What are some alternative ways to represent the golden ratio?

Question

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.

Answer 1

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.

Answer 2

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 $\lfloor k\phi \rfloor$

Answer 3

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.