The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science (Series on Knots and Everything) 1st Edition by Alexey Stakhov (PDF)

Description

Assisted by Scott Olsen (Central Florida Community College, USA) This volume is a result of the author’s four decades of research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the “Mathematics of Harmony,” a new interdisciplinary direction of modern science. This direction has its origins in “The Elements” of Euclid and has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the “golden” algebraic equations, the generalized Binet formulas, Fibonacci and “golden” matrices), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational radices, Fibonacci computers, ternary mirror-symmetrical arithmetic, a new theory of coding and cryptography based on the Fibonacci and “golden” matrices). The book is intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science. Readership: Researchers, teachers and students in mathematics (especially those interested in the Golden Section and Fibonacci numbers), theoretical physics and computer science.

User’s Reviews

Editorial Reviews: About the Author Born and raised in the Ukraine, Alexey Stakhov now lives with his wife Antonina and his children and grandchildren in Toronto, Canada. An absolute genius in both the foundational principles and practical applications of mathematics, he was instrumental in the some of the former Soviet Union’s most profound scientific developments and patents. An outstanding and tireless researcher, brilliant writer, and excellent scholar, Alexey is also a true friend. Over the years he has exhibited superb research skills, organizational abilities and real leadership, at one time overseeing over sixty scientists in the soviet space program. He exudes the highest degree of professionalism, an intense will to succeed, and most importantly, a wonderfully creative and probing mind. His insights in this work are so remarkable that it may well change not only the way we view the history of mathematics, but the future development of mathematics in its applications to the natural sciences. In particular, Professor Stakhov’s knowledge of the Golden Section in both its intricacies and ramifications for the natural sciences is the actual state of the art in academia. And there is no one alive in the world today who understands more fully the real nature and possibilities of the Golden Mean, than Professor Alexey Stakhov.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Nature’s deeper reality lay hidden within its subtle framework of symmetry transformations which generally go without notice by the unobservant who tend rather to witness only their profound consequences. When first introduced to the so-called ‘golden ratio’ at age seven, I could not help but wonder what purpose it served. Its presence made itself known, popping up as it so often did in the most unexpected places. Only later did I began to notice its attendance in nearly every aspect of my studies of Natural Science, from Nature’s simplest organization as presented in Physics, then moving into the more complex study of Chemistry, on up to the most complex of Nature’s studies, that of Biology, esp Genetics. But it was in my more advanced studies of Mathematics where it became abundantly clear, at long last, that the magic of the ratio lay in its ability to lend maximum symmetry and therefore efficiency and flexibility to what we daily experience as natural phenomena and yet simultaneously to impose limits in such a way as to create a kind of unity within the context of diversity. This text served to re-inspire that nearly forgotten sense of what had originally attracted me to the study of Science, Math and Medicine. It is intensely Math-oriented, including any number of elegant proofs of its generalizations provided by some of the most brilliant minds in the world of Mathematics and Science. Of great interest to me is the section concerning the mystery of Phyllotaxis, demonstrating in intricate detail how angular branching is a precision product of differential rates of cell maturation thereby creating the observed twisting of a branch, its angles marvelously approaching the ideal ratio, a type of efficiency so characteristic of Nature’s fashion in obtaining as much surface to volume exposure as possible. Beautiful! A reasonable background in Algebraic Geometry or at least Analysis is required to truly appreciate the deep wealth of information contained therein.

⭐As a child I read “Dr Posin’s Giants- men of science” and found it inspirational. As an adult I read this book by Stakhov and I am bewildered – How did I overlook this book when it was first printed? This book is a masterful exploration of history and the essence of mathematical reasoning…by a master of the subjects.Thanks you, Professor Stakhov, for a marvelous contribution in this discipline.J

⭐Awesome book including extensive info on Fibonacci numbers and many accounts of the history of mathematics

⭐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

⭐.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

⭐and

⭐, supporting the fundamental role of the golden mean. Also the book

⭐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.

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