This question already has an answer here:
- Uses of the 'Golden Ratio' 4 answers
I know the golden ratio is the limit of the ratios of consecutive Fibonacci numbers and that it appears when studying many related combinatorial objects (such as the sequences of zeros and ones with no consecutive ones and the Fibonacci substitution $1 \mapsto 10$, $0 \mapsto 1$). I have also read this answer which states that it is, in some sense, the hardest irrational to approximate by rationals. I'm also familiar with its geometrical definition.
Are there any other interesting properties or applications of this number within mathematics, maybe outside of combinatorics? I'm not interested in its applications within physics, aesthetics or any other discipline.
Edit: This is not a duplicate, since I'm explicitly asking for a different answer than the one given for the other question.