# Golden ratio in contemporary mathematics

A (non-mathematical) friend recently asked me the following question:

Does the golden ratio play any role in contemporary mathematics?

I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess that this is the same for almost every other mathematician working today.

I then began to wonder if I were totally correct in this statement . . . which has led me to ask this question.

My apologies is this question is unsuitable for Mathoverflow. If it is, then please feel free to close it.

• It is used for instance in the study of Fibonacci and related sequences. The recent proof that 144 is the largest perfect power in the Fibonacci sequence used crucially exponential equations with base, you guessed it, the golden ratio. Sep 4, 2021 at 13:07
• Very interesting! Sep 4, 2021 at 13:53
• In music, you can argue that an interval (ratio of frequencies) of the golden ratio is the most dissonant interval. Just type successive Fibonacci numbers into a tone generator here szynalski.com/tone-generator and play them at the same time. Sep 4, 2021 at 13:58
• Probably this is too old to be modern but the Fibonacci numbers were used by Lamé in his proof of the time complexity of the Euclidean Algorithm for finding the gcd. Sep 4, 2021 at 15:13
• The golden ratio is all over the place in geometry involving regular pentagons, and there is no end to research even there. See my MO user picture! Sep 5, 2021 at 11:18

The "Cleary group" $$F_\tau$$ is a version of Thompson's group $$F$$, introduced by Sean Cleary, that is defined using the golden ratio, and it's definitely of interest in the world of Thompson's groups. See An Irrational-slope Thompson's Group ( Publ. Mat. 65(2): 809-839 (2021). DOI: 10.5565/PUBLMAT6522112 ). Very roughly, where $$F$$ arises by "cutting things in half", $$F_\tau$$ arises in an analogous way by "cutting things using the golden ratio". There are lots of similarities between $$F_\tau$$ and $$F$$, but also plenty of mysteries, for example I believe it's still open whether $$F_\tau$$ embeds into $$F$$ (i.e., whether there exists a subgroup of $$F$$ isomorphic to $$F_\tau$$).

• This is a nice example, because it actually involves the golden ratio as a ratio, not just a number. Sep 4, 2021 at 13:24
• This is a beautiful example. Justin Moore and myself show that $F_\tau$ does not embed into $F$ in arxiv.org/abs/2103.14911 Oct 26, 2021 at 7:21
• @JamesHyde Oh, excellent! I somehow missed that, very nice result! Oct 26, 2021 at 9:26

Here's a chess problem. If one greedily places queens on an infinite $$\mathbf{N}\times\mathbf{N}$$ chessboard, column by column, such that at each step no two queens may attack one another, then one obtains a permutation of $$\mathbf{N}$$ (OEIS A065188): 1, 3, 5, 2, 4, 9, 11, 13, 15, 6, ..., and the queens appear to form two lines, of slopes $$\phi$$ and $$1/\phi$$.

While this has not been proven (it is conjectured that every term is either $$n\phi+O(1)$$ or $$n/\phi+O(1)$$), a similar result for queens greedily placed along a square spiral on a $$\mathbf{Z}\times\mathbf{Z}$$ board was established in F. M. Dekking, J. Shallit, and N. J. A. Sloane, Queens in Exile: Non-attacking Queens on Infinite Chess Boards (2020), Electronic Journal of Combinatorics 27, #P1.52. The queens there lie on lines with slopes related to the Tribonacci constant ($$\approx1.83929$$), which is the real root of $$x^3-x^2-x-1$$.

• this is very nice! Sep 8, 2021 at 4:06

I'm not sure how recent "contemporary" mathematics means so I'll mention a few things, some somewhat classical but all connected to the fact that $$\phi=\frac{1+\sqrt{5}}{2}$$ is closely connected to the Fibonacci sequence. I will endeavor to order these roughly in oldest to most recent.

First, in a certain sense, $$\phi$$ is the hardest number to approximate with rational numbers. What do we mean by that? if you have an irrational number $$\alpha$$, and you want to approximate $$\alpha$$ with rational numbers of the form $$\frac{n}{d}$$, then you can get approximations as good as you want by making $$d$$ larger. However, suppose you are interested in getting as close as you can and wanting to know how bad a price you pay in terms of increasing $$d$$, it turns out that by multiple ways of making this precise, $$\phi$$ is the worst possible. This is closely related to the fact that it has continued fraction $$[1,1,1,1...]$$ and so the best possible approximates actually have numerator and denominator Fibonacci numbers.

Understanding the Fibonacci numbers better turns out to be closely connected to the Binet formula, which says that $$F_n = \frac{\left(\frac{1+\sqrt{5}}{2}\right)^n- \left(\frac{1-\sqrt{5}}{2} \right)^n}{\sqrt{5}}.$$

So for example, there's an old result that the Fibonacci sequence distributes over gcd, that is $$F_{\mathrm{gcd}(a,b)}= \mathrm{gcd}(F_a,F_b).$$ It turns out that one of the more enlightening ways of proving this is by using the Binet formula and then looking at how the Fibonacci numbers behave in the ring $$\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$$.

This leads us to the more recent work. Define the order of apparition of $$n$$, denoted by $$z(n)$$, as the least $$k$$ such that $$n∣F_k$$. The last few years have seen extensive work on trying to understand this function, and the closely related function of the Pisano period, which says how long it takes $$F_n$$ to repeat mod $$m$$ for some $$m$$. In the early 2010s a whole bunch of papers on this topic were written by Diego Marques which are of note in this regard. One of the techniques here involves trying to understand $$\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$$ really closely.

In a related note, there's a recent paper in the Journal of Number Theory by Roswitha Hofer which generalizes the golden ration to fields of formal power series. Hofer's paper can probably be used as a jumping off point for generalizing some of the results mentioned above to other contexts.

• @ManfredWeis Fixed. Phrase should not have been repeated. Sorry about that. Sep 4, 2021 at 15:46
• Category theorists might be amused to note that $\mathrm{lcm}(z(n),z(m))=z(\mathrm{lcm}(n,m))$, because of the Adjoint Functor Theorem. Sep 4, 2021 at 22:53
• @OscarCunningham I am very curious about one would use the adjoint functor theorem to prove this. To me it seems just the (much more elementary) statement that the adjoint of a composite is the composite of the adjoints (in the other direction). Sep 5, 2021 at 18:21
• @DenisNardin The point is that $\mathrm{gcd}$ is a limit, and if $F$ preserves limits the $AFT$ says it must have a left adjoint $z$ which will then preserve colimits. Sep 6, 2021 at 10:49
• People often summarize the result in the second paragraph as "$\phi$ is the most irrational real number". However, since Liouville showed that reals that are very well approximated by rationals are transcendental, I say it should really be "$\phi$ is the most rational irrational". Sep 6, 2021 at 13:40

There are several exponential-time algorithms (in theoretical computer science, arguably a subfield of mathematics), whose running time can be bounded by an expression of the form $${n-k \choose k}$$ where $$k$$ is an integer between $$0$$ and $$n$$. Plugging in the $$k$$ that maximizes this term, we get an upper bound $$\sim \phi^n$$ where $$\phi = \frac{1+\sqrt{5}}{2}$$, the golden ratio. This is often in contrast to the naive approach whose running time would be $$\sim 2^n$$.

See for example in Section 3.4 of the book: Exact Exponential Algorithms, by Fomin and Kratsch: https://folk.uib.no/nmiff/BookEA/index.html

The golden ratio is also the order of convergence of the secant method .

edit Dec 2022. It seems it has not yet been recalled the relevance of the golden ratio and Fibonacci numbers in the theory of continued fractions, due to the Hurwitz's theorem.

It should be noticed that in both the examples above, and, it seems to me, in many examples quoted in these answers, the role of the golden ratio is not really due to a certain property in itself, but rather, to the fact that it is the best (or worse) case in a given context, so that it actually gives optimal bounds. I think this explains (and in some sense resizes) the "ubiquity" of the golden ratio and Fibonacci numbers in mathematics and in nature. It is not that simple models like the golden section, or "$$x_n=x_{n-1}+x_{n-2}$$", or the most popular "Fibonacci spiral" really explain so many facts (although it seems that out there most people really thinks so); it seems more true that they give first approximations and sharp bounds to more complex situations.

• Another optimization technique related to the golden ratio is golden-section search. Sep 5, 2021 at 22:14
• @Pietro: I am sort of impressed that you are living a year in the future. :-) Dec 17, 2022 at 2:34
• Oh, you are right and i'm afraid i've been there several other times this year :( Dec 17, 2022 at 7:38

Here is an interesting question that asks whether there is any function that satisfies $$f^{-1}=f'\;$$ i.e. its derivative is equal to the inverse function. In addition to the answers provided there, it can be shown that this can be reduced to solving the delay differential equation $$g'(x) =\frac{1}{g(x-1)}g'(x+1).$$ I am not aware of any practical use of this kind of equations, but it has a nice form nonetheless! And the functional can be defined as $$f(g(x))=g(x+1)$$.

Anyway, here's a fairly simple solution which is a bijection on $$\mathbb R$$ and, lo and behold, is entirely based on the golden ratio: $$f(x)=\begin{cases} \phi (x/\phi)^{\phi } & x\geq 0 \\ \psi (x/\psi)^{\psi } & x<0 \text{ where } \psi =-\phi ^{-1}. \\ \end{cases}$$

In every real quadratic field $$K$$, the unit group of its ring of integers $$\mathcal O_K$$ is known to have the form $$\pm u^\mathbf Z$$ for a unique number $$u > 1$$, which is called the fundamental unit of $$K$$ (really, of $$\mathcal O_K$$). The field $$K = \mathbf Q(\sqrt{5})$$, which is the real quadratic field with smallest discriminant (and the number field overall with the second smallest discriminant, after $$\mathbf Q$$), has $$\mathcal O_K = \mathbf Z[u]$$ and $$\mathcal O_K^\times = \pm u^\mathbf Z$$ where $$u = (1+\sqrt{5})/2$$.

One way to make the Golden ratio more interesting to mathematicians is to generalize the Fibonacci sequence to a non-commutative and possibly non-associative context.

Define the Fibonacci terms $$(t_{n})_{n\geq 1}$$ with respect to a binary operation $$*$$ by letting $$t_{1}(x,y)=y$$ and $$t_{2}(x,y)=x$$ and $$t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$$ for $$n\geq 1.$$

In an associative context, if the operation $$*$$ is the concatenation operation, then the binary words $$t_{n}(0,1)$$ are called the Fibonacci words. For example, $$t_{7}=0100101001001$$. Unsurprisingly, the golden ratio arises from the Fibonacci words too.

Clearly, the length of the Fibonacci words are the Fibonacci numbers and the number of occurrences of $$0$$'s and $$1$$'s are also Fibonacci numbers. Therefore, ratio of the number of 0's in a Fibonacci word to the number of 1's in the same Fibonacci word approaches the golden ratio. The golden ratio also arises from Fibonacci words in many different ways. For example, the $$n$$-th bit in any Fibonacci word except for the first one is $$2-[\lfloor (n+1)\phi\rfloor-\lfloor n\phi\rfloor]$$ (here we start at $$1$$, so $$(abcde)[1]=a$$).

The Fibonacci terms in a non-commutative context arise from large cardinals when obtaining the composition of rank-into-rank embeddings from the application operation on rank-into-rank embeddings. I don't think there are yet any known interesting relations between rank-into-rank cardinals and the golden ratio.

Golden ratio is relevant in most places where you consider Fibonacci numbers. One occurrence where it is particularly visible is the proof by Bugeaud, Mignotte and Siksek of the fact that the largest perfect power in the Fibonacci series is 144 (arXiv version here).

The key is, perhaps unsurprisingly, the Binet formula $$F_n=\frac{\phi^n-\phi^{-n}}{\sqrt{5}}$$. Equating it to a perfect power $$y^p$$ gives us an exponential Diophantine equation which we have to solve. The solution is a combination of methods in which $$\phi$$ occurs prominently: it occurs in determining various congruences on the numbers involved which are further crucial in applying modularity results (like in the proof of Fermat's last theorem), as well as in estimates on linear forms in logarithms using Baker's method.

There's an identity for chromatic polynomials of planar triangulations called the "golden identity" found by Tutte, giving a quadratic relation between the values of the chromatic polynomial at $$\phi+1$$ and $$\phi+2$$. In a fairly recent paper Slava Krushkal and I extended this identity (or rather its dual for the flow polynomial of planar cubic graphs) to the Yamada polynomial, an invariant of spatial graphs. We also have a conjecture that this identity characterizes planar cubic graphs in terms of the flow polynomial.

The golden mean also crops up in matroid theory. A matroid $$M$$ is a golden mean matroid if it can be represented by a real matrix such that every non-zero subdeterminant is $$\pm \phi^i$$, for some $$i \in \mathbb{Z}$$, where $$\phi$$ is the golden mean. Somewhat surprisingly, Vertigan proved that a matroid is a golden mean matroid if and only if it is representable over both $$GF(4)$$ and $$GF(5)$$, where $$GF(q)$$ is the finite field with $$q$$ elements.

See Stefan van Zwam's PhD thesis for more theorems along these lines.

The golden ratio $$\phi$$ is present in category theory and quantum algebra.

It is the smallest possible non-integral Frobenius-Perron dimension of a fusion ring/category, the one with simple objects $$\{1,X\}$$ and $$X \otimes X = 1 \oplus X$$.

Also, $$\phi^2$$ is the smallest possible non-integral index for a subfactor.

The golden ratio is closely related to the Zeckendorf representation which is one of the simplest examples of a numeration system (other than the usual base-$$b$$ represenations). As such, it's an important testing ground for new ideas. For instance, in this paper Drmota, Müllner and Spiegelhofer showed that the parity of the sum of digits in the Zeckendorf representation does not correlate with the Möbius function. Such considerations are interesting for their own sake, and are also connected with morphic sequences thanks to the work of Rigo: One can think of a morphic sequence as an automatic sequence in a non-standard numeration system.

Here is an example which on the surface has nothing at all to do with Fibonacci numbers or continued fractions. Theorem 1.1 of Itai Dinur's 2021 SODA paper Improved algorithms for solving polynomial systems over GF(2) by multiple parity-counting states:

There is a randomized algorithm that given a system $$E$$ of polynomial equations over $$\mathbb{F}_2$$ with degree at most $$d$$ in $$n$$ variables, finds a solution to $$E$$ or correctly decides that a solution does not exist with high probability. The runtime of the algorithm is bounded by $$O(2^{0.6943n})$$ for $$d=2$$ and by $$O(2^{(1-1/(2d))n})$$ for $$d>2$$.

The author adds, "We note that for $$d=2$$, the complexity of our algorithm can be made arbitrarily close to $$O(\varphi^n)$$, where $$\varphi = \frac{1}{2}(1+\sqrt{5})$$ is the golden ratio."

Dinur's algorithm, which builds on recent work by other authors, is highly ingenious and is not just a straightforward re-packaging of the results mentioned by some other respondents where $$\varphi$$ shows up in the asymptotic analysis of the runtime of an algorithm. One indication that the appearance of $$\varphi$$ here is not at all obvious is that the previous best algorithm for $$d=2$$ had an asymptotic runtime of $$O(2^{0.804n})$$.

Clearly there are many possible answers; MathSciNet has 50 entries with "golden mean" in the title and 447 other entries with the phrase appearing "anywhere." Let me mention three in particular.

One of the commonly claimed applications of the Fibonacci numbers in nature is sunflower seeds. While some applications can be disputed, this one seems to be accurate. Michael Naylor wrote an engaging article "Golden, √2, and 𝜋 Flowers: A Spiral Story" for Mathematics Magazine (75(3) (2002) 163-172) that includes a reference to Mitchison's "Phyllotaxis and the Fibonacci Series" in Science (196 (April 1977) 270-275).

A more surprising application may be in symbolic dynamics, where the "golden mean shift" consists of bi-infinite binary sequences the avoid adjacent ones. Similar to $$a(n) = a(n-1) + a(n-2)$$ being one of the most foundational recurrence sequences, this golden mean shift is a basic shift space in the younger field of dynamical systems.

Finally, not so modern but unexpected: Fibonacci numbers arise in analyzing the Euclidean algorithm; they contribute the "worst cases,'' e.g., running the algorithm on 8 and 5 takes four steps, and no pair $$(a,b)$$ with $$a, b \le 12$$ takes more. Knuth says of this result (due to Lamé and other 19th century French mathematicians), "This theorem has the historical claim of being the first practical application of the Fibonacci sequence." (TAOCP 2, p. 360)

• I like the reference to the golden mean shift, but I don't know that TAOCP and the 1977 phyllotaxis article quite count as 'contemporary mathematics' at this point... Sep 5, 2021 at 1:46
• @StevenStadnicki, thanks. For the Euclidean algorithm analysis, there are still details being worked out. A coauthor and I just had "Ties in Worst-Case Analysis of the Euclidean Algorithm" published (Math. Commun. 26 (2021) 9-20) where we worked out the other pairs that take as many steps as the minimal Fibonacci pair. For instance, in $(a,b)$ with $a \le b \le 12$, the pairs $(5,8)$, $(7,11)$, $(7,12)$ and $(8,11)$ all require four steps. The golden ratio came up all over the asymptotic analysis of other families of solutions. (I'm not up to date on mathematical phyllotaxis.) Sep 5, 2021 at 2:05

One of the most clever and amusing, yet deep, presentations I've seen on the interdisciplinary import of the golden ratio and Fibonacci sequences is the video sequence by V. Hart "Doodling in Math: Spirals, Fibonacci, and Being a Plant" and, of course, the comments and refs in the OEIS entry A000045 on the standard Fibonacci sequence and the golden ratio contain a lot of applications.

• It's Vi Hart, not V. I. Hart. Vi is a short form of Victoria. And yes, I agree it's a great presentation :-) Sep 5, 2021 at 6:38
• See also Excursions in Calculus: An Interplay of the Continuous and the Discrete by Young Sep 12, 2021 at 18:09
• Strogatz at opinionator.blogs.nytimes.com/2012/09/24/proportion-control gives a short list of general refs. for the golden ratio. Feb 12, 2022 at 16:22

The golden ratio occurs in the asymptotic growth-rate for the number of numerical semigroups of given genus:

A numerical semigroup of genus $$g$$ is a subset $$S$$ of $$\mathbb N=\{0,1,2,\ldots\}$$ such that $$S=S+S$$ and $$S$$ lacks exactly $$g$$ elements of $$\mathbb N$$.

The number $$n(g)$$ of numerical semigroups of genus $$g$$ is easily shown to be finite and is asymptotically given by $$C \omega^n$$ for some constant $$C$$ with $$\omega=(1+\sqrt{5})/2$$ the golden number. This was conjectured by M.Bras-Amoros, Fibonacci-like behaviour of the number of numerical semigroups of a given genus, Semigroup Forum 76, No 2, 379--384 (2008), proven by A. Zhai, Fibonacci-like growth of numerical semigroups of a given genus, Semigroup Forum 86, No 3, 634--662 (2013). A different and hopefully more comprehensive proof (sorry, self-promotion) is contained in https://arxiv.org/abs/2105.04200 .

Golden ratio appeared in the recent breakthrough A constant lower bound for the union-closed sets conjecture of Frankl's Union Closed Set conjecture by Justin Gilmer, as well as the subsequent optimization Chase and Lovett - Approximate union closed conjecture of the lower bound constant. It relies crucially on the following fact:

Given two iid biased coins with bias $$p$$, the entropy of both being head equals the entropy of a single one being head, only when the bias $$p \in \{ 0, \frac{\sqrt{5} - 1}{2}, 1 \}$$.

This is an easy consequence of the relation $$H(p) = H(1 - p)$$, and that the chance of both coins landing in heads is $$p^2$$.

In numerical analysis, a basic problem consists in approaching a solution $$x^*$$ of $$f(x)=0$$, where $$f:{\mathbb R}\rightarrow{\mathbb R}$$ is at least continuous and more often $$C^2$$-smooth. The basic methods are those of dichotomy, secant and Newton. Dichotomy is more than elementary (based upon the intermediate value theorem) and its convergence is of order $$1$$, meaning that the error $$\lvert x-x^*\rvert\sim2^{-n}$$ decays exponentially. Newton's method is much faster — order $$2$$ if $$x^*$$ is non-degenerate, that is $$f'(x^*)\ne0$$, which means $$\lvert x-x^*\rvert=O(\rho^{2^n})$$ for some $$\rho<1$$. But it requires the knowledge of $$f'$$, which can be questionable in real world applications, where values of $$f$$ are given by measurements. A good compromise is the secant method, which resembles Newton a lot, but the tangent to the graph at $$x_n$$ is replaced by the chord of the graph determined by the abscissæ $$x_{n-1}$$ and $$x_n$$. Then its intersection with the horizontal axis determines $$x_{n+1}$$.

The secant method turns out to be of order $$\phi$$ whenever $$x^*$$ is non-degenerate: $$\lvert x-x^*\rvert=O(\rho^{\phi^n})$$ for some $$\rho<1$$.

The main drawback of the secant method is that its extension to vector fields $$f:{\mathbb R}^n\rightarrow{\mathbb R}^n$$, which can be defined easily, behaves badly when $$n\ge2$$. Notice that the extension of the dichotomy, which involves triangulations and the Sperner Lemma, becomes very slow. Only the Newton method admits an efficient generalization (then called Newton–Raphson).

Yes, it does:

Lyubich, Mikhail; Milnor, John The Fibonacci unimodal map. J. Amer. Math. Soc. 6 (1993), no. 2, 425–457.

and two more papers of the same authors studying what they call Fibonacci map.

Consider the function which is the limit of the sequence $$x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$$ Call this function $$U(x)$$, with domain $$\{x \in \mathbb R \mid x \geq 0\}$$. Notice that for $$x \in [0,1]$$, $$U(x)$$ is equal to the golden ratio. It can be shown that for $$x > 1$$, $$U(x)$$ is greater than the golden ratio and $$U(x) \sim x$$. Finally, this function can be used to show that problem (I) is equivalent to problem (II).

Problem (I): Find the limit, and upper bound on convergence rate, for the sequence $$\sqrt{u_1}, \sqrt{u_1 + \sqrt{u_2}}, \sqrt{u_1 + \sqrt{u_2 + \sqrt{u_3}}}$$, etc. where all $$u_n$$ are non-negative.

Problem (II): Find a way to compute the terms of the sequence $$n \mapsto \sup_{k \geq 0} u_{n+k}^{2^{-(n+k)}}$$.

This is a "constructive" analogue of Herschfeld's Convergence Theorem. Herschfeld's theorem gives a necessary and sufficient condition for an infinite radical to converge. The above gives a necessary and sufficient condition for being able to compute what an infinite radical converges to. The two necessary and sufficient conditions are quite similar.

Postscript: There is a sense in which $$U(x)$$ is a family of transfinite radicals, which generalises the notion of an infinite radical. For more, see the preprint Constructive proof of Herschfeld's Convergence Theorem, arXiv:1907.02700 by R Gutin

• Where is the golden ratio here? Sep 4, 2021 at 20:39
• @SamHopkins $U(0)$ is the golden ratio
Sep 4, 2021 at 20:40
• @SamHopkins In fact, $U(x)$ for $x$ between $0$ and $1$ is the golden ratio. For $x > 1$, $U(x)$ is larger than the golden ratio
Sep 4, 2021 at 20:41
• @SamHopkins Finally, if you know the proof of HCT, then you'll know that the golden ratio plays a key role there
Sep 4, 2021 at 20:42

The golden L is a translation surface that has received much attention recently in the theory of billiard dynamics. It is built out of a $$1 \times 1$$ square with $$1 \times \phi$$ and $$\phi\times1$$ rectangles glued to it along their length-$$1$$ sides. See for example the preprint Davis and Lelievre - Periodic paths on the pentagon, double pentagon and golden L.

See the article "Generalized Dehn–Sommerville relations for polytopes, spheres and Eulerian partially ordered sets" by Margaret Bayer and Louis Billera. These authors aimed to extend the Dehn–Sommerville equations to homology spheres and face lattices of non-simplical convex polytopes. One of their results is that the dimension of the affine hull of a certain set of flag vectors for an Eulerian poset of rank $$d$$ is the $$d$$th Fibonacci number.

In the active field of research called symbolic dynamics (for a nice overview see https://www.southalabama.edu/mathstat/personal_pages/williams/wilshort.pdf by Susan Williams), the canonical example of a subshift of finite type that is not isomorphic to a full shift is the “golden mean subshift”, so called because its entropy is the log of the golden ratio.

I'm not sure whether it qualifies or not, but I found a link between the golden ratio and the Riemann Hypothesis some years ago: Is there a hidden connection between RH and the golden ratio?.

No matter how anecdotic it might be, I nevertheless find this aesthetically pleasant.

Here we have a monotonocally increasing sequence that predicts itself:

$$1,2,2,3,\color{brown}{3},4,4,4,\color{blue}{5,5,5},6,6,6,6,...$$

The $$n$$th term $$a(n)$$ predicts the number of times that term appears in the sequence. For instance, the fifth term is $$3$$ (brown) and thus $$5$$ will appear three times (blue).*

The prediction is implemented through the recursion relation

$$a[n+a(a(n))]=a(n+1).$$

For example, $$a(5)$$ is the second occurrence of $$3$$, and when we march $$a(a(5))=a(3)=2$$ steps forward from there we land at $$a(7)$$, which is the second occurrence of $$3+1=4$$.

Let us plug this into an assumed power-law asumptotic relation:

$$a(n)\approx\alpha n^\beta.$$

Thus

$$\alpha[n+\alpha(\alpha n^\beta)^\beta]\approx\alpha n^\beta+1$$

$$\alpha[n+\alpha^{1+\beta}n^{\beta^2}]^\beta\approx\alpha n^\beta+1$$

Take two terms of the binomial power expansion of the left side:

$$\alpha [n^\beta + \alpha^{1+\beta}\beta n^{\beta^2+\beta-1}]^\beta\approx\alpha n^\beta+1$$

Canceling the identical leading terms leads to

$$[\alpha^{1+\beta}\beta n^{\beta^2+\beta-1}]^\beta\approx1$$

from which $$\beta^2+\beta-1=0$$ and thus $$\color{blue}{\beta=\phi-1=1/\phi}$$. The corresponding value of $$\alpha$$ is then $$\beta^{-1/(1+\beta)}=\phi^{\phi-1}$$. We may elegantly express the result as

$$a_n\approx(\phi n)^{\phi-1}.$$

If we graph $$a_n$$ versus $$n$$ on a log-log plot and draw the straight line represented by $$\alpha n^\beta$$ with the values computed above, we find that the line precisely "threads the needle" through the sequence values. Try it!

*Yes, that is a shout-out to one of our favorite video channels.

Let $$\rho$$ be the binary substitution defined by: $$\rho(00)=\text{empty word}\quad\rho(01)=1\quad\rho(10)=0\quad\rho(11)=01.$$ Let $$R$$ be the self-map of $$[0,1]$$ associating to every $$x=(0.w)_2$$ the number $$R(x)=\left(0.\rho(w)\right)_2$$, taking the binary expansion ending in $$1^\infty$$ in case of dyadic rationals.

In The simplest erasing substitution, Stefano Isola, Riccardo Piergallini, and I proved that the Hausdorff dimension of the fibers of the map $$R$$ has an (optimal) upper bound in $$\log_2\phi$$.