The Fibonacci Resonance

(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: \[
%\boxed{ \textrm{for all integers }n,s
\left.
\begin{aligned}
\ \ F_n \ &=\ \ \phantom{\sqrt{5}}\ F_s\,\phi^{n-s} \ \ \ +\ \ \ F_{n-s}\,(-\phi)^{-s}\quad\\[3pt]
\ \ L_n \ &=\ \ \sqrt{5}\ F_s\,\phi^{n-s} \ \ \ +\ \ \ L_{n-s}\,(-\phi)^{-s}\quad
\end{aligned}
\right\}\quad\begin{cases}
\textrm{for all integers }n,s,\ \ \\
\phi=(1+\sqrt{5})\big /2.
\end{cases}
%}
\]

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.