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 \(\pi\)!

Note by Mohmmad Farhan
1 week, 6 days ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

\(\phi\) 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.)

Gennady Notowidigdo - 1 week, 1 day ago

Log in to reply

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

X X - 1 week, 5 days ago

Log in to reply

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 week, 5 days ago

Log in to reply

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

X X - 1 week, 5 days ago

Log in to reply

@X X I did not see it

Mohmmad Farhan - 1 week, 5 days ago

Log in to reply

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

X X - 1 week, 5 days ago

Log in to reply

@X X HAHA! That was a laugh for me

Mohmmad Farhan - 1 week, 5 days ago

Log in to reply

@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 week, 5 days ago

Log in to reply

@X X I put it on feedback

Mohmmad Farhan - 1 week, 5 days ago

Log in to reply

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

X X - 1 week, 5 days ago

Log in to reply

@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 week, 5 days ago

Log in to reply

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

X X - 1 week, 5 days ago

Log in to reply

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

Mohmmad Farhan - 1 week, 5 days ago

Log in to reply

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

X X - 1 week, 5 days ago

Log in to reply

@X X It's ok

Mohmmad Farhan - 1 week, 5 days ago

Log in to reply

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

Mohmmad Farhan - 1 week, 5 days ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...