scholarly journals The relationship between the sum of reciprocal golden section numbers and the Fibonacci numbers

2017 ◽  
Author(s):  
Haiquan Fang ◽  
Huifeng Xue ◽  
Tiejun Zhou
Keyword(s):  
Golden Section ◽  
Fibonacci Numbers ◽  
The Relationship

scholarly journals Il pentagono come strumento per il disegno delle fortezze

X ◽  
2020 ◽  
Author(s):  
Paola Magnaghi-Delfino ◽  
Giampiero Mele ◽  
Tullia Norando
Keyword(s):  
Fifteenth Century ◽  
Golden Section ◽  
Fibonacci Sequence ◽  
Isosceles Triangle ◽  
Luca Pacioli ◽  
Golden Triangle ◽  
The Military ◽  
The Relationship ◽  
Shape And Size

The pentagon as a tool for fortresses’ drawingStarting from the fifteenth century, the diagram of many fortresses has a pentagonal shape. Among the best known fortresses, in Italy we find the Fortezza da Basso of Florence, the Cittadella of Parma, the Cittadella  of Turin,  Castel Sant’Angelo in Rome. The aim of this article is to analyze the reasons that link form and geometry to the planning of the design and the layout of pentagonal fortresses. The pentagon is a polygon tied to the golden section and to the Fibonacci sequence and it is possible to construct it starting from the golden triangle and its gnomon. This construction of the pentagon is already found in the book De Divina Proportione by Luca Pacioli and is particularly convenient for planning pentagonal fortresses. If one wants to draw the first approximated golden triangle, one can just consider the numbers of the Fibonacci sequence, for example 5 and 8, which establish the relationship between the sides: 5 units is the length of the base and 8 units the length of the equal sides. In the second isosceles triangle, which is the gnomon of the first, the base is 8 units long and equal sides are 5 units long; half of this isosceles triangle is the Pythagorean triangle (3, 4, 5). This characteristic of the golden triangles, that was already known by the Pythagoreans and, in a certain sense, contained in the symbol of their School, allows to build a pentagon with only the use of the ruler and the set square. The distinctive trait of the construction just described makes preferable to use the pentagon in the layout of the military architectures in the fieldworks. We have verified the relationship between numbers, shape and size in the layout of Castel Sant’Angelo (1555-1559) in which the approximate pentagon was the instrument for the generation of its form.

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 On Insolvability of the 4-th Hilbert Problem for Hyperbolic Geometries

2019 ◽  
pp. 1-21
Author(s):  
Alexey Stakhov ◽  
Samuil Aranson
Keyword(s):  
Diophantine Equations ◽  
Hilbert Problem ◽  
Golden Section ◽  
Fibonacci Numbers ◽  
Fundamental Result ◽  
Minkowski Geometry ◽  
Russian Mathematician ◽  
Mathematical Result ◽  
Non Euclidean Geometry ◽  
Elliptic Geometry

The article proves the insolvability of the 4-th Hilbert Problem for hyperbolic geometries. It has been hypothesized that this fundamental mathematical result (the insolvability of the 4-th Hilbert Problem) holds for other types of non-Euclidean geometry (geometry of Riemann (elliptic geometry), non-Archimedean geometry, and Minkowski geometry). The ancient Golden Section, described in Euclid’s Elements (Proposition II.11) and the following from it Mathematics of Harmony, as a new direction in geometry, are the main mathematical apparatus for this fundamental result. By the way, this solution is reminiscent of the insolvability of the 10-th Hilbert Problem for Diophantine equations in integers. This outstanding mathematical result was obtained by the talented Russian mathematician Yuri Matiyasevich in 1970, by using Fibonacci numbers, introduced in 1202 by the famous Italian mathematician Leonardo from Pisa (by the nickname Fibonacci), and the new theorems in Fibonacci numbers theory, proved by the outstanding Russian mathematician Nikolay Vorobyev and described by him in the third edition of his book “Fibonacci numbers”.

scholarly journals Structural-spatial scheme of the Tree of Life in the topographical iconicity of the city–temple–icon–human

2019 ◽  
pp. 178-187
Author(s):  
O. Osadcha
Keyword(s):  
Spatial Structure ◽  
Golden Section ◽  
Tree Of Life ◽  
Human Soul ◽  
Structural Principle ◽  
The Moment ◽  
The Mind ◽  
The Relationship ◽  
The City ◽  
The Temple

The article reveals regularities between the spatial structure of the city-temple-icons and the similar structural principle, which, in the context of Hesychast anthropology, acts in the topography of the human soul. The spatial structure of the Tree of Life, a universal symbol contained in the topographical icons of the level and of the city, temple, icon, and human, is developed and proposed. It is proved that the spatial framework of the Tree of Life is the Golgotha ​​Cross. Considerable attention is paid to the analysis of the main spatial zones of the temple-icons, which have a hierarchical construction. It is assumed that the topographical icon of the city-temple-icon-human is arranged in such a way that it is possible to overcome the ontological gap that was created as a result of original sin. Particularly with the help of distinct geometric constants that determine the structure of the Tree of Life, ancient iconographs tried to restrain/seal the gaping hole, which seemed to be an insurmountable Rubicon, at the moment of the fall between the Spirit and the soul, the mind and heart of man, earthly and divine, profane and sacred worlds. Consequently, the use of sacred numbers was deliberately incorporated into sacred texts, icons, and in the architecture and iconographic programs of the temples. It was analyzed that the internal structure/main sacral energy framework of the icon-temple contains compositional nodes associated with the disclosure of the main semantic load in the iconographic program/plot, and are always constructed on the lines of the golden section. Some regularities in the placement of the central figure in the composition of the temple icon are traced. In the temple, as in the icon, the semantic center of the sacred space is the image of Christ the Almighty, who is placed in a top of an equilateral triangle with a side size corresponding to the width of the temple. The center of the Nimbus passes through the golden section. In the context of the relationship between the topography of the icon-temple and the proposed scheme for determining the topography of the human soul. According to the analogy principle, the structural-spatial scheme of the Tree of Life in the anthropological aspect is associated with the stages of the spiritual perfection of the human soul.

scholarly journals On unique stone beaches on the Arctic coast of the Kola Peninsula

Vestnik MGTU ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 46-56
Author(s):  
Y. N. Neradovsky ◽  
Y. A. Miroshnikova ◽  
A. A. Kompanchenko ◽  
A. V. Chernyavsky
Keyword(s):  
Barents Sea ◽  
Golden Section ◽  
Stable State ◽  
The Arctic ◽  
Solid Core ◽  
Clastic Material ◽  
The Barents Sea ◽  
Beach Sediments ◽  
Arctic Coast ◽  
The Relationship

The results of studies of 11 stone beaches on the coast of the Barents Sea in the area of the Teriberskaya Bay have been presented. The studies were carried out from 2017 to 2019. As a result of the work, the structure of the beaches, their size, the composition of clastic material and the relationship with bedrocks were studied in detail. The genetic link between beaches and sea terraces has been established. Special attention has been paid to the morphology of beach clastic material, the conditions of its formation, and its role in abrasion activity. It has been shown that the clastic material of the beaches mainly corresponds to boulders equal to 100-1,000 mm, to a lesser extent to pebbles 10-100 mm, and rarely - gravel 1-10 mm. Individual boulders reach 2,000 mm. Sandy fractions in the composition of beach sediments are practically absent. The roundness of the fragments is high, semi-circular and rounded grains predominate, the most perfect shape of the rounded fragments is a biaxial ellipsoid or egg. Perfectly rounded boulders and pebbles in some areas account for up to 30 % of beach deposits. Measurements of the parameters of the egg-shaped pebbles have shown that they are close to the parameters of the "golden section" of the egg, i. e. meet the most durable form, resistant to destruction. Thus, the process of abrasion of the beach debris is directed towards their acquisition of the most energetically stable state. This suggests that the original shape of the debris contained a solid core in the form of a biaxial ellipsoid.

Exploratory Calibration of a Retail Location Model Using Search by Golden Section

1971 ◽  
Vol 3 (4) ◽  
pp. 411-432 ◽  
Author(s):  
M Batty
Keyword(s):  
Goodness Of Fit ◽  
Spatial Interaction ◽  
Golden Section ◽  
Fibonacci Numbers ◽  
Location Model ◽  
Fundamental Result ◽  
Calibration Methods ◽  
Retail Location ◽  
Two Parameters ◽  
Parameter Values

This paper presents some experiments in calibrating a retail location model designed for the Kristiansand region in southern Norway. The form and behaviour of the model under extremes of its parameter values is first investigated, and then some calibration methods proposed by Hyman are tested. The need for a more general method is evident and a search procedure based on the Fibonacci numbers is outlined. A generalisation of this method, based on the golden section number, is derived and this method is used to explore the sensitivity of the model's goodness-of-fit to changes in parameter values. A fundamental result of these explorations shows that there is no unique fit for retail models with two parameters, and this result is likely to hold for other models of spatial interaction.

scholarly journals Generalized Fibonacci Search Method in One-Dimensional Unconstrained Non-Linear Optimization

2021 ◽  
Vol 29 (2) ◽  
Author(s):  
Chin Yoon Chong ◽  
Soo Kar Leow ◽  
Hong Seng Sim
Keyword(s):  
Golden Section ◽  
Linear Optimization ◽  
Fibonacci Numbers ◽  
Search Method ◽  
Initial Ratio ◽  
Microsoft Excel ◽  
One Dimensional ◽  
Golden Section Search ◽  
Non Linear ◽  
Number Of Iterations

In this paper, we develop a generalized Fibonacci search method for one-dimensional unconstrained non-linear optimization of unimodal functions. This method uses the idea of the “ratio length of 1” from the golden section search. Our method takes successive lower Fibonacci numbers as the initial ratio and does not specify beforehand, the number of iterations to be used. We evaluated the method using Microsoft Excel with nine one-dimensional benchmark functions. We found that our generalized Fibonacci search method out-performed the golden section and other Fibonacci-type search methods such as the Fibonacci, Lucas and Pell approaches.

Genetic System, Fibonacci Numbers, and Phyllotaxis Laws

2010 ◽  
pp. 207-221
Author(s):  
Sergey Petoukhov ◽  
Matthew He
Keyword(s):  
Mathematical Biology ◽  
Genetic Code ◽  
Golden Section ◽  
Fibonacci Numbers ◽  
Biological Evolution ◽  
Genetic System ◽  
Huge Number ◽  
Matrix Methods ◽  
Long Time

This chapter describes data suggesting a connection between matrix genetics and one of the most famous branches of mathematical biology: phyllotaxis laws of morphogenesis. Thousands of scientific works are devoted to this morphogenetic phenomenon, which relates with Fibonacci numbers, the golden section, and beautiful symmetrical patterns. These typical patterns are realized by nature in a huge number of biological bodies on various branches and levels of biological evolution. Some matrix methods are known for a long time to simulate in mathematical forms these phyllotaxis phenomena. This chapter describes connections of the famous Fibonacci (2x2)-matrices with genetic matrices. Some generalizations of the Fibonacci matrices for cases of (2nx2n)-matrices are proposed. Special geometrical invariants, which are connected with the golden section and Fibonacci numbers and which characterize some proportions of human and animal bodies, are described. All these data are related to matrices of the genetic code in some aspects.

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