Professor Stakhov has written a book that may well change not only the way we view the history of mathematics, but the future development of mathematics in its application to the natural sciences and computer design. This book is the result of the author's four decades of research in the field of Fibonacci numbers and the Golden Mean (also called the Golden Ratio or the Golden Section) and their applications. The volume contains chapters on the history of mathematics, the generalized Fibonaccis and Lucas numbers, the generalized golden proportions, the golden algebraic equations, the generalized Binet formulas, Fibonacci and golden matrices, and hyperbolic models. Stakhov suggests applications of number systems (Bergman's numeral system) with irrational radices in computers, coding theory, cryptography and mobil phone design, because irrational numbers are nothing else than commands for mathematical operations in space and time, see Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being.



Stakhov, of Russian and Ukrainian background, now living in Toronto, Canada, is an outstanding member of a minority of scientists, who has realized that the Golden Mean and its expression through Pascal's Triangle, Fibonacci and Lucas Number lies at the very center of the creative principle of nature itself. Recent scientific developments in physics, chemistry and cosmology have refocused attention to the mathematical properties of the Golden Mean and its meaning on the macrocosmic and microcosmic levels.



However, no scientific work can ever be complete: Stakhov is not aware of some developments in theoretical chemistry by Jan Boeyens, see Number Theory and the Periodicity of Matter and Chemistry from First Principles, supporting the fundamental role of the golden mean. Also the book A New Kind of Science is full of Pascal's Triangles!



Until now, from my point of view "From Golden Mean to Chaos" (printed in Argentina and, it is a pity, not on offer by amazon.com) by Vera de Spinadel is or was the most advanced book on the importance of the Golden Mean. Stakhov cites this book, but does not refer to renormalization and the Kolmogorov-Arnold-Moser theorem, in which the Golden Mean plays a role, too.



Until now, our brains and computers have a quite different architecture. However, by encoding and decoding the information entropy of a system no other wave could be more optimal than a wavelet containing the Golden Mean and its mathematics itself. Indeed, based on empirical data, the paper "The golden mean as clock cycle of brain waves", published in Chaos, Solitons and Fractals 18 (2003) 643-652, suggests that our brain uses these advantages of "The Mathematics of Harmony"! And already Roopun, A. K. et al., Frontiers in Neuroscience 2 (2008) 145-154, confirmed empirically this conclusion of the substantial role of the Golden Mean in our processing of information. Does our brain even work on the basis of a Ternary Mirror-Symmetrical Numeral System, as invented by Stakhov?



The quality of paper and print of this book is far above average. In any case you will buy a book, which will find its place in the history of science.