The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer ScienceAssisted 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 OC Mathematics of Harmony, OCO a new interdisciplinary direction of modern science. This direction has its origins in OC The ElementsOCO 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 OC goldenOCO algebraic equations, the generalized Binet formulas, Fibonacci and OC goldenOCO 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 OC goldenOCO 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. Sample Chapter(s). Introduction (503k). Chapter 1: The Golden Section (2,459k). Contents: 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. Readership: Researchers, teachers and students in mathematics (especially those interested in the Golden Section and Fibonacci numbers), theoretical physics and computer science." |
Contents
Three Key Problems of Mathematics on the Stage of its Origin | xix |
Acknowledgements | xxxix |
Credits and Sources | xliii |
Part I Classical Golden Mean Fibonacci Numbers and Platonic Solids | 1 |
Part II Mathematics of Harmony | 185 |
Part III Application in Computer Science | 359 |
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Common terms and phrases
according algebraic equations arithmetic bers binary combination binary numeral Binet formula binomial coefficients bonacci calculate Cantor's century code combination code matrix coefficients concept connected convolution cryptography digit discovery dodecahedron e₁ elements equal error-correction codes errors Euclid's Euclid's Elements example expression Fibonacci and Lucas Fibonacci Association Fibonacci numbers Fibonacci p-code Figure following form geometric given golden mean golden p-proportion golden section Harmony hyperbolic functions icosahedron identity integer inverse irrational numbers Kepler Let us consider line segment logic Lucas functions Lucas numbers mathe mathematical mathematician matrix measurement algorithms measurement theory method micro-operation minimal form natural number Note number theory numeral system obtain the following optimal p-numbers pentagon phyllotaxis Platonic Solids polyhedra principle problem properties Pythagorean Q-matrix rabbit ratio real numbers recursive relation represented result roots Russian scientific sequence Stakhov summation symmetric hyperbolic Fibonacci symmetry Table ternary mirror-symmetrical Theorem tion triangle values τ²