The “Golden” Non-Euclidean Geometry

Fibonacci numbers and their relation to the golden ratio are among the few mathematical items that gained some publicity among non-mathematicians. The golden ratio ($\phi=1.68033...$) is well known since antiquity and it played an important role in Euclid's Elements and in the work of many other mathematicians. It also shows up in phylotaxis and spirals that appear in nature. And it relates to harmony, another term that has been studied in a mathematical sense since the Greek. The golden ratio has therefore gained some mythical and even mystical status, the latter often has to be understood in a (pejorative) numerological sense.

Fibonacci numbers (denoted $F_n$) is a term coined by Édouard Lucas in the 19th century, who also introduced the sequence of Lucas numbers (denoted $L_n$). Both sequences are solutions of the difference equation $x_{n}=x_{n-1}+x_{n-2}$. The initial conditions for the Fibonacci sequence are $(x_1,x_2)=(1,1)$, while for the Lucas numbers it is $(x_1,x_2)=(1,3)$. The limit of ${x_n}/{x_{n-1}}$ equals $\phi=({1+\sqrt{5}})/{2}$ in both cases. These numbers are defined for all integer indices by $x_{-n}=(-1)^nx_n$.

In a first chapter, the authors give a brief historical survey, summarize some properties of the Fibonacci and Lucas numbers and they introduce hyperbolic functions: the symmetric Fibonacci hyperbolic sine $sFs(x)=({\phi^x-\phi^{-x}})/{\sqrt{5}}$ and cosine $cFs(x)=({\phi^x+\phi^{-x}})/{\sqrt{5}}$. Similarly for the Lucas versions $sLs(x)$ and $cLs(x)$ but these do not have the denominator $\sqrt{5}$. Their graphs look very much like the graphs of the standard hyperbolic functions.

The second chapter is about harmony. The old Greek Music of the spheres was picked up by Pacioli and Kepler. But soon the text comes down to one of Stakhov's pet horses, namely that harmony is a forgotten pillar in mathematics. Counting and classical measure theory are the two other pillars that have resulted in conventional mathematics. However by rejecting Cantor's axiom (a 1-to-1 correspondence between the reals and the points on a line) and a consistent application of the golden ratio and its generalizations, a different measure theory, number system, and geometry can be developed. This is what he calls harmonic mathematics. He considers a delayed version of the above difference equation which leads to the introduction of a new representation of number systems and his $p$-Proportion Codes. However this is soon replaced by another generalized Fibonacci sequence, defined for any real $\lambda>0$ by $F_\lambda(n+1)=\lambda F_\lambda(n)+F_\lambda(n-1)$, with $F_\lambda(0)=0, F_\lambda(1)=1$ and the limiting ratio $\phi_\lambda=(\lambda+\sqrt{4+\lambda^2})/2$ which is a root of the characteristic equation $x^2-\lambda x-1=0$. The above Fibonacci hyperbolic functions can be generalized by replacing $\phi$ by $\phi_\lambda$ and the $\sqrt{5}$ by $\sqrt{4+\lambda^2}$. They are denoted as $sF_\lambda$ and $cF_\lambda$. Note that (up to a factor 2) the classical hyperbolic functions are obtained as a special case of the $\lambda$-Lucas numbers by choosing $\lambda=e-1/e$. For $\lambda=1,2,3,4$ we get the golden, silver, bronze, and copper relations, referred to as the metallic relations.

The third chapter is about Hilbert's fourth problem, in which it is asked to design new forms of non-Euclidean geometry. The formulation was however rather vague and different proposals were made but it remained unclear whether the problem was (completely) solved or not. So the authors have their own interpretation and solve their form of the fourth problem using the hyperbolic functions introduced above. Lobachevsky's hyperbolic geometry is based on classical hyperbolic functions. Replacing the classical ones by the hyperbolic $\lambda$-Fibonacci functions they get different hyperbolic geometries. To obtain a similar generalization for spherical geometry, yet another type of Fibonacci functions are needed. There are of the form $SF_\lambda(x)=c_\lambda\sin(x\ln\phi_\lambda)$ and $CF_\lambda(x)=c_\lambda\cos(x\ln\phi_\lambda)$ with $c_\lambda=2/\sqrt{4+\lambda^2}$. The $\ln\phi_\lambda$ factor appears here for the sake of harmony. A similar form can be obtained in the hyperbolic case giving a true hyperbolic geometry in harmony mathematics. They consider many more relations and formulas in this context and claim that the Clay Mathematics Institute made a mistake by not putting Hilbert's fourth problem on their list of millennium problems. So the authors claim to have actually solved a self declared millennium problem.

The next chapter 4 is about the qualitative theory of dynamical systems based on harmony mathematics. Hence the `golden' and also the other metallic proportions show up again. It is a simple observation that a metallic ratio $\phi_\lambda$ (which is an irrational number) can be approximated from above and below by ratios of successive $\lambda$-Fibonacci numbers. This simple fact is exploited in a complicated framework of foliations of a 2D manifold. First foliations of such a manifold are introduced, which is then specialized to the 2D torus $T^2$. These foliations are characterized by a Poincaré rotation number $\omega$. In the particular case that it happens to equal a metallic proportion, then it can be approximated by ratios of Fibonacci numbers and hence the irrational foliation is approximated by rational ones. Since integral curves for flows of a dynamical system are foliations, this may also be applied in a context of dynamical systems. This chapter is much more mathematical with long mathematical proofs which do not seem to be easily accessible for a general public.

A last chapter is about the fine structure constant in physics. Like the mathematical millennium problems, there is a list of physical millennium problems. The first of these problems is asking whether all dimensionless parameters of the physical universe are calculable. Here the fine structure constant $\alpha$ is declared to be fundamental and hence is the constant to be discussed. The approach taken here is by looking at the Lorentz transform in special relativity theory. It is a transformation of the space-time vector whose matrix can be written as a direct sum of the identity and a hyperbolic rotation over an angle $\theta\in(-\infty,\infty)$. In view of the preceding items it is again a natural thing to replace the classical hyperbolic sine and cosine functions of the rotation angle by the hyperbolic Fibonacci sine and cosine ($\lambda=1$) of an appropriate angle $\psi$ and so obtain a Fibonacci special relativity theory. Here however $\psi\in(-\infty,0)\cup(2,\infty)$ because singularities appear at 0 and 2. Moreover, the speed of light in vacuum has to be made variable. It decreases with the age of the universe. It will be $c^*$ (the classical value) for $\psi\to-\infty$ and it is $c^*/\phi^2$ for $\psi\to-\infty$. The physical meaning is that the Big Bang corresponds to $\psi=0$, the interval (0,2) is the dark age before galaxies were formed (the speed of light is imaginary), and for values larger than 2 this corresponds to the light age, when the stars were formed that created light in the universe. To the left of the origin is the black hole situation with the arrow of time reversed.
In 2000, N.V. Kosinov proposed a formula $\alpha=10^{-43/20}\times\pi^{1/260}\times \phi^{7/130}$. Inspired by this formula, the authors propose to let $\alpha$ depend on $\psi$ by replacing the $\phi$ in this formula by their $\psi$-depending speed of light. The result is an $\alpha(\psi)$ with $\psi=\lambda_0 T$ where $T$ is the age of the universe (in billions of years) and $\lambda_0$ a constant. This $\alpha$ is decreasing with $\psi$ in the black hole range until it becomes 0 at the Big Bang. In the same range the speed of light drops from $1/\phi$ to 0. In the dark age, the derivative is positive and goes from 0 to $\infty$ just like the modulus of the speed of light does, and in the light age it drops from infinity to a little bit below its current value of about $7.29\times 10^{−3}$. Of course as a consequence of the varying $\alpha$, also other values that depend on it will change with the age of the universe. In an appendix allusion to multiverses is made when the $\phi$ in the previous setting is replaced by $\phi_\lambda$ with $\lambda\ne 1$.

The first author Alexey stakhov is a Ukrainian mathematician with a PhD in computer science, who lives in Canada since 2004. He has published many papers and books in which he has proposed many of his original, sometimes controversial, ideas. Chapter 2 clearly summarizes some of his previous work. The second author is Samuil Aranson who is a Russian mathematician, now living in the USA whose domain is differential equations, geometry and topology. It is therefore clear that he must be the main author for chapters 3 and 4, which also explains the somewhat different and more mathematical style. Scott Olson is a professor of philosophy and religion in the USA, who wrote a book on the golden section and who seems to be helping with the English editing of this book.

The first two chapters are elementary with a lot of history and simple mathematical relations. Who wants to read more on Fibonacci and Lucas numbers and generalizations can read Pell and Pell-Lucas Numbers with Applications for a good mathematical treatment and there are of course many popular books on the golden ratio. If you are interested in the golden ratio and harmony, you would certainly want to read The Fibonacci Resonance and other new Golden Ratio discoveries. However chapters 3 and 4 of this book are much more mathematical and create a complicated mathematical framework of foliations, not suitable for a general public anymore, while it only illustrates and applies the fact that the ratio of two successive Fibonacci numbers tend to the golden ratio and hence that this irrational number can be approximated by rationals. The fifth chapter is devoted to physics. The core idea is to replace a classic hyperbolic rotation by a more general one. The physical interpretation is certainly not mainstream and is probably susceptible to critique by theoretical physicists, if they do not consider it to be just numerological mysticism. However, since there is no experimental proof of what is exactly happening at this cosmological scale, it may be another explanation that is as good as many other fantasies. It is clear that the book is mainly collecting results that the authors have published as papers and that are here somewhat streamlined into a more consistent survey. Long lists of references are added after each chapter with many papers of the authors but several are only available in Russian. That this harmony mathematics and Fibonacci numbers and generalizations can solve all these problems clearly adds to the myth of the golden ratio. The typesetting in LaTeX is nicely done. I could spot a few typos but not that serious. For example page 121, a $(dv)^2$ is missing in the equation and on page 232 the Black Hole should correspond to $-\infty<\psi<0$ and not $0<\psi<2$. Also the graphics of chapter 5 are a bit rough and not always very precise. Anyway there are some original ideas to be found in this book. Whether the reader will agree with them or not will depend on who's reading it.

Adhemar Bultheel
Book details

This is a book in which the authors give a summary of some of their work. They study Fibonacci and Lucas numbers and show how these give rise to a new kind of mathematics: the mathematics of harmony. Generalizations of these number sequences and their limits the golden and other metallic ratios are applied to derive a new kind of non-Euclidean geometry, to study foliations and dynamical systems and even a golden Fibonacci version of the special relativity theory in which the fine structure constant from cosmology is analyzed.



978-981-4678-29-2 (hbk)
£98.00 (hbk)

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