An Original Approximation of the Golden Ratio Using an Integral

The Golden Ratio, denoted by the Greek symbol φ, is probably the most fascinating mathematical proportion in the whole world. It arose from a purely geometric manipulation, but also has plenty of intriguing algebraic properties, which justify the adjectives such as "golden" and "divine", with which it has been christened over the last centuries due to its unique properties. This fraction plays the most curious and wonderful role in nature under a special type of sequence, the Fibonacci sequence, which was first introduced by the Italian mathematician Leonardo of Pisa (or Leonardo Fibonacci) in 1202. Curiously, the Golden Ratio has been identified as a "preferred" pattern by nature, appearing in a wide range of natural systems such as the sequential numbers of individuals in the generation of bees, the spiral arrangement of elements along a plant as a phenomenon called phyllotaxis and the spirals exhibited by seashells, horns, hurricanes and galaxies. Much has been published about this intriguing proportion, but there is still a lack of precise and quantitative information on why this works best for certain growth patterns in nature. Whereas some papers go enthusiastically through sorts of curiosities on how it may be featured in natural systems, others actually cover the significance of this ratio as for its physical importance to provide a high rate of stability for growth phenomena. The latter has to be straightforwardly elucidated in order to guarantee a reasonable understanding of its importance.

The Golden Ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the Greek alphabet. What is so intriguing from the Golden Ratio?

The Turkish Online Journal of Design, Art and Communication, 2022

Golden Ratio and Fibonacci numbers have attracted attention from mathematicians, artists, architects, sculptors and musicians for centuries. Golden Ratio was associated with Ancient Greek art and architecture, and named after Greek sculptor Phidias (4 th and 5 th Century BC). Fibonacci sequence which is closely related to Golden Ratio was discovered by an Italian mathematician Leonardo Pisano Bigollo (1170-1250) in 1202. Those two phenomena have created a considerably extensive literature but the studies mainly focused on documenting where we encounter them in human body, in nature, in plants, in animal kingdom, in art and architecture. Golden Ratio was also known as Divine Proportion since commonly observed in nature. Yet, there have been few arguments or attempts to explain real reasons behind the existence of these numbers in nature and in universe. This study documented literature and collected puzzling pieces from various fields such as ancient philosophers, Far East philosophies, Western philosophy, mysticism, economy, sociology, religious scriptures, botany, psychology, astronomy, physics, mathematics and evolutionary approaches. This study aimed to construct a philosophical theory based on a physics phenomenon that widely influenced many fields including art. The information evolved through literature was used to construct an explanation why Golden Ratio and Fibonacci sequence existed in the universe. Starting from Heraclitus (535-475 BC) of Ephesus, there have been claims that all existence is in a constant flow (Panta rhei) motion. This study suggested Golden Ratio and Fibonacci numbers as evidences of a constant universal flow motion which could further be supported by many empirical evidences from the literature.

76.345 Exploring the Hidden Secrets of the Golden Ratio, 2022

Using over 1,000 photographs, the author demonstrates how a unique geometric angle, fundamental to all physical reality, is 'hiding in plain sight' in Nature and architecture. Examples of this architecture date back thousands of years, and are found on all continents and in all major religious traditions. This unique angle is the geometry of Universal Phi Scaling. This angle is found in Sardinian Nuraghes, Scottish Brochs, the Great Zimbabwe, Hindu and Buddhist temples, Gothic and Neo-Gothic cathedrals, churches, mosques, synagogues, obelisks, hieroglyphs, ancient symbols and paintings, castles, forts, government buildings, Mystery School architecture, bell towers, Tesla's Wardenclyffe Tower, Tartarian architecture, contemporary architecture, contemporary Russian Pyramids, DNA, the Torus and much, much more. Prior civilizations used this geometry for healing and free energy, and today it is used in a number of high-tech applications, unbeknownst to the engineers who designed them. The author's research into Sacred Geometry, harmonics and ancient diagrams demonstrates the essential role of 76.345 in cosmology, free energy generation, vibrational healing, and PSI abilities. The book also addresses questions surrounding how mankind came to lose this ancient knowledge. Using concrete examples, the author shows how the history and knowledge of the Golden Ratio, Sacred Geometry, fractals, the Fibonacci Sequence, the Aether, the Platonic Solids, and Western mathematics have all been falsified and suppressed through the coordinated efforts of powerful forces.

2024

Consider the graphs of f(x)=x^x, f(x)=x^1/x, f(x)=1/x^x, and f(x)=1/x^1/x (where y is equivalent to f(x) and its variants) superimposed onto one another. This is for all { x ∈ ℝ ∣ x > 0 }. The stationary points of each of these graphed functions form a rectilinear quadrilateral (referred to as Yaniv’s Rectangle throughout the paper), and our goal is to calculate the area and perimeter of said quadrilateral. To find the stationary points, the 4 functions are differentiated and their gradient functions are set equal to 1. The four stationary points are the four vertices of Yaniv's Rectangle. Upon calculating the area of Yaniv’s Rectangle, a 6-decimal-point value of 1.768801 is yielded, which is approximately equal to sqrt(pi), which has a 6-decimal-point value of 1.772454. The difference in the two values is approximately 0.003853 correct to 6 decimal places (which can be seen as negligible in certain trivial situations). Meanwhile, when Yaniv’s rectangle’s perimeter is calculated, a value of 6.205739 is obtained, which is approximately equal to 2*pi (ie- 6.283185). A difference of 0.0774461 can be seen. As a result, barring the slight differences, the following identities (known in this paper as Yaniv’s First and Second Approximation Identities) can be derived and can be used only for situations in which extremely precise values are not necessary.

Teaching Mathematics and Computer Science, 2014

Viète's formula uses an infinite product to express π. In this paper we find a strikingly similar representation for the Golden Ratio.

European Journal of Physics, 2018

The golden ratio and Fibonacci numbers are found to occur in various aspects of nature. We discuss the occurrence of this ratio in an interesting physical problem concerning center of masses in two dimensions. The result is shown to be independent of the particular shape of the object. The approach taken extends naturally to higher dimensions. This leads to ratios similar to the golden ratio and generalization of the Fibonacci sequence. The hierarchy of these ratios with dimension and the limit as the dimension tends to infinity is discussed using the physical problem.

In the first chapter we will present the definitions,history,relations and arithmetics approximations of Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 220 expressions. In the second chapter we will present exact expressions between Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 127 expressions. In the third chapter we will present approximations expressions between Archimedes' constant π,the golden ratio φ,the number Euler e and the imaginary number i. A total of 112 expressions.