# What are some alternative ways to represent the golden ratio?

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.

• I think your first number is not the same as the others. I think you want $(\sqrt5-1)/2$. – Gerry Myerson May 29 '17 at 12:53
• 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. – David K May 29 '17 at 13:54
• So, Goodra, any thoughts about any of the answers you have recieved? – Gerry Myerson Jun 2 '17 at 7:07
• @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... – VortexYT Jun 2 '17 at 8:43
• 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? – Gerry Myerson Jun 2 '17 at 9:03

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.

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 $\lfloor k\phi \rfloor$

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

$$\sqrt{1+2\sqrt{1+2\sqrt{1+2\sqrt{1+2\sqrt{\cdots}}}}}\to\varphi$$

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.