Chapter 2 is a popular introduction to the “theory of Fibonacci and Lucas numbers”, which actively began to develop in the second half of the 20th century in the works of Soviet and Western mathematicians and philosophers [7–11]. In Chapter 2, we expound the little-known results and applications of the Fibonacci and Lucas numbers, such as Steinhaus’s Iron Table, the connection of Fibonacci numbers with Pythagorean triangles, numerological properties of Fibonacci and Lucas numbers and we also consider the examples of applications of Fibonacci numbers in Nature (pentagonal symmetry, Fibonacci spirals, phyllotaxis phenomenon, and so on).
In Chapter 2, we also describe the original theory of Fibonacci p-numbers, introduced by Alexey Stakhov in the middle of 1960s. The foundations of this theory were expounded by Stakhov in Refs. [6, 16, 17]. Also, the relationship of the Fibonacci p-numbers to the Pascal triangle and binomial coefficients is shown.
Chapter 3 is devoted to the discussion of diagonal sums of Pascal’s triangle to Fibonacci p-numbers. In Chapter 3, we consider a generalization of the problem of the golden section [6, 16, 17, 60] and introduce an important concept of the “golden p-proportion”, which is a positive root of the algebraic equation of the golden p-proportion and generalization of the classical golden proportion. We consider the algebraic equations for the golden p-proportion, based on Vieta’s formulas, and also Binet’s formulas for the Fibonacci p-numbers and for the Lucas p-numbers.
Chapter 4 is devoted to the Platonic solids and Plato’s cosmology and to the discussion of the historical role of the Platonic solids in the two outstanding discoveries of modern theoretical natural sciences, fullerenes and quasicrystals, which were awarded the Nobel Prize. We consider the Archimedean truncated icosahedron as the most important geometric model of the fullerenes, as the mystery of the Egyptian calendar and its connection to the dodecahedron and also Klein’s conception of icosahedron as the main geometric object of mathematics [113]. In concluding part of Chapter 4, we consider the new ideas in the theory of elementary particles, based on the Platonic solids.
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” (Inter-disciplinary 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