# The Fibonacci Resonance and other new Golden Ratio discoveries

The book deals with the Golden Ratio $\phi$, Fibonacci numbers $F_n$ and friends. And there are a lot of friends. The Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ with $F_0=0$ and $F_1=1$ and $\phi=(1+\sqrt{5})/2$ which is the limit of $F_{n+1}/F_n$. For this book, the Binet relation $F_n=(\alpha^n-\beta^n)/\sqrt{5}$ with $\alpha=\phi$ and $\beta=1/\phi=(1−\sqrt{5})/2$ is even more important. Throughout history in human artwork, the Golden Ratio 1:$\phi$ has repeatedly appeared, but also 1:$\phi^s$ with $s$ an integer (positive or negative) or one half. Another important player is the logarithmic spiral (the Golden Spiral) with polar equation $r=\phi^{b\theta}$ with $b=2/\pi$. This means that with every quarter circle, the value of the radius is an integer power of $\phi$ anchored at $\phi_0=1$ for $\theta=0$. Lucas numbers $L_n$ satisfy the same recurrence $L_n=L_{n-1}+L_{n-2}$ but they start with $L_0=2$, and $L_1=1$. Their Binet formula is $L_n=\alpha^n+\beta^n$. It is supposed that $F_{-n}=(-1)^{n+1}F_n$ and $L_{-n}=(-1)^nL_n$ for all integer $n$.

All this is pretty well known, and it can of course be found in this book, but there is much more. It attempts to give a roundup of all that is known about Fibonacci numbers and friends, and adds to this a new insight from the author: the Fibonacci resonance. As the author states in his preface, this volume is bundling three books in one. The first one (here represented by Part I) is conform to what most popular science books mention about these issues. It is an historical survey of how $\phi$ and the Fibonacci numbers appeared in science, in nature, and in artwork in the past. The `third book' (appearing in the form of Part V) is a continuation of the first one, but these elements are less easily found in the literature, at least not in this accessible form, since it discusses several applications of $\phi$ and friends in more recent developments of science. The middle piece (Parts II, III, IV) is more mathematical and develops some original ideas of the author about what he calls the Fibonacci resonance which is based on the elements that were introduced in my first paragraph of this review. But let me start with the historical background.

All the usual suspects appear in the historical survey. Of course the pyramids from Egypt, but also, and these may be less familiar to an average public, the megalithic Sun and the Moon gates in Bolivia, which are extensively discussed. Also the meter of classical Sanskrit involves mathematical patterns. Obviously on the list are also the mathematics of the ancient Greek, re-introduced in the West by the Arabs, and the scientific evolution since the 15th century with the naming of the Fibonacci numbers and the introduction of the symbol $\phi$. The spirals are introduced and their classical appearance in nature (nautilus, pineapple, pine cone, sunflowers, Roman broccoli, etc.). Also in music such patterns can be detected. Bartók, Debussy, and Xenakis are taken as examples. Paris became 'the capital of $\phi$' in the course of the 19th and the early 20th century when artists picked up the Golden Section credo that was propagated by scientists and theoreticians who strongly influenced the Parisian art scene. Among them Charles Henry, friend of the mathematician Édouard Lucas who studied the Fibonacci sequence, Joséphin Péladan who promoted $\phi$ on mystical grounds, Maurice Princet, the mathematician of cubism etc. Extensive discussions are devoted to paintings of Seurat, Toulouse-Lautrec, and Mondrian, the purist `par excellence'. Architecture is represented by Le Corbusier and his Modulor, a system of proportions based on an anthropomorphic scaling.

The `third book' is called '$\phi$ science' and is also 'traditional' in the sense that it lies in the line of expectations. Here we meet the recent applications of the Fibonacci sequence. There we obviously find phylotaxis, not only in plants, but also in DNA, superconductors, and sunlight harvesting. A link not mentioned so far is sphere packing and tiling. Here we meet 3D crystal structures, Penrose tilings, and quasicrystals (extensively discussed), Islamic patterns, superlattices and composites (metamaterials) with the protagonists (Victor Veselago, John Pendry, Dan Shechtman) and applications (cloaking, plasmonics,...). Of course all these applications are practically important, but some topics such as Fibonacci word and Penrose tiling are interesting structures that invite to be studied at an abstract theoretical level.

This leaves us with the most original, most surprising, and most mathematical part of this book. The first idea is to divide the goniometric circle into 32 equal parts. This results in an Ori32 geometry referring to 32 possible orientations. Menhinick got his inspiration from the fact that many applications depend on angles and orientations, rather than the classical point-line-plane approach to geometry. With these 32 parts, the smallest part of the disk is thus a wedge with angle $\pi/16$. This is used as a kind of unit and is called a MIK. It is denoted as $\fbox{1}$ and one can consider multiples of this unit. A quarter disk corresponds for example to $\fbox{8}$. The disk can be divided into wedges progressing in Fibonacci-like manner. Putting together $\fbox{1}$, $\fbox{2}$, $\fbox{3}$, $\fbox{5}$, $\fbox{8}$, and $\fbox{13}$, one get the full disk because $(1+2+3+5+8+13)\pi/16=2\pi$. Because of the periodicity, calculus with these MIK is modulo 32. If the radius of the circle is 1, then the arc length of $\fbox{5}=5\pi/16\approx1$. Similarly the arc of $\fbox{8}=\pi/2\approx\phi$ and for $\fbox{3}$ we get $3\pi/16\approx1/\phi$ etc. All these approximations are too large, so we may forget $\fbox{1}$ (which contributes $\phi^{-3}$) and get $2\pi\approx\phi^{-2}+\phi^{-1}+1+\phi+\phi^2=2\phi+3=4+\sqrt{5}$. Because the arc lengths are only approximately correct, Menhinick points to the analogy of the Pythagorean comma in music theory. His search for the necessary correction resulted in a generalization of the Binet formula: $F_n=F_s\alpha^{n−s}+F_{n-s}\beta^s$ for all integers $n$ and $s$. To visualize this, a golden spiral with equation $r=(F_s/\phi^s)\phi^{b\theta}$ is constructed for each $s$. These are in fact simple transforms of a (standard) golden spiral. The first term $F_s\alpha^{n−s}$ defines points on the spiral at its intersections with the coordinate axes. The second term is a quantized deviation to get $F_n$ from the previous spiral points namely $F_{n-s}$ times a quantum $\beta^s$. Similarly for the Lucas numbers, one has $L_n=\sqrt{5}F_s\alpha^{n-s}+L_{n-s}\beta^s$. The $\sqrt{5}$ rotates the axes over about 150 degrees and the intersections of these rotated axes with the spirals gives again approximations $\sqrt{5}F_s\alpha^{n-s}$ for $L_n$ with a quantized deviation $L_{n-s}\beta^s$. Note that each spiral has its own characteristic quantum $\phi^{-s}$. Then Menhinick considers the finest quantum to be half a wavelength. Since quanta for different $s$ always appear in integer multiples, this can be considered as standing waves of different frequencies that 'vibrate' in resonance: the Fibonacci resonance. It is also investigated whether there is fractal behavior but that issue seems not to be cleared out completely. This is followed by an extra part in which these ideas are applied to generalized Lucas sequences $U_n=PU_{n−1}+QU_{n-2}$, starting with initializations 0 and 1. The Pell and Pell-Lucas numbers are a special case for $(P,Q)=(2,−1)$. The analog of $\phi$ is here the Silver Ratio $\delta=1+\sqrt{2}$. They were recently (2014) studied in the book *Pell and Pell-Lucas Numbers with Applications* by T. Koshy, which is not in the (otherwise quite extensive) list of references of this volume.

All the material that I discussed so far takes about 400 of the approximately 600 pages. The remaining one third of the book consists of appendices with technical and mathematical details, glossaries of terms and symbols used, a collection of formulas, and most of all an overwhelmingly extensive list of references (1004 items!). Also the index is well stuffed and useful for an encyclopedic work such as this book.

There is no doubt that what is described as the first and the third book are useful additions to what is already available in the literature. There are certainly original contributions also there. About the second book, introducing Fibonacci resonance, I am not so sure where all this is leading to, and what to think of all this spiraling number magic. There are obviously interesting, and as far as my knowledge is concerned, new relations derived in this part. What I mean to say is that I can certainly appreciate the formulas underlying the concept, but the resonance interpretation hints to numerological significance that I believe unnecessary. A remark such as the fact that the right-hand side of $\sum_{n=−2}^7\phi^n=\frac{11}{2}(7+3\sqrt{5})$ (formula (12.3)) combines the first 5 primes: 2,3,5,7,11, is of course true, but it is in my opinion pure coincidence and has no further meaning. And there are other examples, including the resonance interpretation, which is amusing and imaginative, but otherwise with little mathematical significance.

The illustrations are plentiful and helpful, except perhaps the 3D model of the different $s$-spirals which is for me only more confusing than what is already in the previous chapter. All the facts and persons of the book are extremely well researched and referenced. Also the pointers forward and backward are detailed and make it so much easier for the reader, and the typesetting in LaTeX is practically flawless (some italic instead of roman *log*'s and *arctan*'s here and there are minor exceptions).

**Submitted by Adhemar Bultheel |

**8 / Dec / 2015