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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.


marked as duplicate by Gerry Myerson, Jonas Meyer, Micah, Mike Pierce, user223391 Jun 21 '15 at 2:57

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    $\begingroup$ Fibonacci heaps ( en.wikipedia.org/?title=Fibonacci_heap ) are a (theoretically) optimal data structure whose runtime estimates involve $\phi$. $\endgroup$ – darij grinberg Jun 20 '15 at 22:40
  • $\begingroup$ See also Golden section search, which has been mentioned a number of times on Math.SE in the context of derivative-free methods for optimization. $\endgroup$ – hardmath Jun 21 '15 at 3:39

The golden ratio comes up in a nice way when one is looking at the rate of convergence of the Secant Method of root finding.

In the Newton-Raphson Method, in good situations, the number of correct decimal places roughly doubles with each iteration. With the Secant Method, the number of correct decimal places, in good situations, gets roughly multiplied by $\frac{1+\sqrt{5}}{2}$.

  • $\begingroup$ I remember in undergrad guessing (on a lark) that the answer to this problem would be $\varphi$ before we proved it and being happily awed when it turned out to be correct! Had forgotten about that. $\endgroup$ – MichaelChirico Jun 20 '15 at 23:36

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