Term Paper: Development and Application of These Concepts in Real Life

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¶ … Real Life

Golden Ratio:

The golden ratio is acknowledged as the divine proportion golden mean, or golden section is represented as a number mostly confronted while considering the ratios of distances in simple geometric diagrams like pentagram, decagon and dodecagon. It is indicated by the symbol 'phi'. The concept 'golden section' was first used by Martin Ohm in the 1835 in his book Die Reine Elementar-Mathematik. The first ever English use was seen in the article of James Sulley in 1875 which appeared in the 9th edition of the Encyclopedia Britannica. The symbol 'phi' was first used by Mark Barr at the inception of the 20th century in commemoration of the Greek sculptor Phidias, who was an extensive user of golden ratio in his works. Phi has surprising linkage with the continued fractions and the Euclidean algorithm for enumerating the Greatest Common Divisor of two integers and is also known as the Pisot Number. (Golden Ratio: mathworld.wolfram.com)

Conventionally, Golden Ratio was well understood by the Egyptians who applied it for construction of their pyramids. However, it was extensively popular under the application by the Greek geometers. Till 1946 the very concept of 'Golden Ratio' was however, not originated, when the mathematician Friar Pacioli published a paper named 'De Divna Proportione' where he made a mention about the ratio as a divine number, one notices everywhere in nature. (Inter.View to George Cardas - Cardas Cables - a brief introduction to Golden Ratio)

Most of us are coming across the number 'Pi' and take it to be the most omnipresent irrational number ever known to man. However, the 'Phi' is taken to be another irrational number and has the same propensity for popping up and is not as popular as 'Pi'. 'Phi' is also visible in several geometrical shapes, however, rather than indicating it as an irrational number, we can visualize it as a ratio in the manner described in the diagram. Taking a line segment we can conveniently split it into two segments a and B. is such a manner that the length of the entire segment is to the length of the segment a as the length of segment a is to the length of segment B. A calculation of such ratios derives a close approximation to the Golden Ratio. Another geometrical diagram related to the Phi is the Golden Rectangle. This particular rectangle has sides a and B. that are in the ratio equivalent to the golden ratio. (Phi: That Golden Number)

Historically, the proportion of length to width of the rectangles of 1.61803 39887 49894 84820 is visualized to be the most enjoyable. The distance between the columns of the structure created by Greek sculptor Phidias and it presently exists in Athens, Greece which forms golden rectangles. Phidas extensively applied the golden ratio in his sculpture. The external dimensions of the Parthenon in Athens, constructed in about 440 BC constituted a kind of perfect golden rectangle. Even the dimensions of great Pyramid of Giza believed to have been built about 4600 years ago much before that of the Greeks were based on Golden Ratio and Golden Rectangle. Many artists who prevailed after Phidias have applied this proportion; Leonardo Da Vinci acknowledged it to be the 'divine proportion' and adopted it in many of his paintings. The famous art 'Mona Lisa' can be examined minutely to find out that the measurements of the rectangles drawn around her face are in golden proportions. (the Golden Ratio: (http://www.geom.uiuc.edu/)

In addition to the application of Golden Ratio in Architecture, it is also applicable to HiFi and Music. To illustrate the standard AES listening room is also known as a Golden Cuboid, where the dimensions are in golden ratio to each other. The cabinets of the most of loudspeakers have golden ratio 'inner' dimensions. More generally, whenever, it is felt to reduce or maximize harmonic resonances the Golden ratio is considered as effective way out. (Inter.View to George Cardas - Cardas Cables - a brief introduction to Golden Ratio)

Fractals:

Fractals refer to geometrical diagrams as like the rectangles, circles and squares. However, the fractals represent unique properties those are not prevalent in case of these geometrical figures. (What are Fractals? A Fractals Unit for Elementary and Middle School Students) Fractals are fascinating images to which the people are quickly attracted. Fractal geometry blends art with mathematics to represent that equations are more than mere collection of numbers. The fractal geometry assists us in modeling visually what is demonstrated in nature; the most acknowledged being coastlines and mountains. The fractals are applied to model soil erosion and to examine seismic patterns also. However, in addition to its extensive application in narrating the complex natural patters the fractals can assist varying the common impression of the students that mathematics is dry and not reachable and may assist to induce the mathematical discovery in the classroom. (the Fractal Microscope: A Distributed Computing Approach to Mathematics in Education)

The fractal geometry was popularized under the Benoit B. Mandelbrot in terms of Mandelbrot set who formulated the concept fractal in 1975 from the Latin fractus or 'to break'. The Mandelbrot set is the set of all the points that continue to be bounded for every iteration of z = z*z + c on the complexity level where the originating value of z = 0 and c is permanent. With the assistance of NCSA supercomputers and two programs written by Michael South and Dr. Robert M. Panroff functioning with the Education Group at NCSA, it is quite feasible to find out many common elementary mathematical principles while exploring the Mandelbrot set. A program has been devised termed as Fractal Microscope that permits anyone to zoom in and out of the Mandelbrot set very quickly and easily by simply indicating and clicking under the Macintosh platform. Another program Starstruck has been developed the path generated through the Madlebrot set by iteration. (the Fractal Microscope: A Distributed Computing Approach to Mathematics in Education)

The prolonged history of fractals reveals that these structures were found out even before the actual coinage of the concept. Karl Weirstrass discovered an example of a function with non-intuitive property in 1872 which is continuous every where but not differentiable anywhere, the graph of which is presently known as fractal. Helge Von Koch being discontented with the very abstract and analytic definition of Weirstrass and propounded a more geometrical view in terms of Koch Snowflake. During 1938 the initiative for construction of self-similar curves was advanced by Paul Pierre Levy in his paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole in terms of Levy C curve. Fractals can be categorised under three groups: Iterated function systems indicating fixed geometric replacements; the fractals generated by recurrence relation at each point in a space; and Random fractals created by stochastic instead of deterministic processes. (Fractal. Wikipedia, the free encyclopedia)

Fractals emerged to be the significant questions twice prior to invention of the computers. For the first time when the British map makers invented the problem with quantifying the distance of coastal area of Britain. It was evident that the more zoomed in the maps were visualized the more detailed and longer the coastline got. However, rarely it was understandable to be a property of fractals- the circumstances of finite area being bounded by infinite line. The second incidence of emergence of the pre-computer fractals was indicated by the French Mathematicians Gaston Julia. He was amazed when a complex polynomial function would visualize like one named after him in form of z^2 + c, where c is a complex constant with numbers which are real and imaginary. (What are fractals: (www.jracademy.com)

The inherent essence behind the formula is that x and y coordinates of a point is considered and connected them into z in the form of equation x +y*i where I is the square root of negative one, this number is squared and then c is added as a constant. The resulting pair of real and imaginary number is plugged back into z, the equation is again run and this is continued until the result is greater than some number. The number of times the equation is run to take out of its orbit a color is assigned and then the pixel (x, y) becomes that color, unless those coordinates cannot get out of their orbit, in that case they are made black. The underlying attribute of fractals represent a large magnitude of self similarity. This indicates that they normally involve miniature copies of themselves included within the original. This also involves minute details. (What are fractals: (www.jracademy.com)

Thus the fractals are taken to be the geometric shapes which have complexity and are minutely detailed. An attempt to zoom in on a section will reveal as much detail as the entire fractal. One of the procedure to think of fractals for a function f (x) is to take into account x, f (x), f (f (x)), f (f (f (x))),… [END OF PREVIEW]

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