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The Fibonacci Resonance

Clive N. Menhinick
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Publication Date: 
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[Reviewed by
Underwood Dudley
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On page xiv, the author says that his book is three books in one, of which the second is “The announcement of a new discovery … The Fibonacci Resonance”. On page 1, he says that he “presents a number of new discoveries, in particular, one that relates Fibonacci numbers, Lucas numbers, the Golden Ratio, and Golden Spirals — the Fibonacci Resonance.”

The second comma in the last quote should be deleted, but never mind.

What, you may ask, is the Fibonacci Resonance? Let \(a=(1+\sqrt{5})/2)\), \(b=(1-\sqrt{5})/2\), and let \(F_n\) denote the nth Fibonacci number. The FR is (page 202) \[F_n = F_s a^{n-s} + F_{n-s} b^s.\]

The author needs approximately \((1+\sqrt{5})/2\) pages for his proof, after having previously devoted several pages to the special case \(s= 5\).

Watch this. As we and the author know, \(F_n=(a^n-b^n)/\sqrt{5}\). Because \[a^n-b^n = a^n –b^s a^{n-s} + a^{n-s}b^s-b^n\] we have \[a^n-b^n=(a^s-b^s)a^{n-s} + (a^{n-s}-b^{n-s})b^s.\] Divide both sides by \(\sqrt{5}\) and we have the Fibonacci Resonance.

It is new in the same sense that \[31415926 + 27182818 = 58598744\] is new: it may never have been written down before, but it is not an advance in mathematics. In the admirable index — a model of what an index should be — there is no entry “Fibonacci Resonance, applications” and I did not see any place where it was put to use. On page 243 the author says that it “provides a way to visualize in detail how Fibonacci and Lucas numbers grow.” Well, we have formulas that do that job very well. In the book’s more than 600 pages I may have missed something. If so, I fault the author for not making it more prominent. By the way, the index, while admirable, is not perfect: “Xenakis, 149” should be “Xenakis, 151”.

Besides the Fibonacci Resonance, what else is in the more than 600 pages? Quite a lot. The author begins with quite a few examples of the occurrence, or seeming occurrence, of Fibonacci numbers and the golden ratio \(a=(1+\sqrt{5})/2\) in the great pyramid of Egypt, stone monuments in Bolivia, the Greek theatre in Epidaurus, and so on. To his credit, he does not say that the builders of the pyramid put \(a\) in it on purpose. He is only reporting what other people have said.

Similarly, he merely reports the finding that the rectangle whose sides have the ratio \(a\) to \(1\) is the esthetically most pleasing rectangle. Why this great nonsense is persistently repeated is a mystery to me. The author’s book has approximate dimensions \(23.7\) by \(16\) centimeters, whose ratio, \(1.48\), is far from \(a= 1.618\dots\) . Books almost never have the shape of a so-called golden rectangle. They are not esthetically pleasing — they are too long and skinny. If you visit your local art gallery, you will find very few paintings shaped like golden rectangles. If artists don’t know what shapes are esthetically pleasing, what hope is there for the rest of us? Personally, I favor rectangles whose ratio of length to height is \(25\) to \(1\). The long horizontal induces a feeling of rest, serenity, and security.

The author gives many examples of art works that contain, or can be construed to contain, golden rectangles or Fibonacci numbers. In some cases, artists consciously inserted them, but in many others, I am convinced, their works are products of talent and intuition whose makers did not break out their straightedges and compasses before creating them. In any event, what difference does it make? People do not line up to see the Mona Lisa because it contains a rectangle similar to one whose sides have ratio \((1+\sqrt5)/2\) to \(1\) and they do not listen to the compositions of Ravel because he sometimes kept Fibonacci numbers in mind. Again to his credit, the author doesn’t report on those writers who would have us believe that Mozart and Beethoven counted bars in their scores to make sure that their works contained the golden ratio.

There are many pages devoted to physics (the subject of the author’s bachelor’s degree) and — I am quoting the author’s section headings — metamaterial superlenses, invisibility cloaking, plasmonics, seismic cloaking, bio-medical metamaterials, quasicrystal nanophotonics and light harvesting, and quasicrystal magnonics. They don’t seem to have anything to do with the Fibonacci Resonance.

In “Part II — Leading to a Discovery” (of the Fibonacci Resonance) the author introduces MIKs, Fibon1, Fibon2, Fibon3, Fibon4, and Ori32 geometry and goes into some detail about their properties. I didn’t know what to make of them, or of much of the content of the book. The book goes on and on, never, as far as I could see, arriving anywhere.

The book is beautifully produced. It includes, at what must have been considerable expense, fifty or more color illustrations. I didn’t notice many misprints. The author sometimes writes “log(n)” and sometimes “log(n)” and “tan” and “tan” both appear, but nothing else struck my eye. Equations are sometimes in different fonts: for example, on page 263, (20.38) and (20.40) are in one font, and (20.39) and (20.41) in another. Other than that, his pages are pleasant to the eye. He writes clearly. He has put in an amazing amount of work: his list of references has one thousand and four entries. There is evidently more to come because one of the subtitles of the book is “Ori32 geometry & cryptochromatology series: Book 1”.

I can recommend the book to those who will find the material in it fascinating. I don’t know who that could be. It isn’t me and, I suspect, it isn’t many members of the MAA.

Woody Dudley retired from teaching in 2004. He is old, and jaded, and when told that \(a^4=(7+\sqrt{5}\cdot 3)/2\) (page 182, 12.3) — the first four primes, in reverse order  says “So what?”

Preface . . . xi
Acknowledgements . . . xiii

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

Chapter 10 Introducing Ori32 geometry . . . 159
Chapter 11 The Ori32 Fibonacci circle . . . 167
Chapter 12 Five Golden Powers . . . 175

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

Chapter 20 Generalization to Lucas Sequences . . . 251
Chapter 21 Pell, Jacobsthal, and Mersenne . . . 265

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

Appendix A Ori32 trigonometry . . . 401
Appendix B Fibonacci hexads modulo 32 . . . 409
Appendix C Continued fractions . . . 421
Appendix D Powers of Phi: Golden Powers . . . 431
Appendix E Binet from a generating function . . . 439
Appendix F Inductive Binet, instant Binet . . . 443
Appendix G Mersenne primes, Hilbert’s 10th . . . 447

Glossary . . . 451
Symbols used . . . 459
Collected formulae . . . 465
References . . . 487
About the author . . . 576
List of Figures . . . 577
Index . . . 591


menhinick's picture

(The author of "The Fibonacci Resonance" thanks Professor Gouvêa for inviting him to respond to the above review.)

This book has been described in one review as an "encyclopedic work about the history, theory, and applications of the Golden Ratio and the Fibonacci numbers". It is therefore a great pity that the present reviewer rather missed both its purpose and its target audience. He forgot that (as noted in the Preface) it is written to high-school maths level. The content is designed to inspire and entertain bright young maths and science students/undergraduates aged 17 upwards. It is also intended for mature recreational readers with lively cultural and scientific interests. Accordingly, the work does not "run on and on" — on the contrary, it is tightly structured. (Even a cursory glance at the Table of Contents will settle this point.)

"New?"---As the reviewer apparently concedes, the Fibonacci resonance is "new" in the sense of "being a previously unknown identity that is now known". Until now, the step-by-step growth of Fibonacci numbers has been understood in terms of a sequence of ratios which converge towards an underlying growth rate (which is irrational in the limit). Johannes Kepler noted this in 1611, pointing out that successive growth ratios \(F_{n+1}\big / F_n,\) (\(n=1, 2, 3, \ldots\)), approximate increasingly well to the Golden Number \((1+\sqrt{5})\big /2\), but never reach it. Instead, the Fibonacci resonance sees each sequence member as having two components:

  1. a steady geometric growth term (simply growing by \(\phi\) each step), and
  2. an "adjustment" term.

As is shown in the book, the size of the adjustments may now be exactly determined.

Further, the Fibonacci numbers have a "companion" sequence viz. the Lucas numbers \(L_n\). These share the same "add the 2 predecessors to get the next" recurrence rule, but they start with 2, 1 instead of 0, 1. It is interesting that as the Fibonacci and Lucas numbers grow, the ratio \(L_n\big / F_n\) tends towards \(\sqrt{5}\), and because of this, it is sometimes said that the Lucas numbers have a "growth lead" of approximately \(\sqrt{5}\) over the Fibonacci numbers. In the book, it is shown that Lucas numbers may too be expressed in terms of "steady growth + adjustment"---so the Fibonacci resonance is actually a formula pair: \[
%\boxed{ \textrm{for all integers }n,s
\ \ F_n \ &=\ \ \phantom{\sqrt{5}}\ F_s\,\phi^{n-s} \ \ \ +\ \ \ F_{n-s}\,(-\phi)^{-s}\quad\\[3pt]
\ \ L_n \ &=\ \ \sqrt{5}\ F_s\,\phi^{n-s} \ \ \ +\ \ \ L_{n-s}\,(-\phi)^{-s}\quad
\textrm{for all integers }n,s,\ \ \\
\phi=(1+\sqrt{5})\big /2.

Or, equivalently stated: \[ \left. \begin{aligned} \ \ F_{s+j} \ &=\ \ \phantom{\sqrt{5}}\ F_s\,\alpha^j \ \ \ +\ \ \ F_j\,\beta^s\quad\\[3pt] \ \ L_{s+j} \ &=\ \ \sqrt{5}\ F_s\,\alpha ^j \ \ \ +\ \ \ L_j\,\beta ^s\quad \end{aligned} \right\}\quad\begin{cases} \textrm{for all integers }j,s,\ \ \\ \alpha=\phi, \ \ \ \beta=-\phi^{-1}. \end{cases} \] Here we see that for all integer \(j,\) both the adjustment terms, \(F_j\,\beta^s\) and \(L_j\,\beta^s\), share the same measure \( \beta^s \).


In other words, by choosing any (non-zero) Fibonacci number \(F_s\) as a start reference (hence fixing \(s\)), all sequence members (with increasing \(j\)) will then deviate from steady geometric growth (i.e. away from the terms in \(\alpha^j\)) by now known distances, and these distances are all multiples of the common measure \( \beta^s \). For the Fibonacci sequence the multipliers form the Fibonacci sequence pattern \(F_j\), and for the Lucas numbers---the Lucas sequence \(L_j\). These patterns are simply visualized in the book — either using abacus beads or as "standing waves on resonating strings" (hence the name "Fibonacci resonance").

The Fibonacci resonance might be described as "neat and elegantly symmetric", but no claim is made that it is a glorious advance in mathematics. Nevertheless, Emeritus Professor Adhemar Bultheel of KU Leuven/EMS describes it as "a generalization of the Binet formula", and it has been confirmed as mathematically correct by the current Editor of the Fibonacci Quarterly. An unexpected consequence of this approach is that in the \(F_n, L_n\) formula pair, the \(\sqrt{5}\) underlying growth lead of the Lucas numbers is now made explicit and exact. (As mentioned, this has previously been regarded as only an approximation, except in the limit). The author therefore has no doubt that the Fibonacci resonance formula pair will join the long list of useful Fibonacci and \(\phi\) formulae, which includes: \[ \begin{aligned} &\text{Binet:} &&F_n \ \ =\ \ \dfrac{\alpha^n-\beta^n}{\sqrt{5}}\\[6pt] &\text{Cassini:} &&F_{n+1}F_{n-1}-F_n^2 = (-1)^n \\[12pt] &\text{MAA, Benjamin & Quinn} \\ &\text{(2003)-Identity 3:} &&F_{n+m}=F_m F_{n+1} + F_{m-1} F_n. \\[9pt] \text{And now,} \\[6pt] &\text{the Fibonacci resonance:} &&F_{n+m}=F_m \alpha^n \ \ \ + \ \beta^m \,F_n. \\[6pt] \end{aligned} \] Such identities are the "bread and butter" of the Fibonacci Quarterly and Ron Knott's excellent website.  

We note in passing however, that the reviewer's (admittedly trivial) outline derivation of the resonance formula is post-factum — that is, easy for him to see, once he was given the result.

Moving on, the reviewer asserts that the author "merely reports" the nonsense that is written regarding \(\phi\):1 (golden) rectangles. But not so! The author actually cites 34 scientific studies and clearly states that any real effect has yet to be confirmed. Here again, in art and music, it is sad that the reviewer misses the point. At the start of the book, the goals are clearly stated: "first [to] gain in-depth understanding of Fibonacci and \(\phi\), and then to examine how these provide excellent tools for artists, musicians, architects, engineers and scientists---should they wish to use them".

Yet the reviewer apparently still believes that artists consciously insert golden rectangles into their works — like talismans. Whereas instead, we find that those artists and musicians (more than a few) who knew the maths and were skilled in its application have used it to partition their canvases and to structure their music. It is the successive choice of sections that can deliver a self-consistent proportional scheme. When using the \(\phi\) ratio, this process is unusually easy and extremely flexible (as Einstein famously noted when praising Le Corbusier's \(\phi\)-based "Modulor"). The Fibonacci Resonance shows how both Seurat and Toulouse-Lautrec partitioned certain \( 3\times 2 \) canvases using golden proportions — e.g. Lautrec's "Aristide Bruant" poster is \( \,3\phi\times (1+1+2\big /\phi)\). Some have sought to hide their workings (Seurat and Bartók), while others (such as Xenakis) — refreshingly — have in print publicly explained their use of maths and the golden proportion.

To summarize, The Fibonacci Resonance was not written for (as the reviewer describes himself) an "old and jaded" retired teacher. It was written instead for fresh and lively intellects. It was designed to take their popular interest in "Fibonacci and \(\phi\)" and (using high-school-level maths) leverage this into an understanding of Fibonacci and \(\phi\) as being very useful tools (in art, music, and architecture), and into an appreciation of how particular linear recurrences (and their Golden and Silver means) are at the core of some very important 21st century science and technology. The book draws upon umpteen "primary source" scientific papers — typically difficult or expensive for the general reader to obtain---and it discusses the latest research. Right now, the $180bn photonics industry is developing Fibonacci and related quasicrystal techniques, thereby significantly improving the performance of cutting-edge optical devices, e.g. photonic crystals for next-generation petabit optical communications networks and computing, and super-efficient photo-voltaic panels for green energy. Exciting stuff! These topics are extremely relevant and of great interest to up-and-coming young mathematicians and scientists.

Indeed, this book will be "a good read" for anyone with a lively curiosity and a high-school-level grounding in maths  anyone enthusiastic to learn about the practical uses of Fibonacci and \(\phi\) and some of the latest discoveries and developments in the arts and sciences. 

Clive Menhinick, Poynton, UK.