The Elegant Golden Ratio

The Golden Ratio (represented by $\displaystyle \varphi , \phi , \Phi$) is one of the greatest discoveries Math has made with applications in but not limited to Architecture (eg. Parthenon) and designs. We will use $\varphi$ for the golden ratio at all times in this note.

To start off, It can be represented in terms of itself and with numerous $1$'s: $\displaystyle \varphi = 1 + \frac{1}{\varphi}$ which makes it slip neatly in the continued fraction and with a set of square roots: $\displaystyle \varphi = 1+ \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$

$\displaystyle \varphi=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$

If a sequence follows a Fibonacci sequence structure, The ratio between a set of 2 numbers is closer to $\displaystyle \varphi$.

$\displaystyle 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \cdots$ We can divide it in sets like this (preferably) $\displaystyle (1, 1), (2, 3), (5, 8), (13, 21), (34, 55), (89, 144) \cdots$

The bigger number divided by the smaller number is getting closer to the $\displaystyle \varphi$ while the smaller number divided by the bigger number is getting closer to $\displaystyle \varphi-1$

Try it out yourself!

But we can also RANDOMLY choose 2 numbers to start off with. Say, $58$ and $6$

$\displaystyle 58, 6, 64, 70, 134, 214 \cdots$

$214 \div 134 \approx 1.6$ which is near $\varphi$'s $1.618$ish

Try it out and see if you got it!

Stay tuned for more updates in the future and please see the my other note about $\pi$!

1 year, 1 month ago

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I remember that $\varphi=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$

- 1 year, 1 month ago

That is in the wiki. I do not want to copy the wiki but I will update the note for your sake

- 1 year, 1 month ago

It's OK. (Though I think $\varphi= 1+ \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$ is also in wiki. )

- 1 year, 1 month ago

@X X I did not see it

- 1 year, 1 month ago

Oh, yeah. (I thought you was saying wikipedia. Sorry!)

- 1 year, 1 month ago

@X X HAHA! That was a laugh for me

- 1 year, 1 month ago

Hmm...I found it here. Very surprised to know that the golden ratio page didn't have the continued fraction form.

- 1 year, 1 month ago

@X X I put it on feedback

- 1 year, 1 month ago

Actually, I think you can just edit the wiki.

- 1 year, 1 month ago

@X X I will try but the last 2 times I tried the entire wiki glitched and there was no more of the wiki

- 1 year, 1 month ago

I edited it (quite roughly and abruptly). I am bad at wiki formatting but good with notes

- 1 year, 1 month ago

Uh, oh~ That's strange.

- 1 year, 1 month ago

@X X Can you help me on my note: Just a little question

- 1 year, 1 month ago

I wish I can answer this...

- 1 year, 1 month ago

@X X It's ok

- 1 year, 1 month ago

$\phi$ is a convergent of the continued fraction; such behaviour is to be expected. A more interesting question would be why the convergent satisfies such relation, regardless of the starting points of the sequence. (HINT: Think generating functions.)

- 1 year, 1 month ago