The mathematics of harmony : from Euclid to contemporary mathematics and computer science

 (Series on Knots and Everything)

 Alexey Stakhov e Scott Olsen

World Scientific Publishing Company | 2009 | 745 páginas | pdf | 8 Mb

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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.

Contents
Three "key" problems of mathematics on the stage of its origin
Classical golden mean, Fibonacci numbers, and platonic solids
The golden section
Fibonacci and Lucas numbers
Regular polyhedrons
Mathematics of harmony
Generalizations of Fibonacci numbers and the golden mean
Hyperbolic Fibonacci and Lucas functions
Fibonacci and golden matrices
Application in computer science
Algorithmic measurement theory
Fibonacci computers
Codes of the golden proportion
Ternary mirror-symmetrical arithmetic
A new coding theory based on a matrix approach 
Dirac's principle of mathematical beauty and the mathematics of harmony : clarifying the origins and development of mathematics
Appendix : Museum of harmony and the golden section.