![]() ![]() ![]() Based on the well-known connection existing between the first two of these interpretations, the authors address the problem of finding out the thread connecting the golden rectangle with the system of equations referred to above. It can, however, also be interpreted as the formulation of the area of a golden rectangle of sides x = 1.618 and 1, and as the system of equations constituted by y = x, and y = 1/(x − 1). The golden quadratic x 2 − x − 1 = 0, when re-expressed as (x)(1) = 1/(x − 1), x = 1.618, can be interpreted as the algebraic expression of division in extreme and mean ratio (DEMR) of a line of length x = 1.618 into a longer section of length 1 and a smaller of length (x − 1). The technique can be considered as an interesting strategy to prove the Equation of Phi. The main contribution of the paper is to study about the validation and substantiation of the Equation of Phi based on classical geometric relations. The paper also explains about the structure and construction strategies of various dynamic rectangles by establishing some relations and dependencies with each other. The basic concept of golden ratio and its relation with the geometry are represented and described in this paper. The ratio also plays an enigmatic role in the geometry and mathematics. Robot sizing especially for the Humanoid Robot, Phi is considered as the key to achieve the human friendly look. Because of its unique and interesting properties, many mathematicians as well as renaissance artists and architects studied, documented and employed golden section proportions in remarkable works of sculpture, painting and architecture. Golden ratio is often denoted by the Greek letter, usually in lower case, Phi (φ) which is an irrational mathematical constant, approximately 1.6180339887. But here in this paper the discussion is limited to the exhibition of mathematical aptitude of Golden Ratio a.k.a. But very few of us are aware of the fact that it is part and parcel for constituting black hole's entropy equations,black hole's specific heat change equation,also it appears atKomar Mass equation ofblack holes and Schwarzschild-Kottler metric-for null-geodesics with maximal radial acceleration at the turning point of orbits. It is inevitable in ancient Egyptian pyramids and many of the proportions of the Parthenon. It is also very prominent on human body as it appears on human face, legs, arms, fingers, shoulder, height, eye-nose-lips, and all over DNA molecules and human brain as well. The frequency of appearance of the Golden Ratio (Φ) in nature implies its importance as a cosmological constantand sign of beingfundamental characteristic of the Universe.Except than Leonardo Da Vinci's 'Monalisa' it appears on the sunflower seed head, flower petals, pinecones, pineapple, tree branches, shell, hurricane, tornado, ocean wave, and animal flight patterns.
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