The golden ratio spiral: visual infinite descent

[@sethgrasse9082]

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

[@hOREP245]

quick maffs

[@bobrobert1123]

Root 3 dummy

[@diamondsmasher]

Seth Person settle down, don't be irrational

[@Cerzus]

Caught that as well

[@cukka99]

They fixed it in the closed captioning

[@hauslerful]

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

[@Mathologer]

What else can it be ? :)

[@hauslerful]

Mathematician's conspiring? o.O

[@flyingmonkeybot]

It's those HI pod guys. CGP did the same to Jake of Vsauce, but no one owns the facts, except maybe Euler.

[@flyingmonkeybot]

As for the picture at 21:36, I've never seen a hurricane hit the US that wasn't spinning in the opposite direction. Where was this photo taken, or is there some trickery here?

[@TrimutiusToo]

They were talking about logarithmic spirals, where each time you add just one square even if it doesn't fill the rectangle, while in this video it can be multiple squares depending on aspect ratio...

[@yakov9ify]

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.

[@Mathologer]

Spot on :)

[@juicyclaws]

yep, the image is horizontally flipped

[@hugo3222]

@@juicyclaws Actually, if it was "vertically" flipped, you won't see the earth., but only some stars or the moon.

[@esajpsasipes2822]

@@hugo3222 i think flipping is meant like mapping all pixels of a photo to other place along a line

[@PASHKULI]

yes, but it is not due the rotation of the earth...

[@eshel56765]

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

[@Nick-ui9dr]

Yeah! Not just art but science fiction... Rather more like math fiction. And I love the way he do transition between fiction and reality... Really a Satyajit Ray calibre movie! 👍 😂👌

[@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.

[@Mathologer]

Spot on :)

[@mallowthecloud]

Yep, exactly. That is what the spiral of a southern hemisphere cyclone looks like.

[@redsalmon9966]

Oh it’s a mirrored image didn’t expect that But now I think about it, flipped the image is easier than changing the direction of the cyclone

[@klausolekristiansen2960]

Changing the direction of the golden spiral would be easy.

[@tallinsmagno4207]

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

[@AttilaAsztalos]

...maybe because today is... (drumroll) phri-day...? (okay, okay, I'll show myself out...)

[@mandolinic]

It's because logarithmic spirals prove the flat earth ;-)

[@dhdydg6276]

theyre good spirals brent

[@tehjamerz]

Mandolinic k.gif

[@GTLx16]

Mandolinic the exact opposite actually.

[@danildmitriev5884]

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.

[@Mathologer]

Spot on :)

[@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!

[@lokvid]

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

[@yakov9ify]

The x solution is the golden ratio, the thing the numbers have in common is that they are all part of the Fibonacci sequence

[@Mathologer]

Which is also not a coincidence :)

[@yakov9ify]

Mathologer well I pretty much guessed it was the golden ratio once I saw the Fibonacci sequence :)

[@digitig]

Because the golden ratio comes up in the closed form expression for the n-th Fibonacci number, of course.

[@yakov9ify]

Tim Rowe well that much is obvious lol

[@digitig]

Probably obvious to anyone who gets this far in a Mathlogger comments section, anyway. Not to everyone. :)

[@ikaSenseiCA]

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

[@Mathologer]

Actually e has a spiral with a nice pattern whereas pi's spiral a bit all over the place. Just have a look at the video I link to at the end which talks about this in terms of continued fractions :)

[@Kris_M]

An elegant presentation of how elegant math(s) can be at times.

[@santolok7662]

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).

[@santolok7662]

Interesting to try to construct 3D spirals of three-tones combinations. For example Major chord is 4:5:6. That will be 1 x 1.25 x 1.5 3D-shape.

[@wyattstevens8574]

​@@santolok7662And minor is 10:12:15 (1/4 : 1/5 : 1/6) in the same way!

[@Ian-nl9yd]

the cyclone is going the wrong way. they spin counter-clockwise in the northern hemisphere, and thats clearly iceland

[@ragnkja]

In order to go clockwise in the northern hemisphere, it would have to be a high-pressure cyclone, but we don't generally get those on Earth.

[@kirkelicious]

This is thematically so close to the last Numberphile videos that I wonder what your inspiration was. I am not accusing you of ripping them off, otherwise your production speed would be amazing. Is something going on in the world of Mathematics that reinvigorated the fascination with the golden ratio?

[@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.

[@Mathologer]

Glad this worked for you :)

[@kibblepickle]

12:05 Euclid! When we divide the large rectangle (A x B) into a set of squares of equal size (B x B) and a smaller rectangle (B x A-B) at each stage, we are basically running a single iteration of the Euclidean algorithm, where the number of squares is the quotient and the new side length (A-B) of the small rectangle is the remainder. The algorithm terminates when we have no remainder left, in other words, when we have found the largest square whose side length is a factor of both the length and breadth of the original rectangle! :D

[@Mathologer]

Exactly :)

[@YaamFel]

At 6:53 you could've just extracted √3 out of the top to get √3(2-√3), then the bottom and the top would cancel out and leave you with √3.

[@h4c_18]

All numbers are terms of the fibonnaci series, and solutions are Phi and 1/Phi.

[@MrRyanroberson1]

You know, everyone hypes about the silver ratio taking two squares off to make a new silver rectangle...what about taking one square from the middle to get two similar rectangles? The sequence goes: 0,1,1,3,5,11,21,43... Where the silver ratio takes a"=2a'+a, this takes a"=a'+2a, and the ratio of successive terms approaches x=1+2/x; x is 2. The general formula for this newly expanded spiral follows a"=na'+ma, r=n+m/r, r²-nr-m=0, and r=(n±sqrt(n²+4m))/2, which for all except m looks just like the golden formula.

[@simon24h]

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

[@Mathologer]

Never thought of (1+root 5 )/2 as an average. Nice :)

[@1oo1540]

Mathologer I wonder if one could define a set of ratios as the whatever mean of 1 and root 5, and whether or not those ratios would have any interesting properties.

[@LudwigvanBeethoven2]

Duh!

[@RazvanMihaeanu]

Mathloger, every time when I see something on KZbin about the Golden Ratio I always wonder why is not shown the relation between the right triangle (with sides 1 and 2 ...while the hypotenuse is √5) and the circle. That puts the sunflower seed spreading/growth into a new light...

[@PC_Simo]

⁠@@Mathologer Me neither. But it’s true. Nice :).

[@אמתילוי-ל3ש]

The final square side’s length is due to the Euclidean algorithm which says that ( if a>b): gcd(a,b)=gcd(a-b,b) A great video!!

[@donaldasayers]

One answer to final question: The spiral should not be not tangential to the lines where corners of squares meet. They are in your picture because you have drawn circular arcs rather than a logarithmic spiral.

[@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

[@xCorvus7x]

Typo around 20:13 1/0.7320... = 1/(√3 - 1) = (√3 + 1)/2 = 1.3660... 16:29 "... one of the usual suspects, Leonhard Euler." To quote 3Blue1Brown on this matter: "It's often joked that in math formulas [and theorems] have to be named after the second one to prove them because the first is always going to be Euler."

[@wiretrees]

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

[@williamboyles9590]

You can go so much deeper than the golden ratio. You can create quadratics for all quadratic irrationals. Let's say we make a quadratic from a continued fraction, like sqrt(3). We'll say x=1+1/(1+1/(1+x)). This is equivalent to x = (2x+3)/(x+2). When a quadratic generated this way of the form x = (ax+b)/(cx+d) and the solution to the quadratic is sqrt(D), then a^2 - Dc^2 = +/-1 (Pell Equation!) and b^2 - Dd^2 = -/+D. In our sqrt(3) example, 2^2 - 3(1)^2=1 and 3^2 - 3(2)^2 = -3.

[@MattiasDooreman]

Have I missed it being 'ratio day' today?

[@Skull218]

Wait, so you're telling me Gyro Zeppeli was lying to me?

[@8316WC]

maybe in their alternate universe it was true.

[@AntonLauridsen]

I know it's a bit late, but if I understood this right then I'm a happy guy now. One of the things I've never liked about the traditional definition of irrational numbers is that it is defined by a negative quality. I.e. it cannot be written as a fraction of whole numbers. If irrational infinite spiral and rational finite spiral then suddenly there exists an equally valid _positive_ definition, namely that irrational numbers are those numbers who has an infinite descend spiral.

[@NesrocksGamingVideos]

The most amazing thing about irrational numbers, to me, is that aproximately ALL numbers are irriational, proportion-wise. Even though there are infinite numbers, most of them are irrational.

[@julienbongars4287]

Great explanation on continued fractions! It's interesting how you use geometric modelling instead of the more common algebraic proofs :)

[@YuzuruA]

loved the trump spiral

[@Mathologer]

Had to be done although I am sure that it will earn this video some dislikes :)

[@PhilBagels]

Nobody makes a better spiral. It's yuge. It's the best spiral ever. That, I can tell you. Make spirals great again!

[@otakuribo]

He's the only thing more irrational than Φ

[@raydeen2k]

Φ on him, I say.

[@danildmitriev5884]

I mean, he has a "very, very good brain", so it is only natural that it produces the best spiral ever.

[@souvik8436]

Sir really i love your explanation..love from india🇮🇳

[@michaelhanford8139]

Final frame of video, The cyclone is over north america but is rotating anticlockwise, ¿que no? ❤️ You make me miss the days in Mr. Olson's maths classes. Small rural school, he farmed & taught euclidean geometry, trig & calc. ❤️ Dankashane! (it was Pennsylvania-Dutch country; so i'm sorry if i misspelled 'thank you'...i only ever heard spoken German😄)

[@OlafDoschke]

Unimportant detail, but before yesterdays numberphile video about the silver ratio, in which @DrTonyPadilla mentioned A4 paper and you now mentioning A4, I thought this was a German only thing, especially as A5 is short for DIN A4 here (DIN being for German what ANSI is for the US, the German Institute for Standardization). Maybe worth a global look: en.wikipedia.org/wiki/Paper_size#/media/File:Prevalent_default_paper_size.svg Funnily Australia is uncharted land here. So is it blue, or did you just import A series Paper for your own usage? Or is it mixed in Australia? Last, not least, I'll not judge which video covers the topic better. @DrTonyPadilla has a nice opener from the fingernail experiment, anyway, it's not an exclusive content war. It's nice to see a topic from multiple perspectives.

[@etymos6644]

It's amazing how easy it is to find the faces you are looking for in clouds...

[@Pageleplays]

17:55 The solution is the golden Ratio x1= (1+sqrt(5))/2 x2= (1-sqrt(5))/2

[@onemadscientist7305]

The two solutions of the quadratic equation shown are the golden ratio and the silver ratio and the integers are fibonacci numbers... Not exactly surprising, but still, that's pretty neat.

[@ΒασίληςΌμικρον]

Pythagoras (Πυθαγόρας) is the first mathematician we know, that is "responsible" for some of these maths. Awesome vid, as usual.

[@TyTheRegularMan]

Once again, you have blown my mind in a way I never thought possible.

[@Mathologer]

:)

[@xwarrior760]

12:02 Oh wait that's Euclid's Algorithm isn' it? For some reason I feel so happy to realize that lol

[@brokenwave6125]

This is one of the best math related videos I've ever seen

[@josephgroves3176]

So... Because you prove that the smaller rectangle is the original sqrt3 rectangle then by induction/recursion the spiral continues infinitely. But if the original side lengths are FINITE integers, then as the side lengths get smaller, they will eventually hit 1, which would stop the spiral. But that contradicts what we proved earlier that the spiral cannot stop, so sqrt3 is irrational. Also neatly ties up with another definition of irrational: that it an infinite non-recurring digits, which is equivalent to saying it can only be expressed as a ratio of infinetly long numbers

[@1975mfa]

I love your videos. I would have loved to have you as my teacher when I was a younger student (which I'm not). I'll suggest my son, who is 17, to watch at your channel. Great job!!!

[@matthewdarocha8243]

Hmm, now im curious what would result from a rectangle who's aspect ratio is a transcendental number

[@JaydentheMathGuy]

RIP PHINATICS

[@Henrix1998]

My guess to the last puzzle is that it is mirrored, the storm should spin the other way around because it is on the southern hemisphere

[@Mathologer]

:)

[@MathOratory]

Puzzle 2 ... Euclid's method of long division of calculating HCF ?? Did he actually visualize it in this way .. I understood the method using the idea of factors .. But this geometrical similarity is beautiful

[@MathOratory]

Last puzzle ... fibonacci series terms as coefficients so x = golden ratio, right ? Beautiful video indeed sir ...

[@Mathologer]

+MathOratory That's it :)

[@Mathologer]

+theo konstantellos Did you watch to the end? :)

[@MathOratory]

Loved it ... I always used to show it like ... f*a - f*b = f*(a-b) ... So common factor and all .. But I'll definitely try this for fun in my next class ... amazing sir ... I was just thinking of something in the sqrrt(3) ... square direction thing ... Don't know if I'm observing too much into it .. The sequence is 1,1,2,1,2,1,2,1,2,... right? Now I was looking into the direction of the arrows and observed something ... (maybe not relevant) ... but if from the first 1,1,2 we take out '1' from each number ... that is taking '1' square in each direction (the first 3 that is)... Then the terms left in the series is again 1,1,2,1,2,1,2 ... So it's like: 1,1,2,1,2,1,2,1,2,1,2,.... = (1,1,1) + (1,1,2,1,2,1,2,1,2,....) Obviously, this will then go recursively ... I just sat with pen and paper to calculate the same for sqrrt (2) and sqrrt (5) ... Dunno if anything is there or just a coincidence .... but it's 3 ones taken out right ,, and it is rt(3) afterall Sorry I didn't have the square root symbol in my keyboard ... :)

[@HakoHak]

I finally finished the video, and found that it got more puzzles xD Puzzle 3 (equation): 1) I got -243 310x3 + 117 493x² - 65 062x + 103 952 = 0 2) And.... I was about to calculate Delta' -- 4(b² - 3ac) -- then I decided to put my equation in a graph calc (graphsketch.com), I thought I was going to see the golden ratio as a solution, but no... I lost faith here x) Puzzle 4: Fibonacci sequence ! Puzzle 5: no idea... I checked the ratios and the squares, they seems roughly good

[@Mathologer]

For puzzle 3 (the solution of this equation kzbin.info/www/bejne/q5Orh35tZqtjZ68m43s) you should get the golden ratio (unless I messed up :)

[@feynstein1004]

Would this work for 3 dimensions as well? i.e. for cube roots? My first thought when I saw the infinite spiral was if pi could be drawn like that. Then I remembered it can't because pi is transcendental.

[@soostdijk]

The Golden Ratio is no mathematical magic but a physical phenomenom. If you add an object to a system in balance and strive to retain balance in the expanded system, that can be expressed as (1 stands for system in balance, phi stands for new physical object): 1/phi = 1+phi. The ratio of forces between the system in balance and the new object should be equal to the forces of the system in balance + new object. Phi is the expression of how the universe retains balance.

[@davidherrera8432]

It’s Euclid’s algorithm to find gcd, substract the smallest from the biggest and repeat until you get 0, the last number is the gcd

[@LudwigvanBeethoven2]

I love your videos. And i havent even watched this yet but i know im gonna like it!

[@whiploadchannel2047]

Multiply by sqrt3 to simplify the fraction

[@saimafa5579]

Youve incorporated this into the MLC logo as well as your youtube pictograph. So interesting.

[@PC_Simo]

18:00 The solution to the equation, there, is X = 1,618033988749895… = φ; and all the numbers, there, are Fibonacci numbers 👍🏻.

[@chthonicone7389]

Your video makes me want to go back in time to the school of Pythagorea in Egypt, but I would probably be thrown from a boat showing them this stuff.

[@frobeniusfg]

The cyclon should spin anticlockwise as the island is actually Iceland situated in northern hemispehre (picture from en.wikipedia.org/wiki/Cyclone).

[@Mathologer]

:)

[@frobeniusfg]

Mathologer The awesomeness of ur videos is spiralling out!

[@BillRuhl001]

Nothing to be squared of! His best t-shirt yet.

[@WhattheHectogon]

@Mathologer the picture is of Iceland (I believe), but is flipped. For some reason no one thought to just flip the spiral instead of the image.

[@bens4446]

So rational nums tile the plane infinite descent wise. In this sense they are "2 dimensional" numbers. Irrational nums also tile the plane, but require infinite time to do so. So they are "2+1 dimensional" (where the 1 refers to the time dimension). I wonder if there are 3+1 dimensional numbers, etc.?

[@TruthIsTheNewHate84]

I love your videos and since subscribing a couple months ago and because of you I have become very interested in mathmatics.

[@richardr3098]

It might be trivial. But I find it interesting that we draw 'irrational sized' shapes. There is no way to draw a rectangle precisely with irrational side lengths. I guess it's no different to drawing circles where the ratio of the diameter to the circumference is pi. It's just weird in a way that these things exist really as thought experiments.

[@dlevi67]

For that matter, there is no way to draw a rectangle precisely with integer side lengths...

[@balajisriram6363]

A thanks right from the heart to mathologer!!

[@michaelhanford8139]

...lemon squeezy 😂👍 I love that you're not stiff& self-important

[@OlafDoschke]

About the final picture of the cyclone. It's what you already showed at the beginning with logarithmic spirals, that would fit even better. Looking at the dark spiral of the cyclone, the gap between cloudy regions, that crosses the edges of the rectangles and arcs bleed over. I assume a real logarithmic spiral will have a smooth change in curvature, not be pieced together from quarter-circle arcs, thus they don't fit in the square regions of rectangles with these specific ratios.

[@JonSebastianF]

*Premise 1:* No ratio of positive integers exists that equals sqrt(2). *Premise 2:* The lengths of the two sides of A-sized paper are positive integers whose ratio equals sqrt(2). *Conclusion:* A-sized paper does not exist.

[@Mathologer]

Actually the aspect ratios of A sized pieces of paper are fractions that approximate the root(2) very well. For example A4: 297/210. So, all's good :)

[@JonSebastianF]

Phew, thank you! :) ...I thought I was going crazy, when my A4-paper just kept existing!

[@marcushendriksen8415]

In answer to your question about the famous Greek mathematician: was it Archimedes? Archimedean spirals are pretty fun. I also don't know of any other one that worked with spirals specifically.

[@Mathologer]

Actually, the mathematician in question is Euclid (google Euclidean algorithm :)

[@marcushendriksen8415]

@@Mathologer damn it! I was thinking too literally

[@KillianDefaoite]

Hi Mathologer, I love your videos. What happens when we try making spirals with cubic, quartic root numbers? How does it change from the quadratic case? And then, how does it then change when we move on to quintic roots, as there is no general quintic root formula? Finally, and most importantly, how does the picture change when we used transcendental numbers?

[@kindlin]

1:17 had my crackin' up! I love this channel.

[@conoroneill8067]

Is there a way to extend the notion of logarithmic spirals from the real numbers to the complex numbers? I know complex numbers can be expressed through infinite fractions, so it could theoretically be possible. Unfortunately, the only way I can think of doing it would be to have a 4 dimensional output - 2 dimensions for the 'a' in the rectangle, and 2 dimensions for the 'b' in the complex number, which is frustrating.

[@avishkathpal4386]

Irrational numbers are number which cannot be represented in a/b form. How did you write root(3) as a ratio of two integers?

[@rizzwan-42069]

This video was gold

[@sabriath]

This is very similar to your triangle production for irrationals.....as describing the lowest common fraction of A/B to equal the questionable value X, if we subdivide the space equally, and still come up with a ratio X, then the fraction A/B is not the lowest common, therefore it has to be irrational. At least that's how you described it in the triangle video.....or something.

[@michalbotor]

i think, that there is unfortunately a slight room for misunderstanding (solely due to notation) regarding what you state at 4:20 and what you prove at 9:19. maybe simply writing '√3 is rational - false' or '√3 in Q - false' or '√3 =/= p/q' in the future would be more unmistakeable. that being said, outstanding math lesson sir. i give it a thought, i narrow my eyes, i tighten my face, i take off my hat, i put on a smile, and i nod in admiration.

[@aheeshhegde2038]

what's relations between golden ratio and Fibonacci series

[@jirayutjujaiboon4777]

Watch this kzbin.info/www/bejne/iZu2eYl6m717h5Y

[@jeffreybernath6627]

At 20:40 we see the same 0.7320... that we saw at 19:25. Not a coincidence!

[@linkspring1287]

Jeffrey Bernath lol.. that's the reason..that spiral is going to repeat...

[@ricardolichtler3195]

I love your explanations!

[@stevenvanhulle7242]

I never understood how so many people can be fascinated by the golden spiral. It's an absurdity. It's a piecewise curve, and its evolute has measure zero. (I wonder if it's even a proper spiral. Mathworld doesn't give a definition for a spiral; the entry is empty.)

[@FredyArg]

I bet this guy told his kids the truth about Santa, fairies, Easter bunny, tooth fairy, and all imaginable creatures, then proceeded to mathematically explain why they don’t exist.

[@PC_Simo]

Since most of these ”Phi-natics” tend to be fundamentalistically religious; I would like to share a top-tip, from my Mathematician-Philosopher-Friend, that these apologists should stray away from arguments relying on Mathematics; since doesn’t Gödel’s Incompleteness Theorem pretty much prove that nothing ”maths-y” can have a divine origin? I’ve given it some thought; and I’ve come to think that, maybe, just maybe, the ”Tower of Babel” -story, in the Bible, is really an allegory of people trying to reach the divine with Mathematics, with the common ”proto-language”, hinted at, standing for the language of Mathematics and Logic, and the Tower of Babel, itself, standing for Mathematics, a human construct; an effort that was doomed from the start. 🤔

[@justwest]

that tshirt is awesome

[@ginnyjollykidd]

In the last picture, phi is crossing over several of the hurricane spirals. So it is completely off!

[@danielrhouck]

Did you and Numberphile conspire to release φ-related videos at around the same time, or is it a coincidence? Either way, this is an interesting video that shows a neat visualization for the continued fractions that they discussed in their video.

[@Mathologer]

Coincidence. This sort of thing happens more often than you would think. Actually never great if you come second as it happened to me with this video. After something happened to me and Infinite series, 3Bue1Brown, Infinite series and exchanged lists of upcoming videos for a while to avoid this from happening. Numberphile was never part of this though.

[@blablablablablablablablablbla]

Who first determined this correspondence of spirals and numbers?

[@myleswinkley9598]

Why is 3A - 5B a positive integer? I know Im being dumb, but I dont know.

[@Mathologer]

Well, since A and B are supposed to be integers 3A - 5B has to be an integer. This integer has to be positive because it is the side length of a rectangle by construction :)

[@lare290]

As we all know, A4 paper is just scaled down when you fold it in half, because the long side is sqrt(2) times the short side. Did you know that the same happens when you fold a paper in n parts and the long side is sqrt(n) times the short side?

[@Minecraftster148790]

12:15 that’s just a visual version of the Euclidean algorithm. Very nice link

[@AliVeli-gr4fb]

thank you and it got better towards the end. I enjoyed it a lot

[@berndfachinger6000]

5:30 : No, it does not repeat. It is at least 1,121212161.

[@Scum42]

9:20 > And so we conclude that... three is irrational. Um...

[@dlevi67]

Ever tripped into a root when walking?

[@aeonsiege1806]

That sqrt(2) spiral seems wrong

[@josephgroves3176]

Aeon Siege. That's cos they showed the compass and straight edge construction (quarter circles drawn through relevant corners, joined together). The actual spiral would be plotted with polar coordinates (r=e^((sqrt2)×(pi×theta/2)) or smth)

[@nikhilpatel6214]

12:03 The rectangle for integers is a visual of the Euclidean algorithm, based on the fact that GCD(A,B) = GCD(A-B, B). So you're going down each time until you get to the GCD. Noice.

[@danielinfinito6304]

Another amazing video that help to connect analytic or algebraic results with visual geometry... Thank you very much. I think that if Erdös would be alive it could talk about "The Channel" as a complement to "The Book". In this sense many of your videos must be in "The Channel".

[@Mathologer]

Nice thought. I'm definitely always trying find the proof in "The Book" and then make it even more accessible with a video like this :)

[@techbizcanada7594]

I'm a little confused... how can you have A/B = root 3 ? Obviously A and B can't be integers? How do you get/calculate the squares?

[@judassab]

I found zero links on this spiral of squares depiction about numbers other that φ. Once you google if you get "spiral of square roots". Are there any?

[@nschloe]

Question about the irrationality proof of sqrt(3): What tells us that the square filling goes on infinitely, i.e., that there is no point at which filling in a square fills the area completely? (as it would be the case if the ratio A/B was deliberately chosen to be rational)

[@trogdorstrngbd]

Look at 12:14.

[@demonsmd]

Rational finite spiral what about -2/7? is the spiral for this number just the same as for 7/2?

[@lexibrancadora4327]

You should solve the anything's possible bottle!!!!!!