Fibonacci Series Research Papers - Academia.edu (original) (raw)

- by
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- Medical Law, MRI, Facial expression, Craniofacial Identification
- by Gregory Martynenko
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- Fibonacci Series
- by Tytyvyllus Flor
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- Mathematics, Matemáticas, Fibonacci Series, Eratóstenes

In studies presented in the literature, relationships between music and mathematics can sometimes be observed. Leonardo Fibonacci (1170-1250) is well known in mathematics with the Fibonacci Sequence and this sequence used to identify numbers in various music elements, too. In related studies, these numbers have been used to demonstrate the existence of the 'Golden Ratio' using methods and theories borrowed from the components of music. Nevertheless, this relationship has subsequently been seen inaccurate. The studies that previously based some works of Chopin, Mozart, Beethoven, Bach and Bartók on Fibonacci Sequence and Golden Ratio are critically examined in the context of musical and mathematical theories in this study. Qualitative and quantitative research methods were used together in this interdisciplinary research in the field of mathematical sciences and critical musicology. It was examined basically the measure or rhythms (sound duration) within the musical works that allegedly used the Fibonacci Sequence and the Golden Ratio, and it was found these studies yielded values close to the terms of the Fibonacci Sequence and the determined values of the Golden Ratio were 0.618, 1.618, and 0.382. It is determined that mathematical, historical and music theoretical data and findings could not provide enough to support the claims of the related studies. Thus, it was determined that the accuracy of the Fibonacci Sequence and Golden Ratio expressed in the works of the related composers are controversial within the framework of the relevant studies.

- by Sumeyye Bakim and +1
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- Mathematics, Applied Mathematics, Musicology, Critical Musicology

В книге рассматривается история возникновения и становления математико-гармонических идей (математики гармонии) от античности до конца XX века. История представлена последовательностью очерков, в которых
в доступной для массового читателя форме обсуждаются основные идеи, касающиеся интерпретации гармонии в философии, математике, естественных и гуманитарных науках, об осмыслении этой категории в различных искусствах: архитектуре, музыке, живописи, дизайне,
художественной литературе. В центре внимания автора — математические и эстетические проблемы. Основной объект исследования — гармонические пропорции, рекуррентные последовательности, симметрийные структуры.
Книга выполнена в широком междисциплинарном аспекте и предназначена для специалистов самых разнообразных областей знания, творческой и практической деятельности: от математики и астрономии до музыковедения и литературоведения.

- by Gregory Martynenko
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- Applied Mathematics, Aesthetics, Art History, History of Mathematics

Golden ratio is often denoted by the Greek letter, usually in lower case, Phi (φ) which is an irrational mathematical constant, approximately 1.6180339887. 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. Robot sizing especially for the Humanoid Robot, Phi is considered as the key to achieve the human friendly look. The ratio also plays an enigmatic role in the geometry and mathematics. The basic concept of golden ratio and its relation with the geometry are represented and described in this paper. The paper also explains about the structure and construction strategies of various dynamic rectangles by establishing some relations and dependencies with each other. 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 technique can be considered as an interesting strategy to prove the Equation of Phi.

- by Akhtaruzzaman Akhter
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- Golden Section, Fibonacci Series, Fibonacci numbers, Phi

Un breve análisis de los primeros preceptos de la belleza, así como puntos de inflexión en nuestra comprensión de la misma hasta un desemboque de los significados en el modernismo del siglo XX con las obra de la unidad habitacional de Marsella de Le Corbusier, la composición Metastasis de Iannis Xenakis y la Stretto House del arquitecto Steven Holl

- by Raúl Medina
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- Arquitectura, Artes, Antropología Social, Le Corbusier

The papers presents design of ladies’ dresses using the Fibonacci series tiling with triangles named Fibonacci rose. As a result of the use of Fibonacci rose for designing of aesthetic, beautiful and harmonic clothing, it can be concluded that in fashion design Fibonacci rose can be used in different ways of colour combinations, proportions toward the clothing sizes, and as a frame of creations of design elements.
Keywords: Fashion Design, Fibonacci Sequence, Fibonacci Series Tiling, Fibonacci Rose.

- by Zlatina Kazlacheva
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- Fashion design, Fibonacci Series, Fibonacci numbers, Fibonacci Sequences
- by Kumar Rahul Verma
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- Fractals, Fibonacci Series, Modularity, Golden Ratio
- by Zlatina Kazlacheva
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- Fashion design, Golden Section, Fibonacci Series, Fibonacci numbers

All the fields of Mathematics-‘Algebra’, ‘Trigonometry’, ‘Geometry’, ‘Calculus’ and ‘Statistics’ are based on the Patterns, Shapes and Numbers and thus we can conclude that Mathematics is based on Pattern, Shapes and Numbers. Defining mathematics is always been keen concern of mathematicians as well as any introduction math class. One way of definition is defining it by “what it does in the era?” and that is the definition of mathematics for the era. So, in this era of analytical science where everyday people are dealing with lots and lots of data analysis in each and every field, the Pattern, Shapes and Numbers have become the limelight of Mathematics-Research and now mathematics has grown from just solving sums to an experimentation and observation, finding relation to the real world and thus it involves science which brings us to conclusion, “Mathematics is the Science of Patterns, Shapes and Numbers.” Thus, Mathematics can be defined as the science of “Pattern, Shapes and Numbers”.

- by Keval Chetanbhai Purohit
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- Mathematics, Philosophy of Science, Numbers, Patterns

Las escalofriantes aventuras de Sabrina. Bienvenidos a un nuevo artículo. Como todos ya saben se acerca el 31 de octubre, la noche de Halloween, víspera del Día de Muertos, de Todos los Santos y fecha también conocida como la noche de las brujas. Y no sin razón, pues esta es una noche muy especial para ellas. La leyenda cuenta que las brujas se reúnen en aquelarres dos veces al año convocadas por el Diablo, el 30 de abril y el 31 de octubre. Fecha en la que se da la bienvenida a un nuevo año de brujería. En esta noche, los poderes satánicos y la brujería están en su nivel más alto. Así que, ahora ya lo saben; en este artículo se hablará de brujas, en concreto de Sabrina Spellan, una de las hechiceras más conocidas de la pantalla. Seguro que muchos de ustedes recuerdan a la encantadora de Sabrina de Sabrina, Cosas de Brujas (Sabrina, the Teenage Witch o Sabrina, la bruja adolescente) interpretada por Melissa Joan Hart. Esta serie de finales de los 90 era una divertida sitcom (serie cómica televisiva) que narraba las historias de una joven bruja, junto a sus tías Hilda y Zelda, su novio Harley Kinkle y su gato Salem. Pues Sabrina ha vuelto y de la mano de Netflix. Las Escalofriantes Aventuras de Aventuras de Sabrina o El Mundo Oculto de Sabrina como se conoce en Hispanoamérica (Chilling Aventures of Sabrina) Se estrenó el 26 de octubre de 2018 en Netflix, serie que es un tanto más oscura que la comedia. Aunque se repiten muchos de los personajes, nos espera un mundo lleno de magia negra, sombras y peligros ocultos con toques de terror, ingredientes perfectos para hacer una maratón de Halloween. La serie relata la oscura historia de la adolescente Sabrina, quien al cumplir 16 años se encuentra en un momento delicado su vida, tiene que elegir entre el mundo de la magia y el de los humanos, ya que ella es mitad bruja, mitad mortal. En esta adaptación, Sabrina luchará por conciliar sus dos universos mientras se enfrenta a las fuerzas malvadas que la amenazan a ella, a su familia y amigos y a toda la humanidad. Las Escalofriantes Aventuras de Sabrina, se desarrolla en el mismo universo en la que está inspirada la serie Riverdale. Los orígenes de la bruja adolescente también se encuentran en el cómic Archie de John L. Goldwater, una serie de historietas que empezaron a publicarse a principios de 1940 en Estados Unidos. Estas narran las aventuras de un grupo de adolescentes llamados Archie, Betty, Verónica y Judhead que viven en la ciudad de Riverdale. Sabrina Spellman es un personaje secundario de este cómic, que vive en un pueblo vecino llamado Greendale. Una joven de su época, valiente, persistente y honesta, dispuesta a luchar por sus amigos. Sabrina debutó en Archie's Madhouse #22 en octubre de 1962 y gustó tanto que finalmente se convirtió en uno de los personajes principales de Archie Comics. manuelverdugo.com

- by Manuel M V L Verdugo
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- Time Series, Time series Econometrics, Time-Series Analysis, Series TV

The Fellowcraft degree, or second degree of Freemasonry, is often underestimated; its secrets sometimes unintentionally dismissed as ho-hum, or over laden with redundant Biblical tales concerning commonplace morality. This study examines the Legend of the Winding Stair, an intriguing and exciting element of Freemasonry, and a central theme of not only the second degree, but the entire masonic system. It suggests the Winding Stair is an esoteric link to the geometric principles of the Golden Ratio and the Phi Spiral, both early mathematical concepts and examples of ancient and sacred geometry. Following this trail, the article explores the involvement and contributions of natural philosophers and scientists such as Sir Isaac Newton, Elias Ashmole, Chistopher Wren and Newton’s secretary and British Grand Master John Theophilus Desaguliers, in the development and transmission of modern Freemasonry and its hidden mathematical allegories.

- by James C. Stewart
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- Mathematics, History of Science, Symbolism, Western Esotericism (Anthropology)

Fibonacci: a natural design, easy to recognise - yet dif cult to understand. Why do owers and plants grow in such a way? It comes down to nature's sequential secret...This paper discusses how and when the Fibonacci sequence occurs in ora.

- by Nikoleta Razkova
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- Nature of Science, Nature, Fibonacci Series, Flowers

How intriguing is the idea that there is a concurrent mathematical pattern that exists in the stars, the human body, the breeding patterns of domestic rabbits, sea shells and plants? Very tantalizing is the idea that math was not just created to quantify existence, but that existence actually quantifies mathematics. Through exploring the Golden Section and the Fibonacci Series, I have learned that mathematics can extend beyond the survival of the high school student. Mathematical patterns can be observed in a plant’s growth patterns, on its quest to maximize its exposure to light.

- by Sarah Schoenfeld
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- Botany, Mathematics, Fibonacci Series, Plants

The design method of Andrea Palladio, the famous Renaissance architect, has yet to be discovered. Although the room dimensions given in his woodcuts are the only data that can be used for this purpose, performing calculations around them only cannot produce any valuable outcome. Instead, to discover Palladio's method, one must focus on the process of the construction, of how workers would have laid out the building.

- by Ates Gulcugil
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- Golden Section, Andrea Palladio, Fibonacci numbers

DESCRIPTION: This document is a research paper I was assigned during an English class at Columbus State Community College. I was to analyze any piece of music that interested me. The purpose was to covey why the chosen song was enjoyable... more

- by Jacob R Wheeler
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- Religion, Spirituality, Alchemy, Spirituality & Mysticism

This article investigates geometry’s importance within freemasonry and discovers the de facto guardianship provided geometry by operative masons through the Middle Ages. Operative masons passed their geometry and their philosophy from generation to generation via initiation. This paper illustrates the transmission and sometimes startling application of those principles while examining the masonic symbolism of the 47th Proposition of Euclid, the Golden Ratio and the Point within a Circle. Consequently, the study reveals an extraordinary path from the operative to the speculative age. By tracing the history of the Master Mason, or third degree, we gain a deeper understanding of this transition. We find a suggestion the third degree came into common usage in the late seventeenth or early eighteenth century, and an implication the resurrection and/or esoteric modification of masonic geometry occurred during this era. We explore the fascinating intersection of Dr John Theophilus Desaguliers and Sir Isaac Newton at the centre of this intriguing period, and contend these men played a part in not only the modification (or re-creation) of the Master Mason degree, but also in the resurrection of those geometric symbols from the operative age that define present freemasonry.

- by James C. Stewart
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- History, Mathematics, Renaissance History, Architecture

The proportions of the Golden ratio and Fibonacci sequence associate harmony and beauty and by this reason they are used in design. The paper presents the use of geometrical forms and tilings, created on the base the Golden ratio and Fibonacci numbers, in fashion and textile design. The forms and tilings are the Golden and Fibonacci spirals, the Golden and Fibonacci series tiling with squares, the Golden tiling with triangles, Fibonacci tiling with triangles – Fibonacci Rose, etc. For creation of successful aesthetic fashion and textile design projects these kinds of forms can be used in different combinations and color decisions.

- by Julieta Ilieva and +2
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- Fashion design, Textiles, Textiles Design, Textile Design

We present 1) a novel unified conception of science, cognition and phenomenology in terms of the Klein Bottle logophysics; 2) as a supradual creative agency based on self and hetero-reference and multistate logic associated to the non-orientable topologies of the Möbius strip and Klein Bottle surfaces; 3) related to the Golden ratio in several areas of biology (particularly genomics), cognition, perception, physics and music, and the multiple biochemical codes of life; 4) semiosis and topological folding in the genesis of life; 5) the torsion geometries and non-orientable topologies, their relation to active time and chronomes, standing waves and cyclical process, providing an ontology for "chance" and apply them to 6) human-bodyplan, neurosciences, music cognition, structure and processes of thinking, particularly Quantum Mechanics, creativity and the logics of the psyche; 7) a universal principle of self-organization and the genesis of life, the π-related visual cortex and holography; 8) as an harmonic principle in the brain's pattern formation, pattern recognition and morphogenesis, and the topological paradigm to neuroscience proposing an explanation for the higher dimensional organization of brain connectomes based on the Klein Bottle as the metaform for patterns; 9) higher-order cybernetics, ontopoiesis and autopoiesis in Systems Biology,the psyche's bi-logic; 10) the supradual nature of phenomenology, its relation to cosmological cycles, and an examination of the forceful omission of supraduality in academic philosophy vis-à-vis the foundations of science and philosophy in ancient Greece; 11) a rebuttal of Dr. Liu et all's PLOS article claiming the appearance of Phi in genomes as accidental, in terms of the supradual ontopoiesis hereby presented and by reviewing several codes of life discovered by Pérez, which elicit their unity already starting at the level of the periodic table of elements and Life compounds atomic mass and 12) the Golden mean in the rituals of whales and the supradual logophysics of social organization.

Se considera la serie de Fibonacci y su efecto en la proporción áurea.

'Golden shapes' are the square, the golden rectangle, and the simple shapes made from these two. For several Palladian buildings, the golden shapes around which they were designed are exposed. This can be considered as evidence that... more

Gli scritti di Antonio Thiery, a cura di Giancarlo Mauri - Federico II, Castel del monte e la conoscenza scientifica - STUDIES IN THE HISTORY OF ART 44 - Center for Advanced Study in the Visual Arts Symposium Papers XXIV - Atti del Convegno Internazionale su Federico II, svoltosi nel gennaio del 1990 alla National Gallery of Art di Washington, nell’ambito del Congresso Intellectual Life at the Court of Frederick II Hohenstaufen
pp. 273-292 - Edited by William Tronzo, National Gallery of Art, Washington - Distributed by the University Press of New England
Hannover and London

- by Giancarlo Mauri
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- Fibonacci Series, Umanesimo, Alto Medioevo, Medioevo

The Menorah Matrix is an integrated numeric structure of two repeating patterns based on the Fibonacci sequence "hidden" within the base-ten number system. Evidence presented here points to the Menorah Matrix to be the foundation of practically all physical laws and phenomena in nature. The inner-dimension of the base-ten and the Menorah Matrix specifically predicts the electromagnetic force, the atomic and subatomic structures (including anti-particles), the DNA molecule and its four bases system, the genetic code, the basis to the limit of 20 amino acids, the Earth♦Moon's blueprint in astonishing correlations and much more.
The Menorah Matrix is the basis for the long-sought "Theory of Everything."
This presentation is a very short version of the entire research which will be published (perhaps in a book format) at a future date.

- by Michael C Grasso
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- Electromagnetism, Philosophy of Science, Philosophy Of Mathematics, Hebrew Bible

The numbers in the so-called Fibonacci Sequence express Euclid’s division in extreme and mean ratio (DEMR), popularly known as the Golden Section. Since the manuscript describing the sequence, Fibonacci’s Liber abbaci, was written in 1202 and since Euclid described DEMR c. 300 BCE, many musicologists have naïvely assumed that composers since 1202 consciously used Fibonacci numbers to express the Golden Section. This is historically misguided. For example, although Euclid’s DEMR was widely-published and discussed throughout maths history, Fibonacci's Liber abbaci (1202) was not. After a brief transmission in manuscript form, Liber abbaci was lost until the mid-eighteenth century and forgotten for a further hundred years until Prince Baldassarre Boncompagni rediscovered it and published it in two volumes in 1857 and 1862. Although there were a few sporadic appearances of a numerical expression for DEMR in the 17th and 18th centuries (unrelated to Fibonacci), real interest in the Golden Section and its aesthetic properties was first awakened in the late 19th century with the golden numberism movement. This paper will examine the historical facts and set out clear principles to guide the analyst.

- by Ruth Tatlow
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- Religion, Musicology, Philosophy, History of Ideas
- by Zlatina Kazlacheva
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- Fashion design, Golden Section, Fibonacci Series, Fibonacci numbers

Breve articolo pubblicato sulla rivista Mistero sulle ricerche dell'astrofisico russo Nikolai Kozyrev. Un tentativo di tendere un parallelo tra la sua Meccanica Causale e quelle teorie che indagano i rapporti aurei, la sequenza di... more

- by Edoardo Segato-Figueroa
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- Russian Studies, Quantum Physics, Æther Physics, Causality

Now, for the first time astronomers have directly imaged the last piece of the puzzle that ties them all together – a dusty donut of material that surrounds a supermassive black hole.
TORROIDS ARE FORMED IN THE AETHERS BY ANUs LINKING UP - TO MANIFEST GRAVITY, VIBRATIONS, LINEAL TIME, THE FIBONACCI SERIES, ALL OF THE PROPERTIES OF OUR 3D REALM ARE A FUNCTION OF TORROID FORMATION DUE TO THE RULE

- by MIKE EMERY
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- Quantum Gravity, Black Holes, Metaphysics of Time, Philosophy of Time

Raw and unedited notes and considerations on R. Penrose and S. Hameroff's Orchestrated Objective Reduction Theory, Dr. Anirban Bandyopadhyay studies, Jürg Fröhlich Bose-Einstein condensation implications, Karthikeyan Marimuthu and Raj... more

The number theory is a branch of mathematics which is primarily dedicated to the study of integers. The number theory, as such, is less applied in engineering compared to calculus, geometry, etc. The problem was that it could not be used directly in any application. But, the number theory, combined with the computational power of modern computers, gives interesting solutions to real-life problems. It has many uses in various fields such as cryptography, computing, numerical analysis and so on. Here, we focus on the applications of the number theory about engineering challenges.

- by Audhithya S V
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- Number Theory, Cryptography, Fibonacci Series, Fibonacci numbers
- by Zlatina Kazlacheva
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- Fashion design, Fibonacci Series, Fibonacci numbers, Fashion

Книга содержит более 30 сонетов, посвященных выдающимся деятелям науки и искусства, причастным к созданию математического учения о гармонии. Это философы, математики, астрономы, искусствоведы, архитекторы, инженеры, языковеды, психологи. Все они творили в разное время в течение последних двух с половиной тысяч лет. Каждый сонет сопровождается краткой характеристикой деятеля науки или искусства. Книга предназначена для ученых, любителей искусства и поклонников поэзии.
ISBN 978-5-905107-16-0

- by Jasna Horvat
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- Art History, Visual Arts, Beauty, Fibonacci Series

The Golden ratio and Fibonacci sequence are used as proportions in design as symbols of beauty and harmony. That symbolism is a result of the strong connections in their mathematical nature. The Golden section is a number, introduced with Greek letter φ, which is found by dividing a line into two parts as the longer part divided by the smaller part is equal as the whole length of longer and smaller parts divided by the longer part. Fibonacci sequence is a series of numbers where every number is equal to the two numbers before it. An investigation of application of proportions based on the Golden ratio and Fibonacci sequence in the fashion design and pattern making of ladies' clothing is the main aim of the paper. Based on the study it may be concluded that in fashion design and pattern making the Golden ratio and Fibonacci sequence can be used in creation of beautiful and harmonic forms directly or with the help of geometrical figures as: In directly use the Golden and Fibonacci numbers proportions can be in one and the same or different directions. In the application with the help of geometrical shapes the Golden and Fibonacci figures combine proportioning and form creation. The Golden and Fibonacci shapes can be used directly as forms or as frames of forms creation of elements and pieces. Its application can be in different directions and location according the bodice. The Golden section and Fibonacci sequence can combine proportions with other principles of design as symmetry, rhythm, etc. 1. Introduction The proportions are one of the most important design principles. The Golden ratio and Fibonacci sequence are used as proportions in design as symbols of beauty and harmony. That symbolism is a result of the strong connections in their mathematical nature. The Golden section is a number, introduced with Greek letter φ, which is found by dividing a line into two parts as the longer part divided by the smaller part is equal as the whole length of longer and smaller parts divided by the longer part, or a/b = (a+b)/a = 1.61803398874989484… [1] Sometimes the Golden ratio is presented in a turned way in which the number is equal to the division of the smaller by the longer part equal to the division of the longer part by the whole length of the line, or b/a = a/(a+b) = 0.61803398874989484… Fibonacci sequence is a series of numbers where every number is equal to the two numbers before it, or xn = xn-1 + xn-2. The sequence starts with 0 and 1 and goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. [2] An investigation of application of proportions based on the Golden ratio and Fibonacci sequence in the fashion design and pattern making of ladies' clothing is the main aim of the paper.

An experimental poem relating the Fibonacci sequence to human aging.

This paper explores properties and applications of an ordered subset of the quadratic integer ring Z[ (1+ √ 5) / 2 ]. The numbers are shown to exhibit a parity triplet, as opposed to the familiar even/odd doublet of the regular integers. Operations on these numbers are defined and used to generate a succinct recurrence relation for the well-studied Fibonacci diatomic sequence, providing the means for generating analogues to the famed Calkin-Wilf and Stern-Brocot trees. Two related fractal geometries are presented and explored, one of which exhibits several identities between the Fibonacci numbers and golden ratio, providing a unique geometric expression of the Fibonacci words and serving as a powerful tool for quantifying the cardinality of ordinal sets. The properties of the presented set of numbers illuminate the symmetries behind ordinals in general, as well as provide perspective on the natural numbers and raise questions about the dynamics of transfinite values. In particular, the first transfinite ordinal ω is shown to be logically consistent with a value whose cardinality is dual: both zero and one. Considerations of these points and opportunities for further study are discussed.

- by Scott Tezlaf
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- Mathematics, Logic And Foundations Of Mathematics, Set Theory, Fractals
- by Amir Shafie
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- IEEE Student Member, Golden Section, Fibonacci Series, Fibonacci numbers
- by Carlos Alberto Riveros Varela
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- Fibonacci Series, Tiempo Continuo

In this paper we probe the interaction between sequential and hierarchical learning by investigating implicit learning in a group of school-aged children. We administered a serial reaction time task, in the form of a modified Simon Task in which the stimuli were organised following the rules of two distinct artificial grammars, specifically Lindenmayer systems: the Fibonacci grammar (Fib) and the Skip grammar (a modification of the former). The choice of grammars is determined by the goal of this study, which is to investigate how sensitivity to structure emerges in the course of exposure to an input whose surface transitional properties (by hypothesis) bootstrap structure. The studies conducted to date have been mainly designed to investigate low-level superficial regularities, learnable in purely statistical terms, whereas hierarchical learning has not been effectively investigated yet. The possibility to directly pinpoint the interplay between sequential and hierarchical learning is instead at the core of our study: we presented children with two grammars, Fib and Skip, which share the same transitional regularities, thus providing identical opportunities for sequential learning, while crucially differing in their hierarchical structure. More particularly, there are specific points in the sequence (k-points), which, despite giving rise to the same transitional regularities in the two grammars, support hierarchical reconstruction in Fib but not in Skip. In our protocol, children were simply asked to perform a traditional Simon Task, and they were completely unaware of the real purposes of the task. Results indicate that sequential learning occurred in both grammars, as shown by the decrease in reaction times throughout the task, while differences were found in the sensitivity to k-points: these, we contend, play a role in hierarchical reconstruction in Fib, whereas they are devoid of structural significance in Skip. More particularly, we found that children were faster in correspondence to k-points in sequences produced by Fib, thus providing an entirely new kind of evidence for the hypothesis that implicit learning involves an early activation of strategies of hierarchical reconstruction, based on a straightforward interplay with the statistically-based computation of transitional regularities on the sequences of symbols.

— This paper deals with an analysis of the different behaviors existing between ladder networks (LN) terminated and not terminated by the characteristic impedance Z0. The main purpose of this work is to find, in both cases, links with the golden ratio and, as a consequence, with Fibonacci numbers. Results of some simulations related to the determination of impedances, node voltages and branch currents are given in order to underline approximation effects on the node voltages and branch currents related to both the presence and absence of Z0.This study has been firstly applied to R-R ladder networks but it is here extended to consider other kinds of LNs, having other types of single cells such as CC ; L-L whose characteristic impedances have also been determined. Double resistive LN have also been investigated for which impedances in any cell, currents in any branch and voltages in any node have been determined in the two cases of presence and absence of the characteristic impedance and with different ratio between longitudinal and transversal impedances. This study has another not less relevant aim: find one of the possible platform for the modelization of biosystems like DNA and RNA, because these structures even if much more complex look like a LN Index Terms—Passive ladder Network, DNA,RNA .

- by WJRR Journal
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- Fibonacci Series, Fibonacci numbers, Fibonacci Sequences
- by Zlatina Kazlacheva
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- Design, Fashion design, Fibonacci Series, Fibonacci numbers

There is an at times heated debate between mainstream archaeologists and historians, and alternative researchers, about the age of the monuments at Giza. Mainstream produces Carbon 14 tests and ancient histories, written long after the... more

- by Ian Douglas
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- History of Mathematics, Fibonacci Series, Giza, Golden Ratio

Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q} = (p^n - q^n)/ ( p - q) = F_n $ coincide
precisely with the Fibonacci numbers (integers), as a result of Binet's formula when
$ p = \tau = { 1 + \sqrt 5 \over 2}$, $ q = { \tilde \tau} = { 1 - \sqrt 5 \over 2 }$ (Galois-conjugate pairs), we extend this result to
the generalizedgeneralizedgeneralized Binet's formula (corresponding to generalized Fibonacci sequences) studied by Whitford. Consequently,
the Galois-conjugate pairs $ (p, q = \tilde p ) = { 1\over 2} ( 1 \pm \sqrt m ) ,intheveryspecialcasewhen, in the very special case when ,intheveryspecialcasewhen m = 4 k + 1$ and square-free,
generalize Binet's formula $ [ n ]_{ p, q} = G_n$ generating integer-values for the generalized Fibonacci numbers GnG_nGn's. For these reasons, we expect that the two-parameter $ (p, q = \tilde p)$ quantum calculus should play an important role in the physics of quasicrystals with 4k+14k+14k+1-fold rotational symmetry.