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 $\pi$ !

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
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"

Math

Appears 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}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered 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"

Math	Appears 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

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

X X - 2 years, 9 months ago

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

Mohammad Farhat - 2 years, 9 months ago

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

@X X – I did not see it

@Mohammad Farhat – Oh, yeah. (I thought you was saying wikipedia. Sorry!)

@X X – HAHA! That was a laugh for me

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

@X X – I put it on feedback

@Mohammad Farhat – Actually, I think you can just edit the wiki.

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

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

@Mohammad Farhat – Uh, oh~ That's strange.

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

@Mohammad Farhat – I wish I can answer this...

@X X – It's ok

$\phi$ 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 - 2 years, 9 months ago

Math	Appears 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}`	$\boxed{123}$