The “golden” algebraic equations (original) (raw)
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Symmetry
The main purpose of this paper is to give many new formulas involving the Fibonacci numbers, the golden ratio, the Lucas numbers, and other special numbers. By using generating functions for the special numbers with their functional equations method, we also give many new relations among the Fibonacci numbers, the Lucas numbers, the golden ratio, the Stirling numbers, and other special numbers. Moreover, some applications of the Fibonacci numbers and the golden ratio in chemistry are given.
Proportiones Perfectus Law and the Physics of the Golden Section
_______________________________________________________________________________ Abstract The proportiones perfectus law is introduced. Let =. By definition, in the spectrum 1 ≤ ≤ , ≥ 1, is a proportione perfectus. With so defined, for an arbitrary positive integer ℎ it is shown that there exists an integer sequence satisfying the quasi-geometric relation ℎ = ℎ , ≥ 1 such that the arithmetic relation ℎ = ℎ + ℎ holds. The golden mean, designated or , becomes the most basic and fundamental of proportiones perfectus. New concepts to the study of the golden section are presented: chirality, number genetics and law of polarity, special numerical harmony, and chemical geometry. A geometrical basis for the fine-structure constant in the golden section is established. Our stating of over forty theorems in this reading serves no other purpose than that of expanding the theory of the golden section while equipping the interested reader with instruments for further research and development of this science of number.
The golden ratio and Viéte’s formula
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.
oalib.com, 2023
This paper attempts to express the golden ratio numbers with nucleotide bases (A T, G, C and U) as regards to Quantum Perspective Model. At first, if you take the exact value of golden ratio numbers after the comma, you can convert this decimal base numbers to binary number base system. Secondly, after converting process of this numbers, you should sequence this numbers as decimal number base system again. Thirdly, sum this decimal base numbers respectively. Fourthly, total adding processes correspond to genetic codes [Adenine (A), Thymine (T) Guanine (G), Cytosine (C) and Uracil (U) ] .Fifthly, the result explanations of golden ratio numbers can be defined like this: [ACATCC].Sixthly, the NCBI (The National Center for Biotechnology Information) search result of this sequences are very interesting model organism consequence just like as "Symphodus melops" (Corking Wrasse) and "Xyrauchen texanus" (Xysmoking texanus).Seventhly, Symphodus melops is a special organism for removing parasites from other fishes. Eighthly, Xyrauchen texanus can create light reflections by using their eyes.Ninethly, defining some irrational numbers such as phi and pi in a ratio or as cyclic numbers may provide a new clue to evaluate irrationality in mathematics. As a result, the expression of golden ratio numbers with genetic codes reaches meaningful consequences to shed lights on novel research method between Mathematics and Biochemistry.
The Golden Ratio(aka “golden mean”, “golden section” or “divine section”: standardly denoted with the capital Greek letter Phi) is the only positive (irrational) quadratic solution of the (minimal polynomial) x^2-x-1=0 equation, so that: Phi=[sqrt(5)+1]/2. The purpose of this paper is to demonstrate some aspects recently discovered by the author of this paper: (0) a recursive geometrical method for constructing Phi (inspired from the Taoist religious iconography), which is more simple and elegant than Odom’s method; (1) an occurrence of Phi in the water molecule geometry; (2) a quasi-Pythagorean property of the (Phi,e,PI) triad; (3) some other interesting quasi-exact relations between the elements of the (Phi,e,PI) triad; (4) the tendency of the human systolic/diastolic blood pressure ratio to stabilize to a value close to Phi with age Keywords: Golden Ratio/section/mean (Phi), Odom’s method, Taoist religious iconography, water molecule geometry, quasi-Pythagorean property, human blood pressure #DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted\_button\_id=AQYGGDVDR7KH2
Mathematical Sanctity of the Golden Ratio
IOSR Journals , 2019
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. 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. It is inevitable in ancient Egyptian pyramids and many of the proportions of the Parthenon. 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 [1, 2, 3, 4].But here in this paper the discussion is limited to the exhibition of mathematical aptitude of Golden Ratio a.k.a. the Devine Proportion.
JOURNAL OF ADVANCES IN MATHEMATICS, 2021
This paper introduces the unique geometric features of 1:2:√5 right triangle, which is observed to be the quintessential form of Golden Ratio (φ). The 1:2:√5 triangle, with all its peculiar geometric attributes described herein, turns out to be the real 'Golden Ratio Triangle' in every sense of the term. This special right triangle also reveals the fundamental Pi:Phi (π:φ) correlation, in terms of precise geometric ratios, with an extreme level of precision. Further, this 1:2:√5 triangle is found to have a classical geometric relationship with 3-4-5 Pythagorean triple. The perfect complementary relationship between1:2:√5 triangle and 3-4-5 triangle not only unveils several new aspects of Golden Ratio, but it also imparts the most accurate π:φ correlation, which is firmly premised upon the classical geometric principles. Moreover, this paper introduces the concept of special right triangles; those provide the generalised geometric substantiation of all Metallic Means.
A Classical Geometric Relationship That Reveals The Golden Link in Nature
Journal Of Advances In Mathematics, 2019
This paper introduces the perfect complementary relationship between the 3-4-5 Pythagorean triangle and the 1:2: √5 right-angled triangle. The classical geometric intimacy between these two right triangles not only provides for the ultimate geometric substantiation of Golden Ratio, but it also reveals the fundamental Pi: Phi correlation (π: φ), with an extreme level of precision, and which is firmly based upon the classical geometric principles.