The Amazing Mathematics of the Golden Ratio

Let me show you some awesome numerical patterns within "the golden ratio", beginning with a story about this number from my childhood, and building to a whole web of connections it has with the "Fibonacci sequence", the "Lucas numbers", and more!

Many people associate this number/ratio with geometry and shapes, but there are tons of numerical traits that make it special too! This episode covers some of my favorite numerical aspects and doesn't cover the geometric aspects, which I will cover at another point in the future. The golden ratio will also play a key role in a future episode about irrational numbers. And in another future episode, we will meet some other "metallic" numbers such as the "silver ratio" and "bronze ratio"!

By the way, the symbol I am using for the golden ratio is the greek letter "phi", which is used for a few things in math, one of which is this number. "Phi" is pronounced more like "fee" according to the sources I've read, although you'll hear some people pronounce it like "fye".

Here's links to two previous episodes that include the Golden Ratio:
(doesn't matter if you watch these before or after this video)

Special thanks to Evan Clark and to all of my Patreon supporters:
Max, George Carozzi, Peter Offut, Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Izeck, Beugul, OmegaRogue, Florian, William Hawkes, Michael Friemann, Claudio Fanelli, and Julian Zassenhaus.

Make sure you're also tuned in to my @Domotro channel for my bonus videos and livestreams and other content!

By the way, I've started putting shorts on this channel but they typically won't go to notifications or subscription feeds (they typically have been on my @Domotro channel and did so well for that channel that I want more people on that shorts page to find this channel too). They will be available on the "shorts" tab on this channel, and I will link the recent ones in video descriptions like this. Here are the ones I've put on this channel so far:

Domotro
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Berkeley, CA 94709

If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com

In case people search any of these words, some topics mentioned in this video are: the golden ratio (phi), square numbers, cube numbers, other integer powers, the fibonacci sequence, the lucas numbers, infinite continued fractions, infinite square roots, and more!

If you're reading this, you must be interested in Combo Class. Make sure to leave a comment on this video so the algorithm shows it to more people :)

DISCLAIMER: Do not copy any uses of fire, sharp items, or other dangerous tools or activities you may see in this series. These videos are for educational (and entertainment) purposes.

[ComboClass]

Thanks for watching! Check the description for some more notes, links, etc. :)

[colly]

if all mathemeticians were this handsome it would be so much easier learning maths

[DiamondRoller]

My mind is honestly blown right now. I love the way certain concepts seem to just weave themselves into another concept in so many different ways. Like, the golden ratio being in the Fibonacci sequence once is crazy enough, but _multiple times?!_ This is what I love about math, and I’m so glad there are passionate people like you out here making videos for people like me to enjoy and expand our knowledge of this wacky number world

[sagewaterdragon]

Domotro, I really can't praise this channel enough. Watching your channel now ignites the same spark of interest and passion for these topics that Numberphile inspired in me over a decade ago. I've fallen in and out of academics (I'm currently on an "extended hiatus" before finishing my engineering degree), but your videos remind me of why I loved math when I was younger and why I should love it again. Thanks.

[Bovineprogrammer]

The connections between phi and the square root of 5 are interesting. Obviously it's in one of the definitions of phi = (1+sqrt(5))/2, but also phi^3 = 2+sqrt(5); also, dividing a Lucas number by its corresponding Fibonacci number tends towards sqrt(5).

[roneyandrade]

I love how you only need 21 and 13 to make a good approximation of the golden ratio (21/13). Which is of course are part of the Fibonacci sequence (0 1 1 2 3 5 7 13 21 ...) Which impresses me even more because less than 10 elemnts and we're already almost there.

[ezzzzie]

OH wow I've only ever seen this channel in a popped out window at work I didn't realize how small it was. this is easily 100k views a video content. thank u for the knowledge, super interesting video

[maynardtrendle]

I'm really liking the more clean-cut, organized, Combo Class. All the older episodes really triggered my cleaning OCD, and made it hard to concentrate.

[Brightblade_Plays]

i particularly like the setting, outside, Chaos happening, and the Cats! keep it going!

[AnonimityAssured]

I'm very much looking forward to your new series on the golden ratio and the intricate connections between the Fibonacci and Lucas numbers. It will be interesting to see whether you explore polynomial expressions for F[2n], F[3n], F[4n], etc., and L[2n], L[3n], L[4n], etc., whose coefficients can be derived from Pascal's triangle and a simple variation on it.

[vampire_catgirl]

You can even express powers of the golden ratio in terms of Fibonacci numbers

Golden Ratio^(n) = F(n) × Golden Ratio + F(n-1)

φ⁰ = 0φ + 1
φ¹ = 1φ + 0
φ² = 1φ + 1
φ³ = 2φ + 1
φ⁴ = 3φ + 2
φ⁵ = 5φ + 3
φ⁶ = 8φ + 5
φ⁷ = 13φ + 8
φ⁸ = 21φ + 13
φ⁹ = 34φ + 21
φ¹⁰ = 55φ + 34

And yes, the 0 case does work, the negative 1st Fibonacci number is also 1. You can work through the Fibonacci sequence backwords by thinking of what has to be added to the current first digit to equal the second digit. To get 1 from 0, you have to add 1 to it. Then to get 0 from 1, you have to add -1 to it, and so on.

[lunafoxfire]

Oooh I learned some stuff. I didn't know the powers of phi got closer and closer to integers, much less the Lucas numbers. Very cool!

[_abdul]

It may be a little off topic but I'm so obsessed with this number I named my daughter after it(in a way) buy assigning numbers to letters (a:1, b:2, etc), for 1618 I got A, F, A, H and AFAH is her name.

[denyraw]

I would enjoy seeing an explananation in your style about why the golden ratio appears all over the place in pentagrams

[jamesdurtka]

A neat observation with the randomly chosen fibonacci-esque sequence is how quickly the first couple digits begin to resemble the traditional fibonacci sequence

[ianweckhorst]

Before I even start this video, I have researched the golden ratio and it is already my favorite irrational number

[_notch]

I did not expect that connection between fibonacci sequenceish and the golden ratio! Amazing lesson, goosebumps.

[themightyripples]

Lmao, I currently work at a bingo hall so the bingo ball cage was a bit of a trip, lmao. Since I've been working there tho I've realized theres a lot of complicated maths to do with bingo odds tho and so Ive been thinking a lot about combinatorics recently :) .

[themightyripples]

Also that was weirdly less chaotic than most episodes, love it still tho

[ivorvp]

When Brady from Numberphile generated a Fibonacci-esque sequence with 2 random numbers, it ended up being known as the "Brady numbers" and it's even in the OEIS. So we should call the sequence from this video the "Domotro numbers".