Fibonacci Numbers, Golden Section, Golden Angle, Golden Rectangle and Golden Spiral

2014 ◽  
pp. 125-138
Author(s):  
Asok Kumar Mallik
Keyword(s):  
Golden Section ◽  
Fibonacci Numbers ◽  
Golden Angle ◽  
Golden Rectangle

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 Symmetry in plants: from Bravais' lattices to an explanation of Fibonacci phyllotaxis.

2014 ◽  
Vol 70 (a1) ◽  
pp. C1420-C1420
Author(s):  
Christophe Gole
Keyword(s):  
Computer Simulations ◽  
Fibonacci Numbers ◽  
Mathematical Work ◽  
Simple Explanation ◽  
Seminal Work ◽  
Plant Samples ◽  
Golden Angle ◽  
Accretion Model

In 1837, fourteen years before publishing his seminal Etude sur la cristallographie Auguste Bravais and his brother Louis wrote an equally seminal work on the arrangement of leaves around the stem of a plant. In this paper, one of the very first truly bio-mathematical work, they introduce and analyze cylindrical lattices and conjecture that only those with the golden angle between successive leaves can exhibit the Fibonacci numbers of spirals predominant in plants. With the advent of the microscope, and following  observations of the plants growing tips by Hofmeister, botanists Schwendener and van Iterson  developed an accretion model of the plant structures. Their work use ideas of what we now would call renormalization of morphogenetic fronts to understand transitions between successive Fibonacci pairs. This gives rise to a  simple explanation of the omni-presence of Fibonacci numbers in plants, which can be verified on digitized plant samples and with systematic computer simulations. This could inform crystallographers in their study of dislocations.

scholarly journals Aesthetic preference of object position on pictures

Psihologija ◽  
2003 ◽  
Vol 36 (3) ◽  
pp. 313-330 ◽  
Author(s):  
Oliver Toskovic ◽  
Slobodan Markovic
Keyword(s):  
Aesthetic Experience ◽  
Golden Section ◽  
Aesthetic Preference ◽  
Golden Rectangle ◽  
Black Circle ◽  
Different Shapes ◽  
The Aesthetic ◽  
The Right ◽  
The Given ◽  
Optimal Effect

In this study three hypothesis were evaluated. The first claims that the golden section position is an ideal position of an object on a picture and that this position does not depend on picture shape, or on the number of objects on it. According to the second hypothesis, the aesthetically optimal effect is achieved when the focus is on the right side of the picture ( for asymmetrically composed pictures). According to the third hypothesis, there is an influence of previous stimulation on aesthetic experience; that is, because of the monotony, the aesthetic preference of observers will change. An experiment was done, with two sections. In the first section, subjects were asked to put a little black circle, on three different shapes of cards (square, golden rectangle and rectangle), in a such way that the given configuration is the most beautiful one in their own opinion. The second section of the experiment was almost identical to the first one, with the exception that the subjects were asked to put two circles on each of the cards. Each one of the three hypothesis was confirmed by the results of this experiment. The preferred position of the circle is the same as the position of the golden section and it does not change with the change of card shape and number of objects. There is a clear preference of the upper-right corner of cards. The preferred position of an object is changed with repetition of the same stimulation (the same shape of cards and the same number of circles).

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.

From the Golden Rectangle and Fibonacci to Pedagogy and Problem Posing

1976 ◽  
Vol 69 (3) ◽  
pp. 180-188
Author(s):  
Stephen I. Brown
Keyword(s):  
Number Theory ◽  
Thirteenth Century ◽  
Golden Section ◽  
Golden Ratio ◽  
Mathematical Journal ◽  
Problem Posing ◽  
Total Height ◽  
Regular Pentagon ◽  
Golden Rectangle ◽  
Relative Placement

There is a wealth of valuable material on the golden section, and “fallout” from it as well, Indeed, the problem of constructing the golden rectangle with straightedge and a pair of compasses, and its relationship to constructing the regular pentagon with similar tools, is documented in Euclid's Elements (Euclid 1956; vol. 1, bk. 2, prop. 11; vol. 2, bk. 4. prop. 11). In the thirteenth century, the Italian mathematician Leonardo Fibonacci found connections between that aspect of geometry and number theory, and today there is a mathematical journal that is devoted exclusively to such issues. The golden section has appeared not only in mathematics but in architecture and art as well. The Parthenon in Greece, for example, has the proportions of the golden rectangle, and various portions of Michelangelo's David, from the joints of the fingers to the relative placement of the navel with respect to the total height, exemplify the golden ratio.

Golden Section’s Numerical Approximations in Terms of Graphics and Application

10.12737/5585 ◽  
2014 ◽  
Vol 2 (2) ◽  
pp. 15-20 ◽  
Author(s):  
Сафиулина ◽  
Yu. Safiulina ◽  
Шмурнов ◽  
V. Shmurnov
Keyword(s):  
Line Segment ◽  
Da Vinci ◽  
Golden Section ◽  
Leonardo Da Vinci ◽  
Problem Solution ◽  
Numerical Approximations ◽  
Circle Problem ◽  
Golden Rectangle ◽  
Gradual Development ◽  
Original Scheme

Golden proportion’s graphic plotting methods have been considered. A history related to gradual development of views on Golden section problem as «law of beauty» is traced. Numerical ratios most frequently used in the art for approximations related to division of a line segment in extreme and mean ratio have been provided. An original scheme for «Golden rectangle» construction based on application offered by Leonardo da Vinci for the quadrature of circle problem solution has been proposed.

scholarly journals Stability of aesthetic preference of object position on pictures

Psihologija ◽  
2004 ◽  
Vol 37 (4) ◽  
pp. 507-525
Author(s):  
Oliver Toskovic
Keyword(s):  
Golden Section ◽  
Aesthetic Preference ◽  
Object Position ◽  
Golden Rectangle ◽  
Different Shapes

The aim of this work is is to give answers to question is aesthetic preference of object position on pictures stabile, or is there a change of aesthetic preference with increase of number of objects and with the change of picture orientation (horizontal-vertical). In conducted experiments subjects had a task to put one, two or three circles on three different shapes of backgrounds (square, golden rectangle, rectangle), in such way that given configuration is the most beautiful one in their own opinion. In some experiments backgrounds were observed horizontaly, and in other verticaly. When the backgrounds were horizontal, aesthetic preference of golden section position did not change with increase of the number of circles. When the backgrounds were vertical golden section position was prefered one in cases with one and two circles, while in the experiment with three circles aesthetic preference of golden section position decreased. In most situations circles were ordered on backgrounds in such way to balance each other. Distance between two circles on same shapes of backgrounds, on repeated situations, is relativly constant in both orientations of backgrounds.

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.

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