaves, petals, or branches is just bound to the golden ratio. Next, the golden ratio is a template for the Milky Way galaxy. Most spiral galaxies in the known universe, including the Milky Way follow the design of the golden spiral. Moreover, it is shown that the divisions formed by the joints in a person’s fingers conceive Fibonacci numbers and the golden ratio. In addition, the golden ratio is also found from the segments created between the human elbow and the hand. This can be drawn to the extent to say the very structure of the human body, such as the human face, was made in golden proportions.2286000412813500Lastly, the golden ratio has been found and used countless times in art and design. First, in Leonardo da Vinci’s famous painting, the Mona Lisa, many golden rectangles can be constructed which align along her arm, chin, eye, upper lip, and nose. Da Vinci, also being a mathematician, is believed to have used mathematics as a template in his art. In the Mona Lisa, he lines up parts of the woman’s face to golden rectangles and the result is a visually appealing portrait. Furthermore, in other numerous other paintings by Da Vinci, he emphasized the golden ratio such as in An Old Man and The Vitruvian Man. Other painters such as Michelangelo in his painting, Holy Family, have displayed the golden ratio with the alignment of a pentagram, or a five-sided star, which is derived from a pentagon. Lastly, in modern day design, one of today’s top companies, Apple, has used the golden ratio and Fibonacci numbers as a guideline of their logos. The construction of various golden rectangles and spirals as well as squares and inscribed circles with dimensions of Fibonacci numbers all contributes to their logos. Figure 10Conclusion & Further ResearchTo reiterate, the golden ratio is quite an interesting, unique irrational number; arguably more fascinating the circle constant pi. Its abundant properties and its ubiquitous nature to appear in math, science, art, architecture, and science are truly remarkable. As for future research is concerned, in depth reviews of appearances in search algorithms, other derivations, appearances in the sciences such as physics, and other branches of mathematics would be an exceptional start and then could be broadened. ................
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