Visualising irrationality with triangular squares

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Get ready for some brand new and very pretty visual proofs of the fact that root 2, root 3, root 5 and root 6 are irrational numbers.
Root 2 being irrational also translates into the fact that the equation x^2+x^2=y^2 has no solutions in positive integers, root 3 being irrational translates into the fact that the equation x^2+x^2+x^2=y^2 has no solutions in positive integers, etc.
What I find very attractive about these proofs is that the destructive core of these proofs by contradiction lead a second secret constructive life, giving birth to infinitely many nearest miss solutions of our impossible equations like for example 15^2+15^2+15^2=26^2-1.
Here is the paper by Steven J. Miller and David Montague which features the basic root 3 and pentagonal root 5 choreographies.
arxiv.org/abs/0909.4913
Footnotes:
-our nearest miss solutions like, for example,
15^2+15^2+15^2=26^2-1
correspond to the solutions of the equation y^2 - n x^2 = 1 with n=2, 3, 5 and 6. This is the famous Pell's equation, which happens to have solutions for all integers n that are not squares.
-there is also a second type of nearest miss solutions like
4^2+4^2+4^2=7^2+1 (a plus instead of a minus at the end). Starting with one of these our choreographies also generate all other such nearest misses.
-the original Tennenbaum square choreography and the first puzzle root three choreography generate both types of nearest misses from any nearest miss solution.
-The close approximations to the various roots corresponding to our nearest miss solutions are partial fractions of the continued fraction expansion of the roots.
-lots more things to be said here but we are getting close to the word limit for descriptions and so I better stop :)
Thank you very much to Marty for all his nitpicking of the script for this video and Danil for his ongoing Russian support.
Today's t-shirt is the amazing square root t-shirt (google "square root tshirt"). Note that the tree looks like a square root sign AND that the roots of the tree are really square.
Enjoy!

[@corpsiecorpsie_the_original]

Mathologer needs a second channel with answers to the homework questions he gives

[@steffen5121]

As a studied graphics designer and 'hobby mathematician' I'm always astonished by Burkard's animations and appreciate the work being put into it in order to make the mathematics look this beautiful.

[@Mathologer]

:)

[@steffen5121]

Fun fact. A calculative way of getting sqrt(3) via the same numbers you got visually from the four triangles is this: a(n) = 4*a(n-1) - a(n-2); for a(0)=1 and a(1)=2 b(n)=4*b(n-1) - b(n-2); for b(0)=0 and b(1)=1 n 0 1 2 3 4 5 ... a(n) 1 2 4 7 26 97 ... b(n) 0 1 4 15 56 209 ... sqrt(3)=a(inf)/b(inf-1) source: oeis.org/A001075 and oeis.org/A001353 I wrote a Bash program to check the results for higher n and it works like a charm.

[@abj136]

Took me 15 minutes (while watching square triangles dance around) to figure out the solution to your shirt. The tree has square roots!

[@Mathologer]

Yes, AND the tree is a square root sign ! :)

[@ongbonga9025]

Here was me thinking he was going to demonstrate irrationality by showing us the infinite orchard problem.

[@TheNameOfJesus]

True, the tree has square roots, but did you also notice it was a "root tree" (root three) and the video was mostly about the root of three?

[@b.clarenc9517]

@@MathologerYour shirt reminds me of this: i.snag.gy/SUZHku.jpg

[@kristendrescher7536]

I need this shirt.

[@Mathologer]

What do you think of the idea of publishing original mathematics on KZbin instead of mathematics journals?

[@manuelnovella39]

Mathologer, it's a fantastic idea

[@Jako1987]

I will likely watch whatever you will publish

[@MarcosLourencoAntonio]

Nice idea!

[@abidurrahman4641]

Its a good idea, keep doing it :)

[@th3officebeefalos456]

Much less formal for one.

[@ecsciguy79]

11:30 "15 and 26 form...a nearest miss solution" You know that there's a name for this, right? It's a Parker Square.

[@michalnemecek3575]

a Parker Triangular Square.

[@washizukanorico]

At first you describe things, then eventually you name them

[@LuizDahoraavida]

@@washizukanorico /r/whoosh

[@earthdog1961]

@@michalnemecek3575 With a PARKER SQUARE you really have to watch those diagonals.

[@squashgoogolplex9392]

aaaaaaaaayyyyyy Matt's gonna be mad tho.. XD

[@partychoosenEDJ]

As for the last puzzle, we didn’t start from the assumption that the solution is the smallest one, hence, finding a smaller solution doesn’t contradict with any assumptions. I have to thank you and your colleagues for sharing these beautifully created math ideas to us. It’s amazing!

[@pokemonjourneysfan5925]

Actually, The supplied answer would not have been shown to be the smallest solution if we had started with that premise. For all we know, the absolute smallest solution might be the next smallest or the one less than that. Despite the fact that root 3 is irrational, this renders the proof invalid. We might also demonstrate that the sum of three triangular numbers could never be a triangular number using this same proof.

[@user-op9de9gc2w]

I had this same concern. The proof mainly depends on our initial assumption.

[@amansparekh]

@@pokemonjourneysfan5925it doesn’t need to be shown to be the smallest solution, just a smaller solution than the original, which then completes the contradiction, and so the proof is perfectly valid

[@pokemonjourneysfan5925]

@@amansparekh So by your logic, I'll try to prove there are no pythagorean triples in pos. integers. Let's assume there are pythagorean triples. The smallest one I found yet is 5^2+12^2=13^2, Using analysis, I can deduce there is a smaller solution 3^2+4^2=5^2. Thus, there are no pythagorean triples.

[@amansparekh]

@@pokemonjourneysfan5925 no, the reason this proof works and yours doesn’t is it shows that given any starting solution, there is a smaller one, whereas if you pick 345, there is no smaller solution

[@Tehom1]

"Why doesn't this prove that 3 triangular numbers cannot add to another triangular number?" - Because you have a base case where there are no leftover hexagons: 3*tri(1) = tri(2). Then you just say that the minimal case is that case.

[@czarlito_]

Tehom Yeah, exactly. Not really sure how T_1*3 being equal to T_2 and T_2*2 being equal to T_3 can 'generate' all equations for integers, proof-wise.

[@andlabs]

Because the question was 3x=y where x and y are triangular numbers, and "T_2*2 being equal to T_3" is 2x=y, not 3x=y =P And of course, this generator is for the strict case of 3x=y, not any arbitrary x+y+z=n (where x, y, z, and n are all triangular)

[@johnrickert5572]

Because they beehive better!

[@subh1]

How do you know that there is not a similar base case solution for 3*n^2=m^2, but where n and m are very large numbers that you cannot easily find/check as you did for the triangle numbers?

[@Tehom1]

subh1, That's not what Burkhardt asked. But anyways, the answer has already been given on this board: because for triangles, the center always falls within a triangle, so you can't divide the big triangle into 3 symmetric parts because you can't subdivide the center triangle.

[@joshuataylor2105]

The geometrical difference between the triangles (square numbers) and hexagons (triangular numbers) is that with the triangles the center point falls within a triangle in all cases, whereas with the hexagons, the center point can fall on a vertex between hexagons. So you can have symmetry around that point and still have all the hexagons only used once, while the triangle in the center can only be used 0 or 3 times.

[@CasualGraph]

Man, it's nice to imagine how a bunch of tetrahedrons demonstrate the irrationality of the cube root of 4 just as easily.

[@SKO_PL]

Casual Graphman We could keep going into extra dimensions this way... Very fun to think about it.

[@15schaa]

Aw, you've just made my bleeding day. So satisfying.

[@josh34578]

Does that work? Don't you get an octahedron left over when you cut tetrahedral corners off of a tetrahedron?

[@SKO_PL]

velcrorex No. Each of the four smaller tetrahedra touches the bigger one with three of its faces and the leftover inside with the fourth face. Since there are four tetrahedra touching the leftover with one face, it must too have four faces. It's just that the small tetrahedra must cover all the surface of the bigger one (so that the inside touches only the small tetrahedra)

[@farissaadat4437]

It's the Desmos man himself! Keep it up, your work is wonderful and even inspiring.

[@JM-vz6ok]

I've never really learn a single thing from these videos but the visual representations of mathematical concepts are nice.

[@doodelay]

"Wth is a triangular square? Well it's definitely not click bait, OK" 😂😂 earned that like

[@Lucroq]

I love those animations where you see an equation that you've used a thousand times visualized so intuitively that it simply 'clicks' and your mind is just blown.

[@aescafarstad]

The answer to the last puzzle is that in case of aa+aa+aa=bb you can prove that a+a > b and triangles will overlap. It follows from the fact that sqrt(3) < 2 because sqrt(4) == 2. In the last example you can't prove that they will overlap. In fact 1+1+1=3 is the simplest counterexample.

[@kaerriss]

I think you're right and that's probably what he expected, good job Maybe we could go a little further, maybe the only case where it does not overlap is with : 1 + 1 + 1 = 3 If this is true, then we could show that we can obtain all the solutions to the equations of the form : 3*T_n = T_m ; by reversing the process. The idea is that if we have a solution, and it's not minimal, then if the minimal solution is the only one where there is no overlap, that means that there has to be an overlap in other cases, and that means that the bigger solution we have can be constructed from a smaller one (the remainder of the overlap) by reversing the process... Using a proof by induction, that means that if we have a solution to the equation, either it is the minimal solution, or we can derive it from the minimal solution by reversing the process and we'll be guaranteed to eventually find all the solutions.

[@hernanipereira]

this video is precisely why Mathologer is one of favorite channels :)

[@jeffreybernath6627]

At 9:00 I was afraid you were going to try to tile the plane with pentagons. That would have been scary! Also, I think the first half of the video would have benefited from a "triangle choreography" showing that the square root of 4 is rational, since you can perfectly pack 4 identical equilateral triangles into one big equilateral triangle with no overlap. (This would look like step 1 of a Sierpinski triangle.) This is kind of the trivial case, but I think showing this complementary case of the proof going the other way would provide a great contrast to the way the proof demonstrates the irrationality of the square roots of 2, 3, 5, 6, etc. But still, fascinating! Thank you!

[@TheLazyVideo]

Number theory is another fun way to show the square-root of any non-square is irrational: Suppose sqrt(n) is rational. Let sqrt(n) = a/b, for relatively prime a and b. Then n = a^2 / b^2, then n * b^2 = a^2. Since n divides the lhs, then n divides the rhs. Since n is not a perfect-square, then let p equal the product of the primes with odd powers in n's factorization, and p != 1. Now, p divides a. So divide both sides by p. (n/p) * b^2 = a * (a/p). Since p still divides the rhs (p divides a), it must divide the lhs. But a and b are relatively prime, so p cannot divide both a and b. Contraction, so sqrt(n) must not be rational. QED

[@podemosurss8316]

7:25 Of course! The rombus there can be divided into two triangular squares by just slicing it by the middle.

[@CrepitusRex]

I don't quite understand it but I love it! Thanks for the great video.

[@Mathologer]

I hope you love it enough to eventually watch it again to really get all of it. Well worth the effort :)

[@CrepitusRex]

Mathologer oh I will. I find myself working them out and I use the internet to find answers. I find the more I watch the more I "see" it. If you know what I mean. I've always had a hard time seeing math. I've got to write it down and then sometimes it comes to me. Sometimes not. But I won't give up. Thanks for your videos.

[@alkankondo89]

I JUST HAPPEN to be giving a presentation this week involving continued fractions, as well as their connection to a certain graph in the hyperbolic half-plane called the Farey Graph, which is built off vertices from the Farey Sequence. Edges connect vertices if they are neighbors in any level of the Farey Sequence. What is interesting is that paths in the Farey Graph to any irrational hit all the vertices that are continued-fraction convergents of that irrational. It's interesting to see that, in this video, there are EVEN MORE ways to visualize approximations to irrationals! Math is full of connections between its sub-disciplines!!

[@randairp]

Woah. Just looked up the Farey Graph. Have you heard of the "Real Projective Line"?

[@alkankondo89]

Thanks for your response! I haven't heard of the Real Projective Line before, but I just looked it up. It looks like it's the same line on which the Farey Graph is built, in that it consists of all real numbers PLUS a point at infinity. EDIT: Ok, I just read that the Projective Real Line is, in fact, the boundary of the Poincare-disk model of the hyperbolic plane. Nice!

[@stefan1024]

This is beautiful! It reminds me of the ratios in music somehow, but maybe that's just me. Your animations really are on fire lately, visual proofs are so strong if made right.

[@maheshsookram4152]

Keep up the great work on these. Brilliantly presented, challenging, and yet still accessible and entertaining to anyone. Truly inspiring.

[@lawsoncrutcher3218]

0:33 he used his shirt as the radical, well done

[@LonkinPork]

14:05 nice little quasi-example of the Four Colour Theorem you got there.

[@MohaMMaDiN55]

My brain became bigger and able to store a lot more information than before just because of your videos and some other videos from other math channels

[@Garganzuul]

Happy to see that the compass and straightedge lives strong on KZbin.

[@Supremebubble]

7:20 the white diamond has an upper half and a lower half that add together to the darkgreen triangle.

[@jkloe]

nope, the upper triangle has base of 7 triangles so there are 49 in there. the white diamond is made of 2 triangles with base of 5 triangles, so 5²+5² = 50.

[@Supremebubble]

Uhm... well of course it doesn't actually add together. Did you watch the video?

[@maskedjessie]

Supremebubble you just said they add together

[@Mathologer]

Exactly. Actually this is a very interesting alternative because the two triangular squares are now empty whereas the ones we started with are filled (green). The same only happens with the original square choreography. This also has an interesting effect on the nearest miss solutions in that they alternate in overshooting and undershooting by 1 (so plus and minus 1 and not just minus 1 :)

[@Supremebubble]

Sigh, of course I meant the triangles that Mathologer is talking about and not the ones that are only there for the sake of visualization.

[@topilinkala1594]

And I must tell that when sometime in primary school our teacher draw the numbers 1, 4, 9, 16, 25 and 36 on the blackboard and asked us what they are. I got to answer and said that they are the sum of consecutive odd numbers. I really can't remember why I did not see them as squares, but I do remember that my teacher said that I was correct and that it was well spotted and then asked again what those numbers are.

[@aguyontheinternet8436]

Yeah, might be what someone would guess if they haven't learned much about exponents, and that person has a good chance to understand squares better than anyone else in the room

[@ahmad97ist]

I love your work! You really put effort in making them. Keep up the good work 👍🏼👌🏼 and thanks for the great content

[@benjanes3675]

I now understand how the decimals places of irrational square roots may go on forever! This is so cool.

[@morkovija]

holy cow, i was not expecting my jaw to drop in the first 2 minutes...

[@micayahritchie7158]

The assumption that our supposed solution was the smallest "solution" with integer values, simply led to a contradiction that meant our smallest "solution" wasn't the smallest in either case, but in the case of the triangle numbers there is no statement that necessitates that the animated image is the smallest solution, however there was such a statement in the squares argument that there has to exist some solution 3A^2= B^2 that is non reducible, however this always leads to a smaller solution. In the case of the triangular numbers there is no such statement that forces any one solution to be the smallest solution so no contradiction is reached by reducing the triangle

[@themobiusfunction]

No, there must be the smallest solutions because no natural number are smaller than 1

[@micayahritchie7158]

@@themobiusfunction tbh. I don't remember what I was on about

[@garthgoldwater5256]

oh man, this might be my favorite mathologer video. absolutely perfect

[@angelogandolfo4174]

The representation when you said “like root 2” at the start (I.e., what was shown on screen) was super nice and subtle!

[@kedrjack4649]

@Danil Dmitriev, я не знаю кто ты, но я тебя обожаю за то что ты переводишь эти ролики(ну и этот канал тоже)

[@untitled8027]

so can you use this to make new fractals?

[@kaerriss]

Yes at 14:03 the drawing is a fractal (actually it's the outline of all the shapes that would be sort of a fractal)

[@jeremysnead9233]

I love your channel. It has put my mind in to math comas, filled algorithms, crystalline and quasicrystalline polymorphism.

[@richardhead8264]

Your opening jingle from *0:00** - **0:04* reminds me of the intro from _"Don't Let Go"_ by En Vogue.

[@Shakespeare563]

At 7:30, the white diamond is made of two identical triangular squares that by definition must be the same area as the dark green. Sorry I know that's super obvious but I never get these challenge questions right

[@Mathologer]

Well you got this one right :)

[@braytongoodall4357]

Just some feedback. I think it would've been great if you could've shown the 1 + 3 + 5 ... is square through filling a square from a corner and adding to a larger square. So close! Also, if you could've shown that 4 is square because 4 smaller equilateral triangles can make one bigger one.

[@Mathologer]

Yes, both the facts you mention (and quite a few more) would have been nice to include. It's always a judgement call where one stops in this respect. For example, I felt that including the standard 1 + 3 + 5 ... would have led us too far off course and it's also something that a lot of people watching this video will be familiar with. On the other hand, the little triangular square proof that I showed was probably new to most people, even those who know a lot of maths :)

[@robertgumpi7235]

brayton goodall just what i thought

[@abj136]

Yes, the 3x3 + 3 + 4 = 4x4; 4x4 + 4 +5 = 5x5 etc.

[@davidmeijer1645]

i love you Burkhardt! You're teaching my post-Covid Grade 9 IB class for me!! Yesterday, I did the infinitely shrinking hexagons on a grid proof. Fantastic production value! Proofs by contradiction and Rational v. Irrational Numbers exquisitely done! It's actually wonderful, under the Hybrid model here in Ontario, I see ONE class from 8:15 to 12:45 with a break for lunch....so lots of time to delve deeper when desired!

[@Bibibosh]

This guy is truely amazing! Yes ! have watch all and continue to watch every video he will make! I am mesmerised

[@beenaalavudheen4343]

The total no of hexagons is a triangular number. That's why its called a triangular triangle

[@Mathologer]

Yes, at least that is why I call it a triangular triangle (and because I think it sounds funny :)

[@CraigGidney]

The reason the "make smaller construction" proof doesn't generalize to triangular numbers is because the construction does not always produce a smaller triangular triangle *on the positive integers*. In particular, applying the construction to 3 T_1 = T_2 produces 3 T_0 = T_0 which is not a smaller solution on positive integers. If you try to fix this by allowing T_0, then the proof would instead fail because applying the construction to 3 T_0 = T_0 produces 3 T_0 = T_0 (which is not smaller). Basically, it is crucial that you prove your construction produces a smaller solution *within the allowed range*. When demonstrating sqrt(3) is irrational, you must show that your construction does not produce the trivial 3 0^2 = 0 solution. You might do so by pointing out that it would require the three smaller triangles to exactly meet in the middle and yet not overlap elsewhere, which is not possible.

[@franzluggin398]

Well said. Your response is the first one I found that gave some insight into how and when exactly a line of reasoning parallel to the triangle case would break down for the hexagon case.

[@diabl2master]

He does mention this at 9:23, w.r.t. root 5 and 6. Maybe he thought the root 3 one was obvious.

[@shyamdas6231]

I had to pause the video and think. It takes time to really appreciate the beauty of mathematics.

[@AlabasterClay]

So pretty! Lovely proof with no numbers and no words. Just the shapes moving around are enough! Love it!

[@BradCozine]

"Now is ze time on Sprockets vhen ve MATH!"

[@wada-wada]

"if you don't agree then there is something really really really wrong with you" lol

[@jkershenbaum]

Personally, I thought that wisecrack was beneath him. He’s a great teacher. He really should know better than to math-shame anyone.

[@wada-wada]

@@jkershenbaum you are right math shaming is not very encouraging for learning

[@jkershenbaum]

word up!

[@reecec626]

I wish I'd had access to these videos back in high school. The explanations and animations make sensical fun.

[@shreyasraut6224]

Those 18 minutes has so much knowledge and home work , some one could probalably write a text book of it... Wat a video #mathologer purely rocks.

[@diegotejada55]

Why can't this method be used to prove that sqrt7 is irrational? Or any other number higher than 6 for that matter

[@andrewpearce6943]

It probably can. But as you saw in the 5 and 6 cases, the animation got really complicated. More triangles -> more complicated to show.

[@KnakuanaRka]

You probably could, but figuring out how to snip up the triangles to make the right number of double covers and gaps while ensuring everything matches up in size would be an utter mess.

[@livingbeings]

Can you explain the Bitcoin selfish miner problem?

[@Mathologer]

I could and I would definitely enjoy it. The problem is that there are so many great things waiting to be explained properly and there is so little time. Having said that the plan is to branch out to topics other than purely math later this year once I've survived my teaching at uni. So, we'll see :)

[@radexicalcradox7225]

Before creation, God did just pure mathematics. Then He thought it would be a pleasant change to do some applied. - J E Littlewood. It's about time for Mathologer aswell. µπΣ

[@niaschimnoski882]

Please‽ In 2013 I thought Bitcoin was cool...😫

[@roeesi-personal]

About the easier proof, it also jumped to me when you showed it the first time. The white rhombus divides to two equal triangular squares that have to equal the overlapping area.

[@superclue]

Thanks I had wondered how you found some of the fraction approximations for roots such as sqrt(2)~24/17. Near miss.

[@mariakhan6090]

I REALLY CARDIOID THIS CHANNEL !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 😍😍😍😍😍😍😍😍😍😍😍😍😍

[@12semitones57]

Friday the 13th.

[@Someone-cr8cj]

Who cares

[@Mathologer]

It's already Saturday the 14th here in Melbourne :)

[@savvyno.7025]

Ryan Truong Mumbiker

[@TobiasBalk]

In Hispanic countries we consider Tuesday the 13th an unlucky day

[@hoodedR]

Dude is your pic convergent?...wait....nvm

[@user-dp2nx8kb1i]

В школе давали готовые формулы, не поясняя откуда они взялись. Этот канал раскрывает всю первичную суть. Просто супер!!!

[@DanBurgaud]

After watching several of your videos, I am starting to understand your methodology. it really opens up a new way of thinking. THANKS!!!!!

[@Mathologer]

Cool :)

[@hindigente]

1:37 "This is an incredibly beautiful proof, and if you don't agree, there is something really, really, really wrong with you". You shouldn't have said that. Not because it is not a really beautiful proof, but because: 1. It's ambiguous. The quote could be understood to refer either to "agreement with the proof" or "agreement with the beauty of the proof". Someone who doesn't understand the proof at first glance could feel offended (rightfully so). 2. It's elitist. There are many people who despite being curious about mathematics, have not yet grown to appreciate a proof's elegance the way we do. It requires some sort of familiarity with mathematics that many people who watch these KZbin videos are still in the process of acquiring. I don't think it was ever your intention to be offensive or elitist, and I'm not even saying that you are/were regardless of your intention. But that it's reasonable to assume that this sentence could be understood that way by a reasonable person. I know all this is very nitpicky, but I wouldn't even bother commenting were I not such a huge fan of your work. I just don't want people to misunderstand what you actually mean and dislike a channel I like so much. That aside, this was a great video, even for Mathloger standards. Keep the great work up!

[@robertgumpi7235]

Felipe Hindi i agree. I thought the same.

[@puneetsingh6782]

seriously, does a linear transform preserves area? how did he even said it?

[@EfHaichDee]

He does this *a lot* in his videos. I think he's just insecure.

[@shy-watcher]

> They might then feel compelled to grab a coat; offense isn't always a bad feeling to have. And the closest thing to grabbing a coat when feeling offended is saying "the way you said that made me feel bad". After all, sometimes jokes don't land, and there is nothing wrong with improving them by listening to feedback. It did rub me the wrong way too. "Something must be wrong with you if you don't agree with me." sounds exactly like a clumsy appeal to consensus/authority, even if it was not meant that way. I happen to prefer proofs that accentuate and formalize the underlying "common sense" assumptions, and I think slick animations hide the axiomatic basis of the proof too much in the name of being elegant. When I saw the triangle proof, my first thought was not "how beautiful" but "why *exactly* is this true?" Does this mean "something is wrong with me"?

[@bgaskin]

True scotsman fallacy: Person A: "No Scotsman puts sugar on his porridge." Person B: "But my uncle Angus is a Scotsman and he puts sugar on his porridge." Person A: "Ah yes, but no true Scotsman puts sugar on his porridge." No true mathematician / truly sane person wouldn't find this proof beautiful...

[@jkloe]

This is kind of weird. What if you just chose the wrong triangles to disprove X²+X²+X²=Y². Maybe you just needed a different set of triangles for X² and Y². What if there is a set of triangles where this equation works? It looks like you chose a wrong example and then say "It does not work. So it's proof by contradiction." 😹

[@hugodiazroa]

+

[@manpreet9766]

Jkl oe actually the set of triangles chosen is just to show that the sides are integers. The main thing to note is that the second set also has integer sides. Why? Because it is created by subtracting integers from integer sides. Additionally this set is smaller than the starting set. Now here is the contradiction, we had assumed at the start that x and y are smallest integers, but we found another set as if x and y had a common factor which got cancelled.

[@chaosredefined3834]

There are only two ways for it to be "wrong": 1) There is leftover space 2) There are integers in your solutions. The "integers in your solutions" thing is why the triangular triangles work. You can go from A=0, B=0 to A'=1, B'=3, and you can't accomplish that if A' is just a sum of As and Bs. And if there are leftover spaces, then if A/B is sqrt(2), then A'/B' is not sqrt(2). The thing is, if a solution is wrong due to leftover space, it's useless. If it's wrong because integers, and you can reverse your operations (so you can go from small solution to bigger solution), then you can find what is the next step up from A=0, B=0, and get your smallest possible solution, which is now a strong proof (It is possible to get A and B that do this, and here it is, a weak proof would be "It is possible to get A and B that do this, but I have no idea how to get them). And if you get a "right" solution, then it is not possible to get A and B.

[@randairp]

Think about the types of triangles you are allowed to use (where you have 3 small "X" and 1 large "Y"). The X triangles can't be too small, otherwise you could snuggly fit 4 of them in, but we're trying to fit exactly 3. X obviously must be smaller than Y, otherwise we could fit 1 of them. This means that the triangles we use to fit 3 X's into Y must overlap with each other. This overlap will always produce the pattern he showed, where the unfilled empty space becomes our new Y, and the 3 tiny overlaps become our new X's. Mathologer chose "near misses" for demonstration purposes, but the actual size of the X's doesn't matter so long as it's smaller than Y and greater than 1/4th the area.

[@jkloe]

Thanks for the explanations, all. I understand it now. I missed the part at 3:13 where Mathologger said "smallest possible" A and B.

[@emilipesta4700]

I have no idea what you’re talking about but I love it

[@hindigente]

This is absolutely gorgeous. I hope to see more of those visual proofs and representations of algebraic numbers. I've really enjoyed Mathlogger videos lately. Really well explained and also beautifully animated, congratulations! Now I have some questions. Is there some geometrical relation between the visual representation of sqrt(p), sqrt(q) and sqrt(pq), p and q prime numbers? Can the procedure be generalized to further dimensions (it surely seems so)? There is a VERY interesting relation between those geometrical shapes and quadratic polynomials in one variable. Can every monic polynomial with integer coefficients be generated this way? Can irreducible 2-degree (or higher, for higher dimensions) polynomials in R be generated this way?

[@Kurtlane]

Wow! This is fantastic! There is enough material here to puzzle for a year. Maybe more.Great job!

[@Jerseyhighlander]

I love when someone explains why it works and why it doesn't work but doesn't at any point differentiate between the two, making the assumption that you can see what is going on in their head.

[@lanjiaojiaozhu2745]

Beautiful proof. I love you. Can you do one (instead of root of 3), what about cube-root of 2??

[@yakkismd5946]

I have found out about this triangle square thing, shown here within the example with 25, many years ago by myself. This is so cool!

[@axadams]

Awesome! Finding new ways to look at the simplest numbers. That's what resonates with me.

[@HighKingTurgon]

Burkhard, I must thank you for your beautiful reductiones, and I must purchase your square roots shirt.

[@moseszewdie6429]

I LOVE YOUR VIDEOS. Best presentation of complex idea. thanks.

[@SecureCottage]

Thanks for that video, the triangular proofs were definitely beautiful. I thought I'd add, as I did in another video for you, that in modular arithmetic irrational numbers can be rational mod a p*q modulus. For instance, while 3*x^2=y^2 is impossible in continuous numbers, this equation is certainly possible in modular arithmetic for certain p*q modulus. For instance 3*8^2=192 mod 61*73 and 192^(1/2) mod 61*73 ===241. Then the square root of 3 is then found by 3*8*241^(-1) mod 61*73 === 1700, and 1700^2 mod 61*73 === 3

[@ethannguyen2754]

7:28 the empty triangles at the bottom can be separated into two large triangles that have to add up to the large overlap triangle.

[@gabor6259]

One of the prettiest math videos on YT, the 3Blue1Brown level of prettiness.

[@DonaldKronos]

@Mathologer - I like the square root T-shirt. Also, cool how you used it in the visuals. :)

[@hymnodyhands]

This was a work of art as well as great math... Thank you!

[@codediporpal]

Great animations. It really makes these proofs quite intuitive.

[@ahobimo732]

This is beautifully elegant. It strains my smooth brain to follow the reasoning, but I managed to get the gist of it (barely).

[@Lorkin32]

I know you showed 5^2 in the beginning, but I'd like to see sqrt(4) or sqrt(9), just to follow the system of counting upwards and showing that it actually works for some.

[@gaboignacio]

I love your content, would It be possible to add some suggested reference books in the description?

[@metanumia]

Great video! I also love your t-shirt with a square-rooted tree! :)

[@HotspotsSoutheast]

I remember the day I discovered that squares are the sums of odd numbers. I was looking at a black and white tile pattern and could see that the squares were made by adding layers of odd numbered tiles. 1 + 3 makes a square + 5 makes a larger square + 7 etc. For a non mathematician it gave me a good feeling that I had discovered something my math teachers never taught me. I even figured out the summation formula for it. I have always enjoyed math “tricks”. Interesting formulas that solved complex problems in simple to understand steps.

[@HotspotsSoutheast]

I also remember my math teacher trying to egg us on for extra credit if we could find a way to trisect an angle. Everyone started trying to use their compasses and angles and I started trying to fold the paper. I figured there had to be some way you could fold the paper to divide the angle into three parts. And years later someone did figure out how to use origami to trisect an angle.

[@Snagabott]

On each row (bar the first one), there was a center triangle. The hexagons had that feature only every other row. With triangular squares, that means there was a center triangle left over. There is no center hexagon on T35. Hence none will be left from the final, smallest case of your choreography.

[@Hogscraper]

'If you don't agree you should really switch to a non-maths channel' This is exactly why I like math so much! When the "truth" of a subject has been proven and settled there's no use arguing it. You either understand or you don't.

[@deucedeuce1572]

Always appreciate the videos. Most of the time I have a question, you answer that question as I'm thinking it or immediately right after.

[@shirtvonnegut3094]

Still trying to wrap my brain around the Euler video… love the content my brother

[@Alfaomegabetagamma]

Fantastic video as always :) Regarding the final puzzle: I think that we can derive correct statements from a correct statement. So if we can prove that any of equations related to one another is true then all of them must be true as they "emerge" from one another. On the other hand side of we prove that one out of the "emerging" equations is false then all of them must be false. To me this is what Logical consequence stands for. But I don't know how to motivate the better.

[@thomaswarren7831]

Thank you Mathologer!

[@ThePharphis]

Amazing video. Maybe I can use the recursive generation of near-misses for a project euler problem sometime. I'll have to look! BTW, have you tried project euler problems before? You would have no problem with at least 100 of them, probably 200 of them since you likely no basic coding and algorithms.

[@SimoneEspositoWeb]

I really LOVE this channel

[@tomburris8380]

Great video as always! Really enjoyed it.

[@Eurley66]

Exceptionnal way of thinking about square roots.

[@danielinfinito6304]

Amazing video. Thank you very much Burkard.

[@xXMatManeraXx]

Hey Mathologer, I'm wondering where we can find those amazing math inspired t-shirts :))

[@aonodensetsu]

I figured out the Tn by myself in class using sigma, while trying to find a solution of the shortest formula for adding integers, glad I was correct!

[@debblez]

17:27 because in the smallest case, the three ‘triangles’ (at that point they are hexagons) fill the larger without any overlap.

[@MrRyanroberson1]

As my other comment may imply, I already made the connection to the square based proof of root two before you proposed the puzzle and was momentarily shocked that you didn't do that equivalent in triangles first

[@joelxrun]

Someone probably already said this but in your original proof, you assumed the smallest solution and showed any solution either admits a smaller solution or isn't a solution (ie: no base case). Here, a smallest solution exists, ie: 3 = (1+2)/1 so contradiction avoided. This video gave me some inspiration to try some weird stuff. Thanks!

[@lbblackburn]

Brilliant! Thank you very much, sir.

[@ranged12345]

Truly amazing proof!

[@gary3ward]

In Galileo's experiments with balls rolling down an inclined plane, he noticed that the increasing distances rolled during equal time intervals was in the ratios of the odd integers and that sums of the distance rolled was equal to the square of the number of intervals. In one experiment he used a straight ramp that had a curve from side to side that was lightly coated with flour. He started the ball out high on one side. Rolling from side to side was like the pendulum of a clock taking equal time intervals leaving a serpentine track that gradually stretched out demonstrating the above relationships.

[@Infinitesap]

As always :) thanks for great stuff and please do more :)

[@Mathologer]

More great stuff coming up :)

[@narmadraval25]

Thank you sir . Can the approximation of pi be found and shown by such a visualization as done here?

[@Mathologer]

Well, it's possible to translate the simple continued fraction of pi into a picture involving nested rectangles (something like the rectangle/square picture underlying the golden spiral). I might actually do video about something closely related soon :)

[@ferdinandkraft857]

But pi is transcendental, so you won't get an algebraic equation like Pell's.

[@samiam7290]

I think what goes missing is the self-similar aspect of the picture.

[@chalkchalkson5639]

17:00 in this case it's not an infinite reduction (3*T_1 = T_2, but when you try to do the propeller thing there is no overlap to construct the next step) so no contradiction arises. With the roots we picked the smallest solution and found a smaller one, here we just picked a solution and constructed another smaller one. In order for the irrationality proof to work you need to show that your smallest propeller image actually does have the overlap needed to construct the next step. For root 2 and 3 that works no problem since 2^2>3 thus the edges of the two smaller triangles must be longer than half the large triangle. As stated above, for the triangular numbers that doesn't work as 1+1+1 = 3 is a counter example