The Fibonacci Resonance and other new Golden Ratio discoveries: Maths, music, archaeology, architecture, art, quasicrystals, metamaterials, ... - Hardcover

Menhinick, Clive N

Hardcover

ISBN 10: 0993216609 ISBN 13: 9780993216602

Publisher: Onperson International Ltd., 2015

Synopsis

A new and definitive reference for the Fibonacci numbers and the Golden Ratio. With Mondrian, Seurat, Toulouse-Lautrec, Tiwanaku, The Great Pyramid, Le Corbusier, Kepler, Penrose, quasicrystals, Pendry, green energy, and the latest light-based technologies, this maths and science book is written to be enjoyed. Explore Bohemian Paris - the capital of phi - in fresh analyses of art, architecture, and music. Lavishly illustrated, this book includes: the history of Fibonacci and Lucas numbers, spirals, sunflowers, pine cones, megaliths, and ornamental tilings. The Fibonacci Resonance discovery is revealed step by step from unique beginnings to abacus-bead visualizations. Comparisons are made with Silver Ratio Pell numbers, and a link is shown to perfect numbers and record-breaking Mersenne primes.

About this title

From the Author

EXAMPLES ANALYSED INCLUDE

Seurat: "A Sunday Afternooon on the Island of La Grande Jatte" and "Circus Sideshow (Parade)".Toulouse-Lautrec: "Ambassadeurs, Aristide Bruant" and "Jane Avril" posters.Steinlen: "Chat Noir" poster.Mondrian: "Composition with Red, Blue, and Yellow 1930".Bartok: "MFSPC".Le Corbusier: "Villa Savoye" and "The Modulor"._________________________________________________________________________________

EMS BOOK REVIEW

"This is an encyclopedic work about the history, theory, and applications of the Golden Ratio and the Fibonacci numbers (and their companions, the Lucas numbers)."
. . . "A remarkable addition to this is the middle part of the book"  . . .
"the most original, most surprising, and most mathematical" . . .
"in which Menhinick develops a theory of Fibonacci resonance that is based on recursion formulas for Fibonacci and Lucas numbers expressing them as deviations from certain points on logarithmic spirals"

Professor Emeritus (KU Leuven)  Adhemar Bultheel---review of 'The Fibonacci Resonance' for The European Mathematical Society (EMS).

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TABLE OF CONTENTS

Preface . . . xiAcknowledgements . . . xiii

I BACKGROUNDChapter 1 Fibonacci numbers and the Golden Ratio . . . 1Chapter 2 Spirals . . . 35Chapter 3 In nature . . . 49Chapter 4 Music---Bartok . . . 61Chapter 5 Paris---capital of Phi . . . 67Chapter 6 Art---Seurat, Toulouse-Lautrec . . . 73Chapter 7 Art---Mondrian, gnomons, and megaliths . . . 89Chapter 8 Architecture---Le Corbusier . . . 123Chapter 9 Binet spirals, Debussy, and a black cat . . . 133

II LEADING TO A DISCOVERYChapter 10 Introducing Ori32 geometry . . . 159Chapter 11 The Ori32 Fibonacci circle . . . 167Chapter 12 Five Golden Powers . . . 175

III THE FIBONACCI RESONANCEChapter 13 The penny drops . . . 191Chapter 14 Proving the theorem . . . 201Chapter 15 Visualizing the effect . . . 207Chapter 16 Hearing it . . . 221Chapter 17 3D model . . . 225Chapter 18 Is it fractal? . . . 229Chapter 19 Fibonacci Resonance summary . . . 243

IV TO PELL AND BEYONDChapter 20 Generalization to Lucas Sequences . . . 251Chapter 21 Pell, Jacobsthal, and Mersenne . . . 265

V PHI IN SCIENCEChapter 22 Phi science introduction . . . 287Chapter 23 Phyllotaxis . . . 289Chapter 24 Phi in quantum mechanics . . . 297Chapter 25 Molecular Phi . . . 299Chapter 26 Penrose tiling . . . 309Chapter 27 Quasicrystals . . . 323Chapter 28 Islamic tiling . . . 343Chapter 29 Superlattices and metamaterials . . . 359Chapter 30 Quasicrystal 'onics' . . . 383

VI APPENDICESAppendix A Ori32 trigonometry . . . 401Appendix B Fibonacci hexads modulo 32 . . . 409Appendix C Continued fractions . . . 421Appendix D Powers of Phi: Golden Powers . . . 431Appendix E Binet from a generating function . . . 439Appendix F Inductive Binet, instant Binet . . . 443Appendix G Mersenne primes, Hilbert's 10th . . . 447Glossary . . . 451Symbols used . . . 459Collected formulae . . . 465References . . . 487About the author . . . 576List of Figures . . . 577Index . . . 591

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ABOUT THE AUTHOR

Clive N. Menhinick is a computer scientist, geometer, and research photographer, based in Cheshire, England. In 2005, he recognized that 'state-of-the-art' developments in DSLR camera technology---particularly the vast increase in colour resolution ('bit depth')---opened up a new frontier for research in natural science. Over a ten-year period he has conducted his own pioneering project, developing digital-photographic techniques and custom software. In his analysis he found that many results appeared to depend on a simple angle-based geometry that he named 'Ori32'. It was the study of this geometry that provided a clear path to the discovery of the Fibonacci Resonance. Clive read physics and mathematics at Liverpool University.

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Edition details

• Publisher: Onperson International Ltd.
• Publication date: 2015
• ISBN 10: 0993216609
• ISBN 13: 9780993216602
• Binding: Hardcover
• Language: English
• Number of pages: 638