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What are some alternative ways to represent the golden ratio? I already know the relatively boring ones compared to the complex ones as well as:

  • $\displaystyle \frac{1+\sqrt 5}{2},$
  • $\displaystyle \frac{1}{1+\frac{1}{1+\frac{1}{1+}} \dots},$
  • $\displaystyle \phi + 1 = \phi ^{-1},$

and also the multiplier of consecutive Fibonacci terms. As some current answers have given, I would not like any formulas that reproduce the above. They are not classed as interesting, as they include repetition. I am looking for formulas that are interesting, and I am hoping to find some without repetition.

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    $\begingroup$ I think your first number is not the same as the others. I think you want $(\sqrt5-1)/2$. $\endgroup$ – Gerry Myerson May 29 '17 at 12:53
  • $\begingroup$ It depends on your definition of "good." I like $\phi^2-\phi -1=0,$ or equivalently $\phi - 1 = \phi ^{-1}$ (check your equations). Rather than boring, I consider it elegant. $\endgroup$ – David K May 29 '17 at 13:54
  • $\begingroup$ So, Goodra, any thoughts about any of the answers you have recieved? $\endgroup$ – Gerry Myerson Jun 2 '17 at 7:07
  • $\begingroup$ @GerryMyerson, yes, but not every question requires accepting immediately. I'm not sure whether that question originates from moderating or just looking at your profile, but questions like this I personally believe that a period of time (approx. 2 weeks) should pass before deciding. Sorry if I misunderstood you intentions... $\endgroup$ – VortexYT Jun 2 '17 at 8:43
  • $\begingroup$ No need to accept anything in a hurry, or ever, but there are three answers, and not even one of them has an upvote. Upvote just means, "this answer is useful". Are all the answers useless? $\endgroup$ – Gerry Myerson Jun 2 '17 at 9:03
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Wikipedia gives $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}$$ $\phi=1+2\sin(\pi/10)$, $\phi=2\cos(\pi/5)$, $\phi=\lim_{n\to\infty}(F(n+1)/F(n))$ where $F(n)$ is the $n$th Fibonacci number, and others.

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Here's a couple

  • $$ \phi = 1 + \sum_{k = 1}^{+\infty} \frac{(-1)^{k+1}}{F_k F_{k + 1}} $$

  • $$ \phi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}} $$

  • Define the alphabet $\{0,1\}$ with the production rules $0\to 01$ and $1 \to 0$

You get

$$ 0 \to 01 \to 010 \to \cdots $$

the locations of the ones occur at locations $\lfloor k\phi \rfloor$

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Here are a few more...

$$ \begin{align} \varphi &=\sqrt{1+\varphi}=\sqrt{1+\sqrt{1+\varphi}}\\ &=\sqrt{1+\sqrt{1+\sqrt{1+\varphi}}}=\cdots \end{align} $$

as well as

$$ \varphi=1+\frac{1}{\varphi}=1+\frac{1}{1+\frac{1}{\varphi}}=1+\frac{1}{1+\frac{1}{1+\frac{1}{\varphi}}}=\cdots $$

and

$$ \sqrt[3]{1+2\sqrt[3]{1+2\sqrt[3]{1+2\sqrt[3]{1+2\sqrt[3]{\cdots}}}}}\to\varphi $$

And just for kicks, let's call attention to the golden sequence

$$\cdots ,\frac{1}{\varphi^3},\frac{1}{\varphi^2},\frac{1}{\varphi^1},1,\varphi^1,\varphi^2,\varphi^3,\cdots$$

UPDATE

Here is a genralization of the root forms

$$ \sqrt[n]{F_{n-1}+F_n\sqrt[n]{F_{n-1}+F_n\sqrt[n]{F_{n-1}+F_n\sqrt[n]{F_{n-1}+F_n\sqrt[n]{\cdots}}}}}\to\varphi $$

where $F_n$ is the Fibonacci number.

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