scholarly journals From the Golden Ratio to the Golden Series and Their Social Application

2021 ◽  
Vol 5 (3) ◽  
pp. 12
Author(s):  
Koumbakis Basilios
Keyword(s):  
Social Sciences ◽  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Normal Series

This paper is about the logic of golden ratio. It is about the calculation of its value and the inverse value, examination of its uniqueness, the relation with Fibonacci sequence and its spiral and the logic of development of an organism. We expand the logic of golden ratio up until the sequence of Zeno from Elea that tends to infinity. We find the differentiate logic of golden ratio coming from ancient years and its unknown relation to the golden ratio. Also, we calculate the values φ of series that follows the logic of golden ratio, reaching the golden (normal) series, as a result of its logic, with its modern applications. Finally, it is criticized the fact that we do not include golden ratio in our education and the consequences that this has, by compare it with the achievements of its era. The application of golden ratio’s logic in social sciences results in possible examples of its use and their advantages.

scholarly journals From Fibonacci Sequence to the Golden Ratio

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Alberto Fiorenza ◽  
Giovanni Vincenzi
Keyword(s):  
Characteristic Polynomial ◽  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Maximum Modulus ◽  
Point Of View ◽  
General Point ◽  
Constant Coefficients ◽  
The Difference ◽  
New Perspective

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

scholarly journals Musica fibonacciana: Aesthetic and practical approach

New Sound ◽  
2017 ◽  
pp. 70-90
Author(s):  
Rima Povilionienè
Keyword(s):  
Music Composition ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Laws Of Nature ◽  
Fibonacci Sequence ◽  
Number Sequences ◽  
Music And Mathematics ◽  
To Come ◽  
Composition Structure ◽  
And Mathematics

In the sphere of musical research, the intersection of two seemingly very different subject areas-music and mathematics is in essence related to one of the trends of music-attributing the theory of music to science, to the sphere of mathematica. It is regarded the longest-lasting interdisciplinary dialogue. The implication of numerical proportions and number sequences in the music composition of different epochs is closely related to this sphere. A significant role in creating music was attributed to the so-called infinite Fibonacci sequence. Perhaps the most important feature of the Fibonacci numbers, which attracted the attention of thinkers and creators of different epochs, is the fact that by means of the ratio between them it is possible to come maximally close to the Golden Ratio formula, which expresses the laws of nature. On a practical plane, often the climax, the most important part of any composition, matches the point of the Golden Ratio; groups of notes, rhythm, choice of tone pitches, a grouping of measures, time signature, as well as proportions between a musical composition's parts may be regulated according to Fibonacci principles. The article presents three analytical cases-Chopin's piano prelude, Bourgeois' composition for organ, and Reich's minimalistic piece, attempting to render music composition structure to the logic of Fibonacci numbers.

scholarly journals Balancing on the edge, the golden ratio, the Fibonacci sequence and their generalization

2018 ◽  
Vol 39 (6) ◽  
pp. 065805
Author(s):  
Gautam Dutta ◽  
Mitaxi Mehta ◽  
Praveen Pathak
Keyword(s):  
Golden Ratio ◽  
Fibonacci Sequence

On a probabilistic analogue of the Fibonacci sequence

1980 ◽  
Vol 17 (04) ◽  
pp. 1079-1082 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Population Models ◽  
Stochastic Version

One of the earliest population models to be studied gives rise to the Fibonacci sequence and has a history dating back more than 750 years. A stochastic version of the model is discussed in this paper, its basic defining property being E(Xn | Xn −1, · ··, X 0) = Xn −1 + Xn −2 a.s. The process {Xn } mimics many of the standard properties of the Fibonacci sequence. In particular, under mild additional conditions, a.s. as n → where α is the ‘golden ratio'

scholarly journals An Efficient Golden Ratio Method for Secure Cryptographic Applications

2018 ◽  
Vol 23 (4) ◽  
pp. 58 ◽  
Author(s):  
Anthony Overmars ◽  
Sitalakshmi Venkatraman
Keyword(s):  
Data Storage ◽  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Ratio Method ◽  
Secure Communications ◽  
Security Attacks ◽  
Security Breaches ◽  
Elliptic Curve Cryptosystems ◽  
Electronic Transactions ◽  
Sequence Method

With the increase in the use of electronic transactions in everyday life, secure communications and data storage to withstand any kind of attack is warranted. The golden ratio, being the most irrational among irrational numbers, can be used in elliptic curve cryptosystems, power analysis security, and other applications. However, in such applications, cryptographic operations should take place very quickly before the keys are extracted or decoded by the attackers. This paper proposes an efficient method of golden ratio computation in cryptography to resist information security breaches. We compare our new golden ratio method with the well-known Fibonacci sequence method. The experimental results show that our proposed method is more efficient than the Fibonacci sequence method. Our golden ratio method with infinite precision provides reliable counter measure strategy to address the escalating security attacks.

Quasiperiodicity and the nanoscopic morphology of some polyurethanes

2015 ◽  
Vol 48 (2) ◽  
pp. 313-317 ◽  
Author(s):  
Norbert Stribeck ◽  
Xuke Li ◽  
Berend Eling ◽  
Elmar Pöselt ◽  
Pieter J. in 't Veld
Keyword(s):  
Chemical Synthesis ◽  
Small Angle ◽  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Real Space ◽  
Morphology Evolution ◽  
Polyurethane Elastomers ◽  
X Ray ◽  
Long Period ◽  
X Ray Scattering

When straining polyurethane elastomers (PUEs), it is often observed that the long-period peak of the small-angle X-ray scattering (SAXS) does not shift normally. An explanation is indicated for some PUEs in the real-space chord distribution. It exhibits a sequence of constant long-period bands. The band positions form a Fibonacci sequence. This relates to the underlying chemical synthesis by polyaddition of hard and soft modules, indicating a nearly quasiperiodic setup in sequences of stringed hard domains. These sequences appear to be the probes provided by SAXS for the study of morphology evolution in such PUEs. Should a regular-as-possible arrangement of physical crosslinks optimize a property of the material, then in the synthesis the mole fractionnHof hard modules should be chosen to benH= τ/(1 + τ) ≃ 0.62, where τ is the golden ratio.

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