Math in the Simpsons: Homer’s theorem

well last weekend i actually did a talk on mats in the simpsons here in melbourne and i'm going to split this up into three different videos so today you get the first one nice and

Short will be familiar some of you will be familiar with this first clip but i think i've got a few twists and turns to add to all this which will entertain most of you all right here we

Go hey there's something you don't see in a toilet every day anybody lose their glasses last chance and some of the square roots of any two sides of an isosceles triangle is equal

To the square root of the remaining side that's a right triangle you idiot this is great anyway so what does it say what does it say well um that and some of the square roots of any two sides of an isosceles triangle

Is equal to the square root of the remaining side which sounds very similar to this guy right pythagoras a squared plus b squared equals c squared well it sounds very similar so what if

If he actually meant to say this what were his mistakes well the guy in the background already pointed out one of them it's not an isosceles triangle it's a right triangle right triangle right

Okay what what else there's there's other things in there that don't work uh any two sides uh any two sides yeah that's that's fine so any two sides got to be those two here in in in pythagoras and then there's one

More which is of course the sum of the square roots the sum of the square roots so some of the square roots doesn't work um any two sides doesn't work so that that's strong

But i got you know it's got me thinking i mean homer knows what an isosceles triangle is this is ridiculous um so maybe he actually did not have uh he did not have pythagoras in mind

Maybe he learned it somewhere else and there's actually another simpsons episode that gives you a bit of a hint where he may have learned it oh there's new marshmallows in the belfast charms

No no that's bart's cereal it's the only way i can get him to take his vitamins the sum of the square roots of any two sides of an isosceles triangle is equal

To the square root of the remaining side that's not right yes it is they're my lines there's a scarecrow in the wizard of oz hmm the scarecrow and the wizard of oz actually if you go to the movie

And uh you find the sketch some of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side oh god i've got a brain so he says this after he's got the brain

Right hmm well like he got a bad deal he got a bad deal but maybe not maybe maybe he's right you know just because this sounds similar to pythagoras doesn't mean that it has to

Be you know it might still be correct right so let's have a really really close look maybe it is correct right um so here's an isosceles triangle so what makes a triangle isosceles we've

Got two sides that have equal lengths okay um or two angles here the same all right now what does he say he says if i do the square root of this and add to the square root of that you get the

Square root of this but he also says you can do this with any two sides so i should also be able to kind of start square of this plus square root of that is square root of that so let's also write down that one here

Now have a look at this one here doesn't mean well it's okay but i mean you've got two times as square of a one on two sides of the equation so what can we do we can just cancel it out so we cancel it out goes and then what's

Left over is square root of b is equal to zero which means that b is equal to zero you get a line right i mean maybe you consider this as an isosceles triangle but usually not right so i mean there's

Definitely something that's not quite right but let's just forget about this any two sides business okay maybe maybe maybe just this was meant right so

Square of a plus square of a is equal to square root of b maybe he meant that one right uh well let's have a look so what we do is uh we pull those two things together so it's 2 times square

Root of a what do we do now pretty obvious we squared to get rid of the square roots so we get this one here and now we'll have a really really close look at it and actually we can do this in pictures here

So what this says is we've got like a and we've got a and b is four times as long as a and it's pretty obvious you can't make any triangle out of anything like this right

So just forget about it no no it's a big no it's not going to work right so it's pretty wrong it's pretty wrong but it can get even wronger so let's just have a look at like any

Triangle whatsoever and have a look where you can actually get a square of a plus squared of b is equal to square root of c so again to get rid of the square roots we just square the whole thing

All right so square here so now what do we get here we get the first times chi squared plus 2 times this times that plus okay so what's this one squared a and then we get plus square root of a

Square b yep that's right and the last one is b okay and then on this side c okay cool now just rearrange this a little bit and you get a plus b plus something c right so basically what

This says is that c is a little bit longer than a and b put together right so there's a there's b and c is a little bit longer and again it's pretty obvious you can't

Make any triangles out of this right so it's pretty wrong also i mean this this this is isosceles triangles in the plane what happens if you actually kind of look at triangles maybe on a

On a sphere or on a weird surface or anything like this can you actually get like the scarecrow theorem or homeless theorem to work anywhere well well in the plane what really

Kills this one here is the fact that in any triangle a and b always add up to something that's bigger than c and basically the scarecrow theorem contradicts that right

So what we're really aiming for is a world in which this guy here is not satisfied it's called a triangle inequality okay and actually turns out that it's very hard to find anything

Reasonable where this doesn't hold so for example if you look at triangles on a sphere they all satisfy this so even on a sphere the scarecrow's theorem doesn't work or

Hormone theorem doesn't work but you know there's lots of mathematicians out there who know about lots and lots of really really strange words so maybe one of you has an idea for a strange world where we can

Actually make this work maybe just in one instance okay um right now i actually also started digging to see what i can actually find pythagoras now pythagoras actually does

Pop up in the simpsons um i know of two instances and actually one of your missions is going to be to figure out where exactly in those clips pythagoras pops up

Maybe some of you actually know of of other instances of pythagoras or the scarecrow or homer's theorem in the sense that i'd really like to to know about it um even other other movies other

Episodes other tv episodes i'd be really interested in finding out um and for the simpsons lovers among you as i said before you know pinpoint all the episodes where these things are from okay let's have a look

At the first clip and see whether you can see pythagoras somewhere in there i want a colossal donut just like the one on the sign down nuts that's false advertising sorry

Sir no refunds i paid for a colossal donut and i'm gonna get a colossal donut that's a tough one you may have to do this one frame by frame but we also get a

Hint in the credit because the second one should be pretty obvious anyway so that's that's that's it for for today you