Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
this new mathematical theory of measurement is the foundation of all traditional positional numeral systems by starting with the Babylonian positional numeral system with base 60 and by ending with decimal, binary, ternary, and other traditional numeral systems. But the most important is that this new theory of measurement generates new, earlier unknown positional numeral systems, such as the Fibonacci p-codes [6, 16, 55, 56, 58, 60, 97], the golden p-proportions codes [6, 19, 53, 97], which are a generalization of the classical binary system, and finally, the ternary mirror-symmetrical system and arithmetic [72, 94, 97], which are a generalization of the classical ternary system and arithmetic.
Volume II consists of seven chapters, which can be divided into two parts. The first part includes Chapters 1 and 2.
Chapter 1 provides the introduction to the Algorithmic Measurement Theory [16], which is based on the constructive approach in modern mathematics.
Chapter 2 is devoted to the Fibonacci measurement algorithms, which generate the Fibonacci p-codes for mission-critical applications.
The second part of Volume II consists of five chapters (Chapters 3–7). Chapter 3 provides a brief statement of the most interesting facts in the history of traditional numeral systems (Babylonian numeral system with the base of 60, Mayan numeral system, decimal, binary, and ternary systems).
Chapter 4 is devoted to the description of the unique positional numeral system with irrational base
Chapter 5 is devoted to the description of the unique ternary arithmetic, the “golden” ternary mirror-symmetrical arithmetic, which opens the new direction in ternary computers.
Chapter 6 is devoted to a study of the Fibonacci p-codes and Fibonacci arithmetic, which are the new scientific results for computer science and can lead to designing of the Fibonacci computers for mission-critical applications.
Chapter 7 is devoted to the study of the general class of the redundant numeral systems. The classical binary system is the partial case of the codes of the golden p-proportions (p = 0), the remaining “golden” codes, corresponding to the cases of p = 1, 2, 3, …, are a generalization of Bergman’s system (p = 1), and for the general cases of p = 1, 2, 3, …, they represent the general class of numeral systems with the irrational bases, which have a fundamental importance for mathematics (as the new definition of the real numbers) and also for computer science (as the basis of “golden” computers) and for the digital metrology (as the basis of the new theory of resistive dividers).
About the Author
Alexey Stakhov, born in May 7, 1939, is a Ukrainian mathematician, inventor and engineer, who has made a contribution to the theory of Fibonacci numbers and the golden section and their applications in computer science and measurement theory. He is a Doctor of Computer Science (1972) and a Professor (1974), and the author of over 500 publications, 14 books and 65 international patents. He is also the author of many original publications in computer science and mathematics, including algorithmic measurement theory [16, 17], Fibonacci codes and codes of the golden proportions [19], hyperbolic Fibonacci and Lucas functions [64, 75] and finally the Mathematics of Harmony [6], which goes back in its origins to Euclid’s Elements. In these areas, Alexey Stakhov has written many papers and books, which have been published in famous scientific journals by prestigious international publishers.
The making of Alexey Stakhov as a scientist is inextricably linked with the Kharkov Institute for Radio Electronics, where he was a postgraduate student of the Technical Cybernetics Department from 1963 to 1966. Here, he defended his PhD thesis in the field of Technical Cybernetics (1966) under the leadership of the prominent Ukrainian scientist Professor Alexander Volkov. In 1972, Stakhov defended (at the age of 32 years) his Grand Doctoral dissertation Synthesis of Optimal Algorithms for Analog-to-Digital Conversion (Computer Science speciality). Although the dissertation had an engineering character, Stakhov in his books and articles touched upon two fundamental problems of mathematics: theory of measurement and numeral systems.
Prof. Stakhov worked as “Visiting Professor” of different Universities: Vienna Technical University (Austria, 1976), University of Jena (Germany, 1986), Dresden Technical University (Germany, 1988), Al Fateh University (Tripoli, Libya, 1995–1997), Eduardo Mondlane University (Maputo, Mozambique, 1998–2000).
Stakhov’s Prizes and Awards
• Award for the best scientific publication by Ministry of Education and Science of Ukraine (1980);
• Barkhausen’s Commemorative Medal issued by the Dresden Technical University as “Visiting Professor” of Heinrich Barkhausen’s Department (1988);
• Emeritus Professor of Taganrog University of Radio Engineering (2004);
• The honorary title of “Knight of Arts and Sciences” (Russian Academy of Natural Sciences, 2009);
• The honorary title “Doctor of the Sacred Geometry in Mathematics” (American Society of the Golden Section, 2010);
• Awarded “Leonardo Fibonacci Commemorative Medal” (Interdisciplinary Journal “De Lapide Philosophorum”, 2015).
Acknowledgments
Alexey Stakhov expresses great thanks to his teacher, the outstanding Ukrainian scientist, Professor Alexander Volkov; under his scientific leadership, the author defended PhD dissertation (1966) and then DSc dissertation (1972). These dissertations were the first steps in Stakhov’s research, which led him to the conceptions of Mathematics of Harmony and Fibonacci computers based on the golden section and Fibonacci numbers.
During his stormy scientific life, Stakhov met many fine people, who could understand and evaluate his enthusiasm and appreciate his scientific direction. About 50 years ago, Alexey Stakhov had read the remarkable brochure Fibonacci Numbers [8] written by the famous Soviet mathematician Nikolay Vorobyov. This brochure was the first mathematical work on, Fibonacci numbers published in the second half of the 20th century. This brochure, determined Stakhov’s scientific interest in the Fibonacci numbers and the golden section for the rest of his life. In 1974, Professor Stakhov met with Professor Vorobyov in Leningrad (now St. Petersburg) and told Professor Vorobyov about his scientific achievements in this area. Professor Vorobyov gave Professor Stakhov, his brochure Fibonacci Numbers [8] as a keepsake with the following inscription: “To highly respected Alexey Stakhov with Fibonacci’s greetings”. This brief inscription because a certain kind of guiding star for Alexey Stakhov.
With deep gratitude, Stakhov recollects the meeting with the famous Austrian mathematician Professor Alexander Aigner in the Austrian city of Graz in 1976. The meeting with Professor Aigner was the beginning of the international recognition of Stakhov’s scientific direction.