Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov

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      Series on Knots and Everything — Vol. 65 MATHEMATICS OF HARMONY AS A NEW INTERDISCIPLINARY DIRECTION AND “GOLDEN” PARADIGM OF MODERN SCIENCEVolume 1: The Golden Section, Fibonacci Numbers, Pascal Triangle, and Platonic Solids

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      ISBN 978-981-120-710-5 (hardcover)

      ISBN 978-981-120-637-5 (ebook for institutions)

      ISBN 978-981-120-638-2 (ebook for individuals)

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      Printed in Singapore

      In fond memory of Yuri Alekseevich Mitropolskiy and Alexander Andreevich Volkov

       Contents

       Preface to the Three-Volume Book

       Introduction

       About the Author

       Acknowledgments

       Chapter 1. The Golden Section: History and Applications

       1.1 The Idea of the Universeal Harmony in Ancient Greek Science

       1.2 The Golden Section in Euclid’s Elements

       1.3 Proclus Hypothesis and New View on Classic Mathematics and Mathematics of Harmony

       1.4 Some Simplest Mathematical Properties of the Golden Ratio

       1.5 The Golden Ratio and Chain Fractions

       1.6 Equations of the Golden Proportion of the Nth Degree

       1.7 Geometric Figures Associated with the Golden Section

       1.8 The Golden Section in Nature

       1.9 The Golden Section in Cheops Pyramid

       1.10 The Golden Section in Ancient Greek Culture

       1.11 Golden Section in the Art of the Renaissance

       Chapter 2. Fibonacci and Lucas Numbers

       2.1 A History of the Fibonacci Numbers

       2.2 The Sums of the Consecutive Fibonacci Numbers

       2.3 Cassini’s Formula

       2.4 Lucas Numbers

       2.5 Binet’s Formulas

       2.6 Steinhaus’s “Iron Table”

       2.7 Pythagorean Triangles and Their Presentation Through Fibonacci and Lucas Numbers

       2.8 Fibonacci Numbers in Nature

       2.9 Fibonacci Numbers and Solution of Hilbert 10th Problem

       2.10 Turing and Fibonacci Numbers

       2.11 Role of the Fibonacci Numbers Theory in Modern Mathematics

       Chapter 3. Pascal Triangle, Fibonacci p-Numbers and Golden p-Proportions

       3.1 Binomial Theorem

       3.2 Pascal Triangle

       3.3 Diagonal Sums of Pascal’s Triangle and Fibonacci p-Numbers

       3.4 The Extended Fibonacci p-Numbers