So here's a little bit of mathematical magic. Let's just have a look at it five pieces Move them around you combine them, and we've turned one square into two squares tada not bad huh, so one square out of two squares but of course there's one thing that actually doesn't change when you do this sort of thing and area doesn't change, so if you calculate the area of the square that we started with and then the total area of the two squares that we end up with it's going to be the same and obviously you expect similar sort of things to happen whenever you cut something into pieces recombine the area shouldn't change and right Except there's this really really famous theorem in in Mathematics which is called the Banach-Tarski Paradox and Vsauce just an amazing video on this just a couple of weeks ago It says that you can take a solid ball. Well, it's got to be mathematical ball It can't actually do it was a real ball it's got to be a mathematical ball solid and you can split it up into a couple of different pieces again and Then you can just kind of move around these pieces in space. You don't do anything with them and we combine them like this and You get two solid balls of exactly the same size as the one that you started with So the same thing as before you know take one thing and turn it to suit two of the same shape But the difference is that actually the volumes of both these resulting spheres are exactly the same as the one that you started with Where are you doubling up the volume it's really crazy. You know. I mean if you have to say this is only possible in the World of Mathematics When we are talking about dimensionless points and all kinds of other things so Nothing to do is you know atoms or whatever you know it's in math, in maths you can you can do the sort of thing Right so it is a really really good one. Now when I first came across this one This paradox I actually wasn't that impressed Why? Because I already knew quite a bit of mathematics and I played around with infinities and all this other stuff So what I thought, I'd do today is I'll do a prequel to the Vsauce video okay which kind of puts you into the mind of a young Mathologer when he first encountered Banach-Tarski There's quite a few things that you won't find anywhere in textbooks here in this video. So it should be good okay, so When you think about this, what are we doing? We kind of take a shape and we kind of split it into bits And then we move the bits around and all of a sudden We've got something that's got double the area volume or whatever you want okay, and actually you can do this very very easily Okay, like this, so I just take this this little segment there and the bits I'm using are actually individual points that the segment is made up off okay? And then I'm going to now teach every single point where to go, okay. Let's go, so I need some point up there and Some auxiliary line down there, and now I'm going to teach this point. Where to go okay, so this point is going to go there Okay, then this point is going to go There and every point goes the same way right so every point goes the same way so now all the points Know where to go they all go to different places right and say I'll just say Everybody now all your points you all go. I want to all go together and cut our Line segment of double the lengths that you had before and really what you're doing here is kind of the stretching the whole thing right? But when you think about it in terms of Banach-Tarski, you know just think of the individual points that the individual points They all move and do anything else and then reassemble themselves into longer bits right and so if you do this one here Or that one here or anything else You can think of this as the same points? Rearranging themselves into these different shapes of different lengths, which is pretty amazing to start with right? That's the thing, there is no empty space Okay, and you can do the same sort of thing here with maybe you know Squares I kind of stretch it out, and you can think of like the individual points. You're kind of moving rearranging themselves and so it's Always the same number of points and actually that's also why we say in Mathematics That like this shape and all the ones we have before, they have the same infinite number of points right so like all actually all Plane Shapes with an area have the same infinite number of points you can see. On all linear Shapes have the same number of points But so in that circle same number of points but get yeah They have the same number of points. Yeah, they have the same number for they have the same number It's same the same infinite number of point that's right, but it gets worse it gets worse, How can it get worse well it actually turns out that this square and this segment here have the same number of points? And actually even the mathematicians who first found this you know you know they couldn't believe their eyes. So I'll just quickly show you how this roughly goes so basically you have to teach the points in here Where to go in there, okay, and it goes like this so let's say this is like 1 This is one unit right and so we take any point in the square like that guy here and that now has coordinates ok so it has coordinates like zero point two something in the x direction and Zero point six in the y direction, okay? and now what we do is we kind of You know do this? so we take the two two numbers here and Interleaf them like this and that gives you a new number around there right and that new number down there Well, it's somewhere in this interval here. It's right there. So now we've taught this point. Where to go, okay? We've taught this point over and every other point also now some ways to go right now. We basically say go and all the points from the square rearrange themselves entered interval and Well, there's a little bit of a problem in here because some numbers have like two different decimal expansions but can be fixed up nicely that's the basic idea for rearranging all the points in the square into points of an interval and You can do the same [sort] of things for cubes right for cubes You've got like three coordinates, and you also zip them together like this, and so what all this means really is that? our [block] infinity Rules So if you've got any blob of area or volume or lengths or whatever all right it all Consists of the same number of points, it's the same infinity. They were talking [about] so I think that's pretty amazing. That's pretty amazing Yeah, yeah into a line [see] the same number of point you can make this thing as short as you want as long as you Want infinitely long doesn't matter all right? So our whole space like if you've got this little little [tiny] little line segment contains exactly the same number of points as the infinite three dimensional space that we live [in] or the infinite four dimensional space We live in or whatever all right? So you know pretty much anything which somehow tangible in mathematics with a volume or area or [line] like this? but Not all infinities are the same you would think well after this all infinities are the same we've just been able to kind of Collect everything together But actually there's infinitely many different infinities to this infinities that are a lot you jur than what we've been talking about here But you never see them In real well you can you can [make] them up, but like you know in school or [Uni] you really don't see them very often But there's one below At least one, and that's [actually] the one that people usually talk [about] it's accounting infinity, and so whenever you've got an infinity Where you can actually count the objects with like 0 [1] 2 or 3 4 or 5 then it's countable infinity And that's actually really really really different from the blob infinity ok which well, we'll call it uncountable right although uncountable also contains all those other infinities, but when we talk about You know uncountable unity we just mean the blob infinity ok so just [in] terms of points where we've got these points on the number lines it Correspond to 0 1 2 3 So those points you cannot Kind of reassemble into a line segment cannot be done and actually there's a really quick way of seeing this and it's also um It's not a way. That usually it's usually done But it also shows you how how much huger the blob infinity is then this infinity ok so here we go So we'll do is we'll just take a segment here. Which is 8 units long, okay? We take half of it and [put] the first point on it call it [2-0] pound, right? You take half of the remaining bit and put point on and call it a one bit And we just keep on going [like] that. It's like to infinity 1 to infinity gets shorter and shorter line segments Everyone in the Middle [heiser] has one of our points here, okay, and now what we'll do is we'll just take all these line segments, okay, and Distribute them like on the infinitely long line, okay number line you just put them on somewhere, okay? So any way you want any way they want even they can overlap? So maybe we want to avoid at the points that the red points come together, okay? But apart from that we can do anything you want right anything whatsoever now What's absolutely clear is that since these line segments start out adding up [to] well eight units? No matter how we distribute them here They're still going to add up to eight units and eight units can't cover an infinite long line so what it [means] is no matter what you try here you go always going to have a lot of Empty space there where nothing is right where you don't get anything right so [with] these you see segments You can't cover the whole line what it also means that with individual points which are contained in line segments, you can't you know do the whole thing but you can't do the whole thing and Actually when you [think] about it. You know there's like heaps of space left over as infinity, and there's there's eight And actually that eight you can make as small as you want right? So you could even ask well I mean if I actually wanted to assemble these these Countably many points like you know into something that can be measured in terms of lengths, you know Well at the moment [would] be less than eight, but since we could also started out with an interval of lengths You know it's one over googleplex Or even smaller so the logical conclusion is really so if you actually wanted to measure this in terms of lengths it would be zero So it's not much. It's [huge] right, there's a huge difference of in it and when you talk [about] different infinities. It's really huge okay So there's and these are these basically the two two infinities [that] you get in in maths, right? so these are all kinds of objects there see and they fall into two different categories here in terms of so this is the countable infinity and Is the uncountable infinite serves as the blob infinity okay? So this is a blob infinity and this yes the other the other guys yeah, and well Pretty pretty neat right so I mean I knew all this stuff before I started looking at [burn] [our] task now [burn] [of] Taski, so what's the difference? So there was the resource the full thing right and then we had our Version [baby] kind of you know so what's the difference? Well? There's a difference here in terms of We just take the point see right here. They take big chunks What it is but we can actually do this in terms of big chunks, too? So just imagine that we don't move the points, but we we have removed these line segments that the square consists of okay? So we can do this All right, so the line segments kind of rearrange themselves, so it's big chunks or like in a three-dimensional way You can have slices through this cube right you kind [of] do this and double over the amount Okay, so that's big chunks moving around in space. You know so [yeah], here's very similar. There's still a difference, right? there's still a big difference here [and] So here. We sort of like uncountably many pieces that move [but] They have got finitely many That's a big difference There's also something else which it's quite important when you look at the slices or the bits that are kind of moving around here they are What are very nicely modified you know so that is basically something that you think well? It's very close to your meat cutting cutting off a slice of cheese or something like this if you look at these guys here They're very fuzzy. They are very fuzzy even mathematically even just to kind of make them up you can't use scissors or anything What comes out of here doesn't have any sort of? Properties like area or anything like this and even to define them you have to bring it I can really really really big gun in mathematics. Which is actually a bit controversial which is called axiom of choice, so [I'm] not going to go into this, but just met just there there's something really weird here Okay, and how can we see what's what's really amazing here And what really makes this thing aways well finitely many pieces so just quickly What [rotors] vSAuce2 well he cuts the ball apply this okay? then eventually We kind of go like this. We just take one out We rotate this piece by 90 degrees, and then we rotated by 90 degrees it turns into those four four pieces together Which is really weird one, and then we take where [whatever's] left over here put that in and that gives us the first ball There's lots of stuff left over okay? then take that guy he split it up into two pieces rotate that one here by 90 degrees and Then all of those two together correspond to those three guys Throw those two leftover ones back in just need this one And that one which actually small pieces are countable pieces we kind of you can make them up from nothing. There's another trick and we're And we get these these two solid balls again Pretty amazing you're ready now to watch this video. You should really watch this video. Haven't watched it yet But now what is amazing [also]? Is that actually this doesn't work for plain figures for example? [so] if you tried the same [thing] for a circle isis a solid circle and you were to try and pick like split this up into finitely many parts and Recombine them in this way into two other circles that actually doesn't work on so there's a lot more work there And there's a lot more amazing stuff there. You know when you really go all the way down to two finite in this respect [because] It's infinite, but this one here you really need kind of this extra Extra dimension to move around and you'll actually see this if anybody watched the resource video that you need something extra, [and] yeah That's basically it
It seems that nothing except for actually reading the theorem (some day, when it's more accessible) will make me actually appreciate this paradox.
Hang on, I just watched a video where he said it is impossible to sum two equal squares to make a larger square.