The Elegant Golden Ratio

The Golden Ratio (represented by φ,ϕ,Φ\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 φ\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 11's: φ=1+1φ\displaystyle \varphi = 1 + \frac{1}{\varphi} which makes it slip neatly in the continued fraction and with a set of square roots: φ=1+11+11+1\displaystyle \varphi = 1+ \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}

φ=1+1+1+\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 φ\displaystyle \varphi.

1,1,2,3,5,8,13,21,34,55,89,144,\displaystyle 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \cdots We can divide it in sets like this (preferably) (1,1),(2,3),(5,8),(13,21),(34,55),(89,144)\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 φ\displaystyle \varphi while the smaller number divided by the bigger number is getting closer to φ1\displaystyle \varphi-1

Try it out yourself!

But we can also RANDOMLY choose 2 numbers to start off with. Say, 5858 and 66

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

214÷1341.6214 \div 134 \approx 1.6 which is near φ\varphi's 1.6181.618 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 π\pi!

Note by Mohmmad Farhan
1 year, 2 months ago

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Comments

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

X X - 1 year, 2 months ago

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

Mohmmad Farhan - 1 year, 2 months ago

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It's OK. (Though I think φ=1+11+11+1\varphi= 1+ \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}} is also in wiki. )

X X - 1 year, 2 months ago

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@X X I did not see it

Mohmmad Farhan - 1 year, 2 months ago

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

X X - 1 year, 2 months ago

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@X X HAHA! That was a laugh for me

Mohmmad Farhan - 1 year, 2 months ago

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

X X - 1 year, 2 months ago

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@X X I put it on feedback

Mohmmad Farhan - 1 year, 2 months ago

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@Mohmmad Farhan Actually, I think you can just edit the wiki.

X X - 1 year, 2 months ago

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

Mohmmad Farhan - 1 year, 2 months ago

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

Mohmmad Farhan - 1 year, 2 months ago

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@Mohmmad Farhan Uh, oh~ That's strange.

X X - 1 year, 2 months ago

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

Mohmmad Farhan - 1 year, 2 months ago

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@Mohmmad Farhan I wish I can answer this...

X X - 1 year, 2 months ago

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@X X It's ok

Mohmmad Farhan - 1 year, 2 months ago

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ϕ\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.)

A Former Brilliant Member - 1 year, 2 months ago

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