TITLE: Is a visual proof to $\varphi = 1 + \tfrac{1}{{1 + \tfrac{1}{{1 + \tfrac{1}{{1 + \,\ddots\,}}}}}}$?
QUESTION [9 upvotes]: Every body knows some thing about $\phi$ golden ratio
One of definition for $\phi $ is like below 
$$\varphi  = 1 + \cfrac{1}{{1 + \cfrac{1}{{1 + \cfrac{1}{{1 + \,\ddots \,}}}}}}$$ 
My question is about this definition , Is there a visual proof for this formula?  
In general, can we bring some visual sense for continued fractions?
I am thankful about you hint in advance.
(I looked for this object on the Internet,but I can't find ...)

REPLY [3 votes]: $$φ = \left\{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} \ \ \mid \ φ = \frac{1+\sqrt{5}}{2} = \text{Golden Ratio}\right\}$$
$$\implies φ^2 = 1 + φ$$
$$\implies φ^2 - φ = 1$$
$$\implies φ(φ - 1) = 1$$
$$\implies φ - 1 = \frac{1}{φ}$$
$$\therefore φ = 1 + \frac{1}{φ} \lor \frac{1}{φ - 1}$$
$$\therefore φ^2 = \frac{1}{2 - φ}$$
$$\therefore φ = \frac{1}{2 - φ} - 1$$
This is why $φ$ can be expressed as a continued fraction, but a visual proof can be obtained in the following instructions:
$$1) \qquad \text{Lay down an A$4$ sheet of paper sideways on a flat surface. It has a length $L$ and a}$$
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{width $\mathcal{W}$}.$
$$\ \ 2) \qquad \text{Fold the top left corner down so it touches the bottom length of the sheet of paper}$$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{dividing it into a right-angle triangle and a rectangle. The rectangle has a}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{length $λ$ and a width $ω$}.$
$\ \ \ 3) \qquad \text{If $\frac{L}{\mathcal{W}}$ is equal to $\frac{λ}{ω}$ then their ratio is both equal to φ, the Golden Ratio}.$