TITLE: Golden ratio in contemporary mathematics
QUESTION [55 upvotes]: 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.

REPLY [3 votes]: 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 $|x-x^*|\sim2^{-n}$ decays exponentially. Newton's method is much faster - order $2$ if $x^*$ is non-degenerate, that is $f'(x^*)\ne0$, which means $|x-x^*|=O(\rho^{2^n})$  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 ressembles a lot Newton, 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:  $|x-x^*|=O(\rho^{\phi^n})$  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).