# Manual The mathematics of harmony. From Euclid to contemporary mathematics and computer science

Readership: Researchers, teachers and students in mathematics especially those interested in the Golden Section and Fibonacci numbers , theoretical physics and computer science. Acknowledgements xxxix. Classical Golden Mean Fibonacci. Fibonacci and Lucas Numbers. Regular Polyhedrons.

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Generalizations of Fibonacci Numbers and the Golden Mean. Application in Computer Science. Fibonacci Computers. Lifespan David Sinclair Inbunden. Inbunden Engelska, Spara som favorit. In the theory of architecture, it is well-known the book "Proportionality in Architecture", published by Prof.

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Grimm in [47]. The purpose of the book has been formulated in the "Introduction" as follows:. Taking this general importance of the Golden Section in all aspects of architectural thought , the theory of proportionality, based on the proportional division of the whole into parts corresponding to the Golden Section, should be recognized as the architectural basis of proportionality at all. In the 20th years of 20th century, Pavel Florensky wrote the work "At the watershed of a thought. In the second half of 20 th century the interest in this area is increasing in all areas of science, including mathematics.

The Soviet mathematician Nikolai Vorobyov [44], the American mathematician Verner Hoggatt [45], the English mathematician Stefan Vajda [46] and others became the most outstanding representatives of this trend in mathematics. Reviving the idea of harmony in modern science is determined by new scientific realities. The penetration of the Platonic solids, the "golden ratio" and Fibonacci numbers in all areas of theoretical natural sciences crystallography, chemistry, astronomy, earth science, quantum physics, botany, biology, geology, medicine, genetics, etc. Recently on the website "Academy of Trinitarizm, Institute of the Golden Section" has published the article by well-known Russian Fibonacci-scientist Victor Shenyagin [48] with the following appeal:.

Of course, it is pleasant for the author to get a high appreciation of his research from his colleagues. Similar estimations were given also by other well-known scientists: the Ukrainian mathematician academician Yuri Mitropolsky [7], the American philosopher Scott Olsen [49], the Belarusian philosopher Edward Soroko [50], the Russian philosopher Sergei Abachiev [51], and many others.

Their feedback is the highest award for the author.

## Dr. Scott Olsen interview – “The Golden Ratio” – #108

Unfortunately, this scientific discipline, called the "Mathematics of Harmony" [1] until now is not a generally recognized part of the conventional mathematics. However, this scientific direction became widely known in modern science after the publication of the book [1], as well as the author's publications in the Ukrainian academic and English scientific journals [3, ]. The algorithmic measurement theory is the first mathematical theory, created by the author during 70th years of 20 c. The foundations of this theory are described in the books [12, 13].

First of all, the author have excluded from consideration the abstraction of actual infinity as an internally contradictory concept the "completed infinity" , because still Aristotle protested against this. This contradiction implies automatically the contradictions in the foundations of mathematics.

Such a constructive approach to the algorithmic measurement theory, based on the abstraction of potential infinity, allowed the author to solve the problem, which was never been considered in mathematics. We talk about the problem of the synthesis of optimal measurement algorithms, which are a generalization of well-known measurement algorithms: the "counting algorithm," which lies at the basis of Euclidean definition of natural numbers, and the "binary algorithm," which lies at the basis of the binary system, the basis of modern computers.

Each measurement algorithm corresponds to some positional number system. From the algorithmic measurement theory, it follows all the known positional number systems. Thus, the algorithmic measurement theory is a source for the new positional number systems. Note that mathematics was never seriously engaged in number systems. And this is the mistake of mathematics. New positional number systems in particular, introduced by the author Fibonacci p-codes can become an alternative to the binary system for many critical applications.

In this book Polya showed an unexpected connection of Pascal's triangle with.

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The study of the so-called "diagonal sums" of Pascal's triangle, conducted by many mathematicians including the author of this article , has led to the discovery of an infinite amount of the recurrent sequences, called the Fibonacci p -numbers [12]. By studying the Fibonacci p -numbers and considering the limit of the ratio of neighboring Fibonacci p -.

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The positive roots of the equation 8 form a set of the new mathematical constants. Within several millennia, starting since Pythagoras and Plato, the mankind used the widely known classical Golden Proportion as some unique number. And here in the end of the 20 th century the Ukrainian scientist Stakhov generalized this result and proved the existence of infinite number of the Golden Proportions! And all of them have the same right to express Harmony, as well as the classical Golden Proportion.

Clearly, such mathematical result is of fundamental importance for the development of modern science and mathematics.

## Stakhov, A. P. (Alekseĭ Petrovich) [WorldCat Identities]

The redundant binary positional number systems the Fibonacci p -codes. Fibonacci measurement algorithms, based on the Fibonacci p -numbers [12], led to the discovery of the new positional number systems, called the Fibonacci p -codes:. In , the author worked during 2 months as Visiting-Professor of the Vienna University of Technology.

At the final stage of the stay in Austria, the author made a speech "Algorithmic Measurement Theory and Foundations of Computer Arithmetic" at the joint meeting of the Austrian Computer and Cybernetics Societies. The success of the speech was stunning. It was recognized that this speech provides new informational and arithmetical foundation of computers. Japan, England, France, Germany, Canada and other countries are official legal documents, which confirmed a priority of the author in this area.

In , the young 12 year old American wunderkind George Bergman in one of the U. In , the author had generalized the number system 11 and introduced in mathematics a wide class of. Theory of the codes of the golden p -proportions was set forth in the book [14]. This book attracted the attention of the Soviet scientificpopular Journal "Technology for young people. The main idea of the article is reduced to the following.

The codes of the golden p -proportions, which are connected with Pascal's triangle, can be considered as the beginning of the new number theory, the "golden" number theory. Indeed, with the help of the codes of the golden p -proportions we can represent all real numbers, including natural, rational and irrational.

The codes of the golden p-proportions change our ideas about the relationship between rational and irrational numbers, because the special irrational numbers the golden p-proportions are. It is proved [24] that the sum of 14 for any natural number N is finite always, that is, any natural number N can be represented as the finite sum of the powers of the "golden proportion".

Since all powers of the "golden. To our surprise, we find [24] that this sum is equal to 0 for any natural number.

This property has been called the Z-property of natural numbers [24]. Since this property is valid only for natural numbers, this means that in [24] we found a new property of natural numbers, via 2. Scott Olsen. New York: Walker Publishing Company,. Petoukhov S. Metaphysical aspects of the matrix analysis of genetic code and the golden section.

Metaphysics: Century XXI. World Scientific , Stakhov A. Automation and Computer Engineering , , No 6, Russian. The Golden Ratio in Digital Technology.

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## Mathematics of Harmony

Automation and Computer Technology. Applications of Fibonacci. Numbers , Kluwer Academic Publishers, Vol. Introduction into Algorithmic Measurement Theory. Algorithmic Measurement Theory.