## 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." |

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### Contents

1 | |

6 | |

Fibonacci and Lucas Numbers | 60 |

xiv | 83 |

Regular Polyhedrons | 137 |

Generalizations of Fibonacci Numbers and the Golden Mean | 186 |

Application in Computer Science | 359 |

Fibonacci Computers | 416 |

Codes of the Golden Proportion | 476 |

Ternary Mirror Symmetrical Arithmetic | 523 |

xviii | 569 |

Epilogue Diracs Principle of Mathematical Beauty and the Mathematics | 615 |

Three Key Problems of Mathematics and a New Approach | 632 |

the Golden Information | 646 |

Hyperbolic Fibonacci and Lucas Functions | 255 |

StakhovRozin Definition | 277 |

Fibonacci and Golden Matrices | 317 |

Matrices and their Powers | 343 |

661 | |

Museum of Harmony and Golden Section | 675 |

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### Common terms and phrases

according algebraic equations angle Archimedean Solids arithmetic bers binary numeral Binet formula binomial coefficients bonacci century cFsx classical hyperbolic functions code combination code matrix coefficients concept connected digit discovery dodecahedron Egyptian elements equal Euclid’s example expression Fibonacci and Lucas Fibonacci Association Fibonacci code Fibonacci numbers Figure following form geometric given golden mean golden pproportion golden rectangle golden section Harmony hyperbolic functions hyperbolic Lucas icosahedron integer Kepler Leonardo line segment logic Luca Pacioli Lucas functions Lucas numbers mathe mathematical mathematician measurement theory minimal form natural number number theory numeral system numerical sequences numerological numerological value obtain the following Pacioli pentagon phyllotaxis Platonic Solids problem properties proportion Pythagoras Pythagorean rabbits ratio real numbers rectangle recursive relation regular polyhedra representation represented result roots sFsx square Stakhov summation symmetric hyperbolic Fibonacci symmetry Table Theorem tion triangle wellknown τ τ τ