Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science
Volume 1
The Golden Section, Fibonacci Numbers, Pascal Triangle, and Platonic Solids
Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science
Volume 1
The Golden Section, Fibonacci Numbers, Pascal Triangle, and Platonic Solids
Alexey Stakhov
International Club of the Golden Section, Canada & Academy of Trinitarism, Russia
Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Control Number: 2020010957
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
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
Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd.
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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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Desk Editor: Liu Yumeng
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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
2.7 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
3.3 Diagonal Sums of Pascal’s Triangle and Fibonacci p-Numbers
3.4 The Extended Fibonacci p-Numbers