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).
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