Visualising irrationality with triangular squares

by birtanpublished on August 17, 2020

welcome to another mythology video it's something that only a few experts among you will be aware of but pretty much every single mythology video features fresh maps made up just for you every once in a while I even sneak in some of

My own little theorems and proofs today's video is one of those videos what I want to show you today are some new beautifully visual ways to prove the irrationality of some small integer roots like root 2 as well as some really

Cool and closely related mathematical gems and it's all based on what I usually refer to as triangular squares what the hell is the triangular square well it's definitely not clickbait okay what I mean by triangular squares are

Equilateral triangles that are made up of mini equilateral triangles like that one over there why triangular squares because the number of mini triangles is always a square namely the total width in terms of mini triangles squared in

This case the width of the big triangle is five minute wrangles and I claim they're exactly five squared equals 25 of these mini triangles okay and I've prepared a really nice animated proof that this is the case in general ready

To be dazzled here we go okay yeah and it's getting squarish there and done there v squared this is an incredibly beautiful proof and if you don't agree and there's something really really really wrong with you and it's probably

Time to switch to a non mass channel k also hiding just around the corner there's some more beautiful stuff too good to pass over so let's just savor that too before we go all irrational note that the layers of a triangular

Square are consecutive odd numbers so one yellow triangle in the first layer three orange in the second layer then 5 7 and 9 and so one plus three plus five plus seven plus nine the sum of the first five or numbers is five squared

Doing the same for general triangular square of which end proves that the sum of the first n odd numbers is N squared also super beautiful isn't it anyway for what follows you only need to remember that a triangular square of which n

Contains n squared mini triangles I now want to use triangular squares to prove that the square root of three is irrational that is that the number root three cannot be written as a fraction as a ratio of integers to prove this let's

Assume that it's in fact possible to express root three as such a fraction okay and let's also assume that a and B are as small as possible okay then squaring both sides and simplifying gives this here okay but

Then of course 3b squared equals a squared that's the same as saying that B squared plus B squared plus B squared equals a squared so what does this mean this means that if root three were rational then the equation x squared

Plus x squared plus x squared equals y squared would have positive integer solutions and that B squared plus B squared plus B squared equals a squared would be the smallest such solution but this would also mean that combined the

Three Green triangular squares of width B would contain exactly the same number of mini triangles as a triangular square grid of whit's a on the right and this would mean that all the mini green triangles on the left would fit exactly

Into a large grid on the right and here's a super stylish way to begin an attempt to fit them pretty anyway this attempt results in three dark green triangle overlaps and an empty triangle patch in the middle

This patch and the overlaps will also be triangular squares right however since there are just enough mini green triangles the three dark green overlap triangles together must exactly fill the white triangular grid but that's

Absolutely impossible why because we supposedly started with the absolutely smallest way to express a square as the sum of three other squares that run up there and of course no solution can be smaller than the smallest one well we've

Played this game a couple of times already big last couple of videos so you should be okay with that now the only way to resolve this contradiction is to conclude that the assumption we started with namely that

Root three as a fraction is false in other words root three is irrational or equivalently the equation x squared plus x squared plus x squared equals y squared has no solutions in positive integers how slick is that now using

Similarly stunning triangle choreography we can also show that root 2 root 5 and root 6 irrational now before I do that let me hammer the crux of our proof by contradiction just one more time ok so root 3 being a fraction is equivalent to

There being three identical triangular squares adding to another triangular square right and the contradiction hinges on us showing that if this was really possible for some triangular squares then the same would be possible

Using even smaller triangle squares so now let's see how to prove that root 2 is a rational root to being a fraction is equivalent to there being two identical triangular squares adding to another triangular square like that and

To make the contradiction work we then simply have to show that this equality implies another equality involving even smaller triangular squares and here's how we can capture those smaller triangular squares let's chase them down

Okay pretty pretty pretty pretty very nice at this point the dark green overlap would be exactly as large as the white empties area now let's shift over lap down this would fill some of the empty area and

Create new overlaps there to smaller triangular squares heading to another triangular square which then unleashes the contradiction and proves that root 2 is irrational as well hmm here's a little puzzle for you we have to have a

Password there will be more have another look at this picture here can you see an even easier way to arrive at a contradiction that is can you see two other small triangular squares that add to one large

Triangular square it's really jumping out at me but see where you can see it too let me now show you the route five and six triangular choreographies followed by some choreographies usual actual squares and Pentagon's first root

Six with six triangular squares adding to another triangular square there we go beautiful you must agree right two things before I move on when using the root five and root six

Choreographies to show the irrationality of these numbers one actually also has to make sure that the various dark overlaps and animations always exist and are of the same size I leave filling in the details as

Another puzzle for you the root five choreographies are particularly tricky to pin down for example in the Pentagon root five choreography it's not even clear where the squares are again consider filling in the details as a

Puzzle or check out the write-up in the description of the video longtime mythology fans may remember that in the early days of the channel I did a video on proving that root 2 is a rational using the simple square choreography in

Fact it was this beautiful proof by a mathematician Stanley Tannenbaum popularized by the great John Conway that started this whole line of investigation a paper expanding on this idea was authored by Stephen Miller and

David Montague who discovered the root 3 triangle and the root 5 Pentagon choreographies my contributions are the triangular square root 2 5 & 6 choreographies and using triangular squares to illustrate things anyway all

This is just the star of much more really really beautiful mathematics such as nearest miss solutions to our impossible equations best rational approximations to our small integer roots and a really nice paradoxical

Inside of our triangular numbers to finish things off well let's go remember by proving that root 3 is a rational we also prove that the equation x squared plus x squared plus x squared equals y squared has no positive integer

Solutions this means that the three green triangular squares together and the white triangular square that I used to illustrate the proof actually cannot have the same number of mini triangles in fact I chose the numbers on the left

And right to be as close as possible differing by just one and so 15 and 26 form what I want to call and nearest miss solution to the equation X square plus X square plus X Y because Y squared the next best thing to an

Integer solution among other things this means that 26 over 15 is an extremely good approximation of rule 3 sort of the root 3 counterpart of PI's 22 over 7 have a look pretty good right when we run our choreography we start with this

Propeller formation here and after our triangles finish their dance we end up with another propeller formation even better the new triangular squares correspond to another nearest miss solution that one down there let's

Unleash the choreography on this small propeller ok ok there yet another propeller and another nearest miss solution one there now clearly something goes wrong if we try this one more time and I'll leave it as yet another puzzle

For you to figure out what but if we can't go on forever in one direction let's go the other way let's run the choreography in Reverse here we go so everything going reverse now larger even larger now we're back to where we

Started from but why stop here let's just keep on going right ok and now we won't step up it's probably not so surprising it's also not hard to show that this new propeller also corresponds to a near-miss like this ok so the

Numbers pan out like this here and you can check that this is really true in fact it's not too hard to prove that if we keep on going we'll always get nearest miss solutions and in fact we get all nearest miss solutions to our

Equation in this way anyway here's a list of the smallest of these near miss solutions ok now if we take the ratios from top to bottom this will give better and better approximations of route 3 let's just check this ok so 2 over 1 7

Over 4 26 over 15 and there I've highlighted how many digits we get correct in fact the sequence of these fractions converges to root three so if we combine all our propellers containing all of

These ratios into one large picture in some sense this overall picture captures root 3 is root 3 so this is root 3 have you ever seen root 3 this is it very pretty right and as you've probably already guessed everything I just said

About root 3 can also be shown to work for the root 2 root 5 and root 6 choreographies now just for fun here's a picture of root 2 corresponding to the original Tenenbaum choreography also super pretty right now there are some

More footnotes that I really should add about a second type of nearest miss solution about irrationality of integer roots in general Pearl's equation and some interesting connections with infinite continued fractions for those

Of you interested I'll put some footnotes in the description of this video but for the finale of this video let me tell you about a freaky alternate number reality to which our triangle choreographies can be applied as well as

A very paradoxical puzzle for you that pops up in this context have a look at this I call this a triangular triangle well nobody in the universe does that except for me but who's going to stop me right I'm the mythology puzzle for you

Figure out the motivation behind this strange name it's not hard anyway the number of hexagons in a triangular triangle of which n is called the ends triangular number hit T n so T 1 is just 1 T 2 is 1 plus 2 equals 3 T 3 is 1 plus

2 plus 3 equals 6 and so on and so the nth triangular number TN is just the sum of the first n positive integers which as many of you will know is equal to n times n plus 1 2 over 2 so for example T 5 the number of hexagons in this

Triangular triangle is equal to 15 now just like the integer squares the triangular number also feature in number theory and a lot of famous mathematicians have proved theorems about them for example up there is one

Of the most famous entries in Gauss's mathematical notebook it says that every positive integer is the sum of at most three triangular numbers you can tell that Gauss was really excited about having found a proof for this just have

A look at the Eureka and really bold letters preceding the statement okay are now important for us is the related factor unlike squares there are instances where three times a triangular number equals another triangular number

Here's an example so T 20 plus T 20 plus T 20 is equal to T 35 let's unleash our root three choreography on this set up and see what happens so obviously this propellor correspond to a smaller triangular number some that

Also works this one here T 5 plus 2 5 plus T 5 is equal to T 9 in fact we can run the choreography forwards and backwards to generate all instances of three identical triangular numbers adding to another triangular number now

Here's my main puzzle for you forget about the other ones if you just want one this is it the smaller equation follows from the larger one right so why doesn't this prove just as in the case of triangular

Squares that three identical triangular numbers cannot add to another triangular number that's a tricky one let's see whether anybody can figure it out anyway that's it for today


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