The golden ratio spiral: visual infinite descent

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So you all know the golden (ratio) spiral. But did you know that not only the golden ratio but really every number has such a spiral associated with it? And that this spiral provides key insights into the nature of a number. Featuring more proofs by contradiction by infinite descent (my current obsession), infinite continued fractions, etc.

Here are some articles that debunk a lot of the golden spiral in nature nonsense:
Clement Falbo

George Hart

Keith Devlin

Here is a very good website devoted to everything to do with the golden ratio and Fibonacci

Thank you very much to Michael (editing), Marty (nitpicking) and Danil (translating).

Enjoy!

[mallowthecloud]

21:35...

Well, it depends if the satellite image is from the northern or southern hemisphere. That island kind of looks like a mirrored Iceland, which would make sense, since that spiral is only cyclonic in the southern hemisphere. The image would have to be flipped for the spiral to by cyclonic in the northern hemisphere. And that is a low pressure system (hence the clouds), so it must be associated with a cyclone, not an anticyclone.

[sethgrasse]

9:24 "So we conclude that 3 is irrational."
Whoa, that's quite the jump there.

[hauslerful]

Is it a coincidence that Numberphile talked about this as well at the same day? :)

[yakovify]

For the final puzzle, the land mass on the top left looks like iceland so this is the north hemisphere, hurricanes in the north hemipshere always go counter clockwise because of the rotation of the earth.

[eshel]

every video you make is a work of art! please upload more ♥

[danildmitriev]

Awesome, as always! :)
My guess would be that the fact about the greatest common divisor at 12:03 is due to the Euclidean algorithm (speaking of Greek mathematicians :) ). The construction of the spiral is essentially a visualisation of this algorithm, which is quite an efficient way of computing GCD.

[temshasanaie]

I love how you use geometry to explain things. My math skills are not what they used to be，but some of your videos really bring a smile to my face

[mjt]

These videos are really beautifully put together. I really appreciate all of the work of writing and animating that went into them, as well as their entertaining deliveries!

[tallinsmagno]

Why are so many people talking about logarithmic spirals all of a sudden?

[Neophlegm]

Even by the standards of your channel, this was an absolutely exceptional video. It's a masterful example of clear explanations. Awesome.

[SpencerTwiddy]

Literally a combination of the topics in the 2 most recent Numberphile videos, but with a lot added and done in classic Mathologer style. I'm not complaining at all, just makes me a tad suspicious! ;)

[therealpinktea]

im a phinatic myself and was excited to see the debunk portion of the video. love the vids mathloger!

[GhostlyGorgon]

The square spiral for rational numbers is a great visualization of the Euclidean algorithm! Which explains why the rational square spirals must terminate and why the final square has side lengths of the gcd of the two sides. Great video!

[ikaSenseiCA]

What about non-quadratic irrationals like pi and e? What are the properties of their spirals?

[santolok]

Wow! Thanks to you, I have an idea for another visualization of musical consonances (besides Lissajous).

I will try to programmatically depict a smooth increase of the 1x1 rectangle to the size of 1x2 with "spiral squares".

One side (x1) is the frequency of the main sound. The other side (from x1 to x2) is the frequency of the second sound.

I hope it will show the difference between "good" and "bad" two-tones.

Just intonation dictates that:
1:1 – prima, unison.
1:1.33.. (3:4) – natural "fourth".
1:1.5 (2:3) – natural "fifth".
1:2 – octave (e.g. 440 Hz and 880 Hz simultaneously).

Other ratios are more dissonant. One of the most dissonant is the triton (1:√2).

[MichaelHokefromCO]

Great video - thanks! Keep up the great work! I've finally gotten around to learning about continued fractions, and came across the square-cutting algorithm about a month ago. It's such a beautiful way to visualize continued fractions. Your explanation here is clear and enjoyable. I am envious of kids today who have at their disposal such wonderful ways to learn and explore interesting topics in math early on.

[simonh]

So, the golden ratio lies between 1 and sqrt(5).

[lokvid]

Wow! This was again a very amazing video by Mathologer. Math is so magic.

[joshinils]

I like this themed bunch of videos. This should happen more often. You all should talk to each other and do some sort of themed week as a collaboration

[wiretrees]

Thanks for your videos sir. You are a kind human and great teacher. I love your use of visual devices in these videos.