Golden Section

В докладе рассматриваются статистические закономерности распределения ритмических масс по строкам и строфам русского классического сонета. Основной материал – 200 сонетов из сборника К.Д. Бальмонта «Сонеты Солнца, Меда и Луны». Сформирован усредненный геометрический профиль динамики ритма русского сонета и установлена корреляционная связь между ритмической структурой и смыслом сонета. Исследована связь между ритмическими и лингво-полиграфическими характе-ристиками сонета: размером слова, консонантным коэффициентом и др.
          Исследование рассчитано на широкий круг филологов-русистов, стиховедов, искусствоведов, специалистов в области структурной, прикладной и квантитативной лингвистики.

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.

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.

Introduction – Geometry and Architecture in Milan, design between art and science


Printing an architectural guide on paper in this “tablet” era, might seem anachronistic.
The idea of preparing this volume, which is not meant to be comprehensive and contains only a selection of texts dedicated to the search for the hidden mathematics in Milan’s Architecture, was prompted by the fact that the organisation of the 9th edition of the Nexus Conference - Relationship between Architecture and Mathematics –will take place at the Milan Polytechnic.
The foreign guests, just like any other guest, will be able to read the city with slightly different eyes than the average tourist; the scholars might find a series of suggestions for further investigations related to the relations that, in time, have bonded this city with mathematics, leaving indelible marks on its monuments.
Milan has always been a dynamic and active city. The political importance of the Roman capital, the medieval township, the princely dukedom, has triggered a vivid development of sciences and arts, and provided an application field that has insured the birth of a technical culture always ahead of times, even when the political dominance gave place to the economical power.

One of the reasons why Milan is nowadays still economically relevant in the European life, as one of its main business centres, is because the city, from the Middle-Ages on, has always invested in economy, attracting important scholars and promoting scientific researches that had a relevant impact on arts, enriching the city with great artworks hidden beneath the hectic urban context, looking towards the future whilst proudly preserving its past.

During the Middle Ages, as much as in the Renaissance, art and science were profoundly bound to one another. Art was influenced by science and, often, Architecture expressed an embedded knowledge that went far beyond the mere building techniques requirements.
Technique - the relation between knowing and doing – as an application field of science, brought art into the every day life, promoting the technological progress from which industry was born and that has always been one of the strengths of western culture.
At the peak of the Renaissance, Cesariano stated that architecture was a science, and as such it had to rely strictly on theory. The fundamental scientific fields were arithmetic and music, expressed by proportional ratios between numbers.
After a while, the union between art and science dissolved in the contemporary predominance of technique with the enhancement of technology as a unique value of scientific research.
Art parted from science and technique but its playful relation with theoretical knowledge and exact sciences, especially mathematics, never ceases to fascinate.
Among the arts, architecture is the one that developed the strongest relation with mathematics, especially in the use of geometry as a control tool of design.
The reference to geometry in historical monumental architecture is not exceptional: however, Milan’s architecture shows precise recurring original features, such as, for example, the high number of churches and other Renaissance buildings with an octagonal floor plan, which have never been studied in their entirety.
This small collection of significant examples of relationship between mathematics and architecture offers, in the whole, a different image of the city and its monuments, which acquire a different meaning. Furthermore, the research of the mathematical features of existing and planned monuments underlines the paramount part that science had in the technical development (and therefore economical) of the city.

The present collection of texts is incomplete due to the fact that it reports the results of a work in progress, and sums up researches that were brought along at a larger scale.
In fact, these preliminary considerations on some of the many case-studies that the city offers aim at promoting a greater attention towards this kind of investigation that focussed so far only on a few examples.
Our purpose is to try to unveil, widely speaking, the historical grounding of mathematics in the architecture of a city, and the thin common thread that crosses the centuries from the Middle Ages to the Rationalism, searching for the rules of the perfect design.
This volume does not report the results of a finished investigation, but intends to be the starting point of a new one.

In the theory of architecture, there is probably no more hotly debated and controversial issue than the use of the golden section as a tool for governing the proportions of forms and spaces. In this article, the author shows that the work of the Venetian architect Carlo Scarpa has its roots in the classical theory of proportions. He examines two drawings by Scarpa, demonstrating their application of harmonic proportions to the museum space and the close ties between it and the art works on display. Unlike Le Corbusier, perhaps the most important modern master to have used the golden section in his designs, Scarpa employs this proportional system in a pragmatic and experimental way, applying it only in places of special importance. This is true of the “small masterpieces” gallery in the Gallerie dell’Accademia and of the Main Lecture Theatre at IUAV, again in Venice. Scarpa thus reveals two important principles of his work: that small size is an essential premise for attempting perfection and, more generally, that the architectural project is a matter of visual perception based on the quest for the “right proportion”.

Background in analysis. In an earlier study (Harper 2007) the direct occurrence of the crux in phi (φ) proportion was observed in a large percentage of two contrasting bodies of Scarlatti sonatas (Essercizi and Cantabile sonatas). In the Sonata in E Major, K. 380-a sonata with distinctive folkloric characteristics-direct occurrence of the crux-phi relationship is also found in exact mathematical proportion in both halves of the sonata. While it is not known if this proportion is deliberately conceived, Scarlatti's structural construct is evidenced in and is the basis of this work. Background in performance. Ten different recorded interpretations (Horowitz, Asperen, Smullyan, Browning, Pletnev, Coleman, Fadini, Li, Lipatti, Gilels) were chosen for comparison in K. 380 with performers on harpsichord, fortepiano, and piano. The digital audio editor Audacity 1.3.3 was used to study the performances. After repeats and extra audio materials being removed, timings ranged from the longest (Horowitz and Asperen) at 3'10"9 to the shortest (Gilels) at 2'31"5. Expressive elements, such as variations in dynamic contrasts and rubati, are clearly discernible. The crux was compared in the ten performances first in real time and then in equalized time using a time-based analytical approach.

The Golden ratio and Fibonacci sequence are used in proportions in design as symbols of harmony and beauty. The paper presents the use of tilings in spiral forms, created on the base the Golden ratio and Fibonacci sequence, in fashion design. The tilings are the Golden spiral tilings with Golden rectangles and Golden triangles and Fibonacci spiral tilings with rectangles, squares and triangles. They are the Golden spiral tilings with Golden rectangles and Golden triangles and Fibonacci spiral tilings with squares and equilateral triangles.For creation of successful aesthetic fashion design projects these kinds of tilings can be used: in different situation according to the clothing and the body; in different proportions according to the clothing and the body; In two, three or four colors combinations; and in rhythm on the base mirror or translating symmetry. In fashion design the Golden and Fibonacci spiral tilings can be used directly, as frames for real geometrical spirals or entered elements.

In accordance to studies by G.S.Hawkins, the author of this paper continued decoding of ancient monument Stonehenge in Great Britain. The method of decoding by J.-F.Champollion was used with success. The author discovered mathematical code and several images which look like Egyptian hieroglyphs. The author worked out the vocabulary of Stonehenge. It became possible to decode several words and short message with the help of the vocabulary. Mathematical theory of Hydrogen atom and atomic mass (1.0079...) were coded in Stonehenge. A part of message looks like the phrase: "Eternally living Atom". Where Atom (or Atum) is well-known name of Egyptian solar god. Also knowledge of partial differential equations is demonstrated by builders of Stonehenge. The author considers that Stonehenge was distant Egyptian solar temple and ancient center of science and higher education simultaneously. Probably Stonehenge was used for military purposes and for forecasting of results of wars with the help of mathematical modeling too. It explains military power of ancient Egypt. Also the author describes process of building of Stonehenge with the method of hydraulics.

Intentionally or unintentionally, from ages, architects, builders and construction experts have used mathematics as a very basic yet important tool for the soulful purpose of design, execution and finalization of building projects. In the history, architects were mathematicians and also some mathematicians were architect too. Vitruvius was a very well-known architect as well as famous mathematician. Mathematical readings of Pythagoras were later used in building proportions. Well known worker and user of golden ratio Leonardo Da Vinci along with many achievements was an architect too. The approach of this research paper is to come up with findings on importance of mathematics in architecture, as in geometry, from very important site analysis to final design of elevation or façade. Aim of the whole research is to come up with mathematical functions related to mensuration of building construction and Architectural Engineering. This paper is an initial part of the same research.

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.

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.

Golden Proportion or Golden Ratio is usually denoted by the Greek letter Phi (φ), in lower case, which represents an irrational number, 1.6180339887 approximately. Because of its unique and mystifying properties, many researchers and mathematicians have been studied about the Golden Ratio which is also known as Golden Section. Renaissance architects, artists and designers also studied on this interesting topic, documented and employed the Golden section proportions in eminent works of artifacts, sculptures, paintings and architectures. The Golden Proportion is considered as the most pleasing to human visual sensation and not limited to aesthetic beauty but also be found its existence in natural world through the body proportions of living beings, the growth patterns of many plants, insects and also in the model of enigmatic universe. The properties of Golden Section can be instituted in the pattern of mathematical series and geometrical patterns. This paper seeks to represent a panoptic view of the miraculous Golden Proportion and its relation with the nature, globe, universe, arts, design, mathematics and science. Geometrical substantiation of the equation of Phi, based on the classical geometric relations, is also explicated in this study. Golden Ratio and its chronicle, concept of Golden Mean and its relations with the geometry, various dynamic rectangles and their intimacy with Phi, Golden Ratio in the beauty of nature, Phi ratio in the design, architecture and engineering are also presented in this study in a panoptical manner.

For over 150 years, many people have claimed that the number popularly known as the Golden Section has particularly appealing aesthetic properties when incorporated into the proportions of visual artworks. Although the empirical evidence for these claims has been equivocal, in the 1980s an important mathematical argument seemed to show that many, perhaps nearly all, empirical discoveries of the legendary number in art and nature were probably mathematical artifacts. This article overturns that argument as being, itself, the artifact of an unrealistic assumption.

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.

The Golden ratio and Fibonacci sequence are often used in the design for creation of beautiful and harmonic forms. The papers presents design of ladies' dresses using the both ways of constructing of Fibonacci series tiling with squares. As a result of the use of the both versions of Fibonacci series tiling with squares for designing of aesthetic, beautiful and harmonic clothing, it can be concluded that in fashion design Fibonacci squares can be used in different position, combinations, proportions toward the clothing sizes, and color decisions.

This article deals with quantitative and typological approaches to Fibonacci Sequences. The presented typology is based on methods used in mathematical linguistics. It has a shape of triangle, in the apex of which there is a classic golden section equation. The triangle presents equations of arbitrary power. Coefficients of these equations form a strict mathematical system based on the Golden section and Fibonacci numbers.

В статье рассматривается квантитативно-типологический подход к последовательностям Фибоначчи. Типология основана на структурном подходе, принятом в математической лингвистике. Она имеет вид треугольника, вершиной которого является классическое уравнение золотого сечения. В треугольнике представлены уравнения произвольных степеней, коэффициенты которых образуют строгую математическую систему, основанную на золотом сечении и числах Фибоначчи.

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

The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more "Metallic" than the Pentagons, Octagons or any other (n 2 +4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means.

The Golden triangle is an isosceles triangle with the ratio of the base to the leg, which is equal to the Golden section or Golden ratio. Two quantities are in the Golden ratio (φ) if the ratio of the smaller quantity to the larger one is equal to the ratio of the larger quantity to the sum of the both quantities, or φ = 0,618. It is considered that the proportion of Golden section is a symbol of the beauty and harmony. By this reason this ratio is used in art and design for creation of aesthetically beautiful and harmonic forms. The paper presents the application of the Golden triangle in fashion design and pattern making of lady"s clothing. In fashion design and pattern making, as a precondition of aesthetic and harmonic forms in clothing, the Golden triangle can combine proportioning and forms creation and it can be used directly as shapes or frames of designing of details and elements. The use as frames is a possibility of creation of interesting models in successful proportions. The pattern making of clothing with the Golden triangle is more successful with the use of CAD systems about easily creating the triangle by angles and easily designing of new shapes of the details and elements on base of the use of the triangle as a shape.

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.

The Golden Section is a geometric ratio often touted for its aesthetically perfect proportions and its use as a design blueprint found in both nature and in iconic works of architecture throughout history. The Parthenon is often cited as the prototype par excellence of Golden Section architecture, though such assertions are not supported by empirical studies. Among twentieth century architects it is most commonly associated with Le Corbusier and his Modulor system. However, two of his lesser known contemporaries - Dom Paul Bellot, a Benedictine monk-architect, and Sigurd Lewerentz, an enigmatic Swede - also created works in which the Golden Section was applied, but with adjustments and in combination with other dimensions as part of unified systems of proportion. With much less fanfare, they show how the Golden Section can be applied in brick and mortar, but also reveal the limits of this divine proportion.

Considering the Madonna with the Child by Bartolomeo Vivarini kept in the throne hall of Palazzo Colonna in Rome and examine this for the first time in its entirety, is a very delicate operation today due to the vast bibliography dedicated to the venetian artist. This tempera on wood, signed on the cartouche on the base - 1471 -, shows the artist's "sculptural" intentions to use colors to recreate refined compositions of Venetian gold carpentry, present in other works by him. Until now little considered by critics, the panel brings to light new reflections on the use of gold, which in this case are no longer veterans of the International Gothic but new ideas for perspective and light in two-dimensional works: gold is not only synonymous with the Holy Spirit and Celestial Glory but becomes a technical tool and material that gives shape to the figures.

This paper intends to concentrate on the artists applying the Golden Ratio in the contemporary art, however, indicates, a quick inspiration for some of artists applied the Golden Ratio in the past and the modern era art. accordingly, the research argues the ways which the artists in the contemporary art applying the Golden Ratio on their artworks, which is known that the Golden Ratio is a highlighted path for any artists looking for success in his artwork.

The special case of the (р+1)th degree algebraic equations of the kind xр+1= xp+1 (р=1, 2, 3, ...) is researched in the present article. For the case р=1, the given equation is reduced to the well-known Golden Proportion equation x2=x+1. These equations are called the golden algebraic equations because the golden p-proportions р, special irrational numbers that follow from Pascal’s triangle, are their roots. There is researched the general properties of the roots of the golden algebraic equations in this article. In particular, formulas are derived for the golden algebraic equations that have degree greater than р+1. There is reason to suppose that algebraic equations derived by the authors in the present article will interest theoretical physicists. For example, these algebraic equations could be found in the research of the energy relationships within the structures of many compounds and physical particles. For the case of butadiene (C4H6), this fact is proved by famous physicist Richard Feynman

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.

It's about a graduation conclusion work (in Portuguese) to obtain a bachelor's degree in Industrial Design (2011 – 2015) at Paulist Stadual University (UNESP).

“Harpsichord, one of the ancestors of the modern piano along with the clavichord, flourished in the 15th to 18th century in Europe, having previously its construction planned in a artisanal/experimental way.
In mid-20th century several musicologists and experts as Frank Hubbard and Wolfgang Zuckermann, among other personalities, decided to revive the instrument when compiling his former joinery methods, maintaining, however, its manufacturing traditions.
Once the rapid prototyping (RP) is a technology that allows savings in time and more accurated precision in manufacturing products, there's the possibility to devise a new method which allows exploring domestic materials (about Brazil) and making the instrument accessible to students or admirers of Early Music, besides generate new literature on the subject.
By owning the multidisciplinary nature, this project demanded a bibliographical study on the history and construction of instruments, concepts of Golden Section, sketches for a new product, studies of tuning, music theory and Wave Physics. Finally, there's the presentation of pieces of a prototype for review before public.”

Klang ist hörbare Bewegung. Erschallt Klang im Raum und trifft auf dessen Begrenzungen, entsteht Echo: ein hör­barer Vorgang in Verbindung mit räumli­cher Vorstellung. Steigert man die Kom­plexität dieser Vorgänge und lässt aus Klang Musik und aus Raum Architektur werden, so sind weiterhin "Echos" erkennbar, die Klang und Raum, die Musik und Architektur miteinander ver­binden. Unsere Arbeit soll einen thema­tischen Einstieg in die Vielfalt dieser gemeinsamen Punkte vermitteln, mit der Mathematik als Ausgangspunkt. Abschluss stellt eine seh- und hörbare Verknüpfung von Musik und Architektur mit Hilfe des Computers dar, basierend auf der Analyse der Arbeit des Architek­ten und Komponisten lannis Xenakis.

This research provides more than 35 measurements rules derived from the perspectives of Vitruvian Man and Neufert and their basis of the golden proportion, to build a human body model on computers for the use of multimedia. The measurements are based on 25 proportional rules derived from 15 proportions given by Vitruvian Man and 29 golden proportions in Bauentwurfslehre by Ernst Neufert. Furthermore, the research will suggest two algorithms to calculate the 67 measurements with precision; assuming that the algorithms output will be used as guideline to human body modelers in simulation, gaming, plastic surgery, as well as the world of biometrics or wherever human body measurements and calculations is needed like prosthetic limbs, spatial design, and machine learning of human biometrics. Furthermore, building proportional models creates visual harmony in measurements and visual parity model. Hence, the chapter facilitates and explains for the human modeler the process of human modeling from within an algorithm. This research is an expanded work based on two published conference papers listed in the references section.

Modern natural science requires the development of new mathematical apparatus. The generalized Fibonacci numbers or Fibonacci р-numbers (р=0, 1, 2, 3...), which appear in the “diagonal sums” of Pascal’s triangle and are assigned in the recurrent form, are a new mathematical discovery. The purpose of the present article is to derive analytical formulas for the Fibonacci р-numbers. We show that these formulas are similar to the Binet formulas for the classical Fibonacci numbers. Moreover, in this article, there is derived one more class of the recurrent sequences, which is defined to be a generalization of the Lucas numbers (Lucas p-numbers).

This publication deals with the metamorphoses of the Brisbane Victoria Bridge. It includes an Appendix Palladiana where the ideological "matrix" of the Palladian Manifesto Villa "La Rotunda" is said to be the "Tetraktys" (the upper numbers/part of the Pythagorean Lambda). Furthermore the root two rectangle, Serlio's "proportione diagonea", is also said to be at the basis of the Villa's proportionality patterns (see Appendix Palladiana - with figures - Palladian proportionality patterns), "along all axes" (Morgan, 1960), making the masterpiece "triaxially symmetrical", with a section a la "doron" (see Vitruvius and March, 1998), which, according to the author, has given origin to the famous name of the Golden Section (in Italian: Sezione D'Oro - originally allegedly called a "Sezione 'Doron' ").

The goal of the present atricle is to develop the “continues” approach to the recurrent Fibo-nacci sequence. The main result of the article is new mathematical model of a curve-linear space based on a special second degree function named “The Golden Shofar”.

The field theory of gravitation is developed on the basis of Kepler's third law. A formula with a golden section is obtained under the condition of considering complex solutions as real ones. The article shows the correspondence of the formula with the golden section to tabular data. The exact formula is obtained by considering a three-dimensional space as imaginary with a real plane perpendicular to the observer's attention vector.

This work consists of a preliminary look at the relationship between Phi and the spiral scaling structures that that can be seen in DNA and the cosmos. It is pointed out that fractals have a built in structural self similarity as part of their make-up. The fractal dimension isn't an integer but is a number that has a fractional part. The mathematical properties of Phi are studied in various ways as series and product series. Series expansion of Phi using fractional exponents were determined in several ways in an attempt to determine the difficulty in handling such expansion

In this study the efficiency of heat engines and the coefficient of performance of heat pumps are reconsidered following the idea of the golden section ratio. Two models are analyzed: the golden section Carnot cycle and the golden section endo-reversible cycle. The observed expressions for efficiency and coefficient of performance are cast in terms of the golden section ratio. Single (steam turbines[40Wilfried,

Геометрический анализ пропорций древнерусских христианских храмов X – XV вв. при помощи символической задачи «квадратура круга» выявил тесную связь с философско-богословскими понятиями и позволил описать общий принцип разбивки внутренних габаритов планов, подкупольного квадрата и построение основных высотных отметок, как в трёхнефных церквях, так и в многонефных соборах. Геометрические решения задачи «квадратура круга» дали возможность систематизировать и описать чаще всего встречающиеся отношения ширины к длине в интерьерах трёхнефных церквей: 8:9, 10:13, 2:3, 5:8, и объяснить пропорции основных символических делений в храме относительно ширины: «мир видимый» от входа до иконостаса; «мир невидимый» за иконостасом в алтаре, строящийся половиной круга символического круга «неба» из геометрического решения задачи «квадратура круга» в храмах с отношением ширины к длине как 10:13, 2:3, 5:8. В планах многонефных соборов и церквей с притворами выявлено структурное ядро, представляющее простой тип девятиячеистого средневизантийского храма с пропорциями ширины к длине в интерьере 10:13. Геометрический анализ позволил подтвердить существование в центральном подкупольном пространстве «Животворящего столпа», о котором имеются упоминания в богословских, летописных и иконографических источниках. Вертикальные пропорции столпа до центральной точки свода центральной главы кратны половине ширины трёх центральных нефов и составляют пропорции 1:3, 1:4, 1:5 или относительно полной ширины 5:8. Площадь же круга являющегося горизонтальным сечением этого столпа равна площади подкупольного квадрата в 75% древнерусских храмов X – XV вв., а в остальных 25% храмов диаметр столпа равен диагонали подкупольного квадрата. Пространственный образ «Животворящего столпа» предстаёт перед нами как река Света Отца льющаяся с небес в центральном подкупольном пространстве, которая разделяется на два рукава по описаниям Святителя Григория Богослова (IV в.), и образует внутреннюю ширину трёх центральных нефов. Геометрическая запись этого христианского понятия Святой Троицы через сплетённые три круга обнаруживается в построении пропорций древнерусских храмов X – XV вв. и триконхе на горе Нево в Иордании IV в. Подтверждение существования геометрического способа начертания планов христианских храмов мы получаем и при анализе храма св. Иоанна Крестителя в Иерусалиме к.VI – VIII вв., где в изначально построенном сооружении и в последующей перестройке, визуализируются триадологические споры и борьба с ересью арианства.
The geometric analysis of proportions of ancient Russian churches of the Tenth–Fifteenth centuries by solving the symbolic problem of the “quadrature of circle” revealed close connection with philosophical and theological concepts and made it possible to describe the general principle of layout of internal plan dimensions, the dome square and main heights both in three-nave and multi-nave temples. Geometric solutions of the problem of the “quadrature of circle” enabled to systematize and describe the most common width-to-length ratio in the interior of three-nave churches: 8:9, 10:13, 2:3, 5:8, as well as to explain the proportions of the main symbolic separation in the church relative to width: “visible world” from the entrance to the iconostasis; “invisible world” behind the iconostasis in the altar, built as half the symbolic circle of the “heaven” based on the geometric solution of the problem of the “quadrature of circle” in temples with the width-to-length ratio of 10:13, 2:3, 5:8. Plans of multi-nave cathedrals and churches with vestibules show a structural core representing the simple nine-cell type of the Middle Byzantine temple with width-to-length interior proportions of 10:13. The geometric analysis confirmed the existence of “The Life Giving Pillar” in the central dome space, as evidenced by doctrines, chronicles and icons. Vertical proportions of the pillar to the central point of the vault of the central dome are multiple of the half the width of three central naves (1:3, 1:4, 1:5) or relative to the full width (5:8). The square of the circle, being the horizontal section of the pillar, is equal to that of the dome square in 75% of ancient Russian churches of the Tenth–Fifteenth centuries. The pillar diameter in the remaining 25% temples is equal to the diagonal of the dome square. The spatial pattern of “The Life Giving Pillar” is represented as the river of the Holy Father, which flows from the heavens in the central dome space and is divided into two arms, according to the description provided by Gregory the Theologian (Fourth century). It forms the inner width of the three central naves. Geometric recording of the Christian concept of the Holy Trinity through the three bundled circles is revealed in proportioning of ancient Russian churches of the Tenth–Fifteenth centuries and the triconch on Mount Nebo in Jordan (Fourth century). The existence of the geometric outline of plans of Christian temples is also confirmed during the analysis of the Church of Saint John the Baptist in Jerusalem (late Sixth–Seventh centuries): both the original and the renovated structure visualize triadological disputes and the struggle against the Arian heresy.