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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.

Another remarkable scientist, who had a great influence on Stakhov’s research, was the Ukrainian mathematician and academician Yuri Mitropolskiy, the Head of the Ukrainian Mathematical School and the Chief Editor of the Ukrainian Mathematical Journal. His influence on Stakhov’s researches, pertinent to the history of mathematics and other topics, such as the application of the Mathematics of Harmony in contemporary mathematics, computer science and mathematical education, were inestimable stimulus for Alexey Stakhov. Thanks to the support of Yuri Mitropolskiy, Stakhov published many important articles in the prestigious Ukrainian academic journals, including the Ukrainian Mathematical Journal.

In 2002, The Computer Journal (British Computer Society) published the fundamental article by Stakhov, “Brousentsov’s Ternary Principle, Bergman’s Number System and Ternary Mirror-Symmetrical Arithmetic” (The Computer Journal, Vol. 45, No. 2, 2002) [72]. This article by Stakhov created great interest among all the English scientific computer community. Emeritus Professor of Stanford University Donald Knuth was the first outstanding world scientist, who congratulated Prof. Stakhov with this publication. Donald Knuth’s letter became one of the main stimulating factors for writing Stakhov’s future book The Mathematics of Harmony From Euclid to Contemporary Mathematics and Computer Science (World Scientific, 2009) [6]. Professor Stakhov considers this book as the main book of his scientific life.

Stakhov’s arrival to Canada in 2004 became the beginning of a new stage in his scientific research. Within 15 years (2004–2019), Prof. Stakhov had published more than 50 fundamental articles in different international English-language journals, such as Chaos, Solitons & Fractals, Applied Mathematics, Arc Combinatoria, The Computer Journal, British Journal of Mathematics and Computer Science, Physical Science International Journal, Visual Mathematics, etc. Thanks to the support of Prof. El-Nashie, the Editor-in-Chief of the Journal Chaos, Solitons & Fractals (UK), Stakhov published in this journal 15 fundamental scientific articles that garnered great interest among the English-speaking scientific community.

The publication of the three fundamental books The Mathematics of Harmony (World Scientific, 2009) [6], The “Golden” Non-Euclidean Geometry (World Scientific, 2016, co-author Prof. Samuil Aanson) [52] and Numeral Systems with Irrational Bases for Mission-Critical Applications (World Scientific, 2017) [53] is one of the main scientific achievements by Stakhov during the Canadian period of his scientific creativity. These books were published thanks to the support of the famous American mathematician Prof. Louis Kauffman, Editor-in-Chief of the Series on Knots and Everything (World Scientific) and Prof. M.S. Wong, the famous Canadian mathematician (York University) and Editor-in-Chief of the Series on Analysis, Application and Computation (World Scientific). A huge assistance in the publication of Stakhov’s books by of World Scientific was rendered by the American researcher, Scott Olsen, Professor of Philosopy at the College of Central Florida, and Jay Kappraff, Emeritus Professor of Mathematics at the New Jersey Institute of Technology. Prof. Scott Olsen, who was one of the leading US experts in the field of Harmony and the golden section, was the English editor for Stakhov’s book mentioned above and the Emeritus Professor Jay Kappraff was the reviewer of Stakhov’s book, The Mathematics of Harmony (World Scientific, 2009).

The prominent Ukrainian mathematician and head of the Ukrainian Mathematical School, Yuri Mitropolskiy, praised highly Stakhov’s Mathematics of Harmony. Academician Mitropolsky organized Stakhov’s speech at the meeting of the Ukrainian Mathematical Society in 1998. Based upon his recommendation, Stakhov’s articles were published in the Ukrainian academic journals, in particular, the Ukrainian Mathematical Journal. Under his direct influence, Stakhov started writing the book, The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science [6], which was published by World Scientific in 2009 following the death of the academician Mitropolsky in 2008.

Scientific cooperation of Alexey Stakhov and Samuil Aranson

Samuil Aranson’s acquaintance to the golden section and the Fibonacci numbers began in 2001 after the reading of a very rare book “Chain Fractions” [107] by the famous Russian mathematician, Aleksandr Khinchin. In this book, Samuil Aranson found results, related to the representation of the “golden ratio”in the form of a continued fraction.

In 2007, Prof. Aranson read a wonderful Internet publication, Museum of Harmony and Golden Section, posted in 2001 by Professor Alexey Stakhov and his daughter Anna Sluchenkova. This Internet Museum covers various areas of modern natural sciences and tells about the different and latest scientific discoveries, based on the golden ratio and Fibonacci numbers, including the Mathematics of Harmony and its applications in modern natural sciences. After reading this Internet Museum, Samuil Aranson contacted Alexey Stakhov in 2007 through e-mail and offered him joint scientific collaboration in further application of the Mathematics of Harmony in various areas of mathematics and modern natural sciences. Scientific collaboration between Alexey Stakhov and Samuil Aranson turned out to be very fruitful and continues up to the present time.

New ideas in the field of elementary mathematics and the history of mathematics, developed by Stakhov (Proclus’s hypothesis as a new look at Euclid’s Elements and history of mathematics, hyperbolic Fibonacci and Lucas functions [64, 75] as a new class of elementary functions and other mathematical results) attracted the special attention of Prof. Aranson. Scientific collaboration between Stakhov and Aranson began in 2007. From 2007, they published the following joint scientific works (in Russian and English), giving fundamental importance for the development of mathematics and modern theoretical natural sciences:

Stakhov and Aranson’s Mathematical Monographs in English

1.Stakhov A., Aranson S., The Mathematics of Harmony and Hilbert’s Fourth Problem. The Way to the Harmonic Hyperbolic and Spherical Worlds of Nature. Germany: Lambert Academic Publishing, 2014.

2.Stakhov A., Aranson S., Assisted by Scott Olsen, The “Golden” Non-Euclidean Geometry: Hilbert’s Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant,World Scientific, 2016.

Stakhov and Aranson’s Scientific Papers in English

3.Stakhov A.P., Aranson S.Kh., “Golden” Fibonacci goniometry, Fibonacci-Lorentz transformations, and Hilbert’s fourth problem. Congressus Numerantium 193, (2008).

4.Stakhov A.P., Aranson S.Kh., Hyperbolic Fibonacci and Lucas functions, “golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem. Part I. Hyperbolic Fibonacci and Lucas functions and “Golden” Fibonacci goniometry. Applied Mathematics 2(1), (2011).

5.Stakhov A.P., Aranson S.Kh., Hyperbolic Fibonacci and Lucas functions, “golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem. Part II. A new geometric theory of phyllotaxis (Bodnar’s Geometry). Applied Mathematics 2(2), (2011).

6.Stakhov A.P., Aranson S.Kh., Hyperbolic Fibonacci and Lucas functions, “golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem. Part III. An original solution of Hilbert’s fourth problem. Applied Mathematics 2(3), (2011).

7.Stakhov A.P., Aranson S.Kh., The mathematics of harmony, Hilbert’s fourth problem and Lobachevski’s new geometries for physical world. Journal of Applied Mathematics and Physics 2(7), (2014).

8.Stakhov A., Aranson S., The fine-structure constant as the physical-mathematical millennium problem. Physical Science International Journal 9(1), (2016).

9.Stakhov A., Aranson S., Hilbert’s fourth problem as a possible candidate on the millennium problem in geometry. British Journal of Mathematics & Computer Science 12(4), (2016).