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.



Golden ratio in contemporary mathematics
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](https://arxiv.org/abs/1806.00108)* ( 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 $F\_\tau$).


