scholarly journals Chromatic zeros and the golden ratio

2009 ◽  
Vol 3 (1) ◽  
pp. 120-122 ◽  
Author(s):  
Saeid Alikhani ◽  
Yee-Hock Peng
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Chromatic Polynomial ◽  
Chromatic Roots

In this note, we investigate ?n , where ?=1+2?5 is the golden ratio as chromatic roots. Using some properties of Fibonacci numbers, we prove that ? n (n ? N), cannot be roots of any chromatic polynomial.

scholarly journals Density of Chromatic Roots in Minor-Closed Graph Families

2018 ◽  
Vol 27 (6) ◽  
pp. 988-998 ◽  
Author(s):  
THOMAS J. PERRETT ◽  
CARSTEN THOMASSEN
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Classical Result ◽  
Golden Ratio ◽  
Chromatic Polynomial ◽  
Closed Graph ◽  
Small Interval ◽  
Chromatic Polynomials ◽  
Graph Families ◽  
Chromatic Roots

We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around τ + 2 where τ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at τ + 2. Our results lead us to conjecture that τ + 2 is the only such number less than 4.

scholarly journals On Generalization of Fibonacci, Lucas and Mulatu Numbers

2020 ◽  
Vol 1 (3) ◽  
pp. 112-122
Author(s):  
Agung Prabowo
Keyword(s):  
Continued Fraction ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Binet Formula

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

The Grand Periodic Function

2018 ◽  
Author(s):  
Jan C. A. Boeyens
Keyword(s):  
Number Theory ◽  
Periodic Function ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Magic Number ◽  
Self Similarity ◽  
Covalent Interaction ◽  
Great Pyramid ◽  
Relationship Of ◽  
The Relationship

The discovery of material periodicity must rank as one of the major achievements of mankind. It reveals an ordered reality despite the gloomy pronouncements of quantum philosophers. Periodicity only appears in closed systems with well-defined boundary conditions. This condition excludes an infinite Euclidean universe and all forms of a chaotic multiverse. Manifestations of cosmic order were observed and misinterpreted by the ancients as divine regulation of terrestrial events, dictated by celestial intervention. Analysis of observed patterns developed into the ancient sciences of astrology, alchemy and numerology, which appeared to magically predict the effects of the macrocosm on the microcosm. The sciences of astronomy and chemistry have by now managed to outgrow the magic connotation, but number theory remains suspect as a scientific pursuit. The relationship between Fibonacci numbers and cosmic self-similarity is constantly being confused with spurious claims of religious and mystic codes, imagined to be revealed through the golden ratio in the architecture of the Great Pyramid and other structures such as the Temple of Luxor. The terminology which is shared by number theory and numerology, such as perfect number, magic number, tetrahedral number and many more, contributes to the confusion. It is not immediately obvious that number theory does not treat 3 as a sacred number, 13 as unlucky and 666 as an apocalyptic threat. The relationship of physical systems to numbers is no more mysterious nor less potent than to differential calculus. Like a differential equation, number theory does not dictate, but only describes physical behavior. The way in which number theory describes the periodicity of matter, atomic structure, superconductivity, electronegativity, bond order, and covalent interaction was summarized in a recent volume. The following brief summary of these results is augmented here by a discussion of atomic and molecular polarizabilities, as derived by number theory, and in all cases specified in relation to the grand periodic function that embodies self-similarity over all space-time.

scholarly journals Fibonacci, Golden Ratio, and Vector Bundles

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 426
Author(s):  
Noah Giansiracusa
Keyword(s):  
Lie Algebra ◽  
Moduli Space ◽  
Vector Bundles ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Theoretical Physics ◽  
Stable Curves ◽  
Exceptional Lie Algebra ◽  
Summation Formulas

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

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 On the l.c.m. of shifted Fibonacci numbers

2021 ◽  
pp. 1-10
Author(s):  
Carlo Sanna
Keyword(s):  
Rational Number ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Random Variables ◽  
Periodic Sequence ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Number Formula ◽  
Dilogarithm Function

Let [Formula: see text] be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that [Formula: see text] where [Formula: see text] is the least common multiple and [Formula: see text] is the golden ratio. We prove that for every periodic sequence [Formula: see text] in [Formula: see text] there exists an effectively computable rational number [Formula: see text] such that [Formula: see text] Moreover, we show that if [Formula: see text] is a sequence of independent uniformly distributed random variables in [Formula: see text] then [Formula: see text] where [Formula: see text] is the dilogarithm function.

The Divine Proportion, matrices and Fibonacci numbers

2008 ◽  
Vol 92 (523) ◽  
pp. 14-21
Author(s):  
H. Brian Griffiths
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
The Other ◽  
Matrix Approach ◽  
Mathematical Development ◽  
Divine Proportion

A friendly engineer recently sent me a version of Figure 1 below, and asked me to explain the connection between it and the well-known Fibonacci Problem (FP) about calculating a population of rabbits. He knew that Figure 1 was related to the ‘Divine Proportion’ or ‘Golden Ratio’ ϕ ( = ½(√5 - l), which also occurs in the solution to FP, and wondered how such different problems could be related by such a number. (He unfortunately regarded ϕ as 0.618 exactly, thus missing a lot of stuff to arouse curiosity.) I knew of various references that I could recommend, but none covered all the things my engineer mentioned, so I constructed the following mathematical development, leaving the relations with biology and architecture to be explained in the books referenced later, see for example [1]. A matrix approach is used here, and may be new to those readers of the Gazette who may be quite familiar with the other material.

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