Alright this week we've got something extra special for you People often ask me you're mathematician. What sort of mathematician are you well? I'm a geometer so here geometry special t-shirt today and today, I want to show you a new part of my mathematical toy collection which is mostly geometrical. Okay so it's Mainly about these sorts of you know contortionist cubes that probably everybody here is familiar with So have a look at one of those things I mean just in case you don't remember this one here is kind of a maths themes, it has a times table on it and what it can do is, it can transform like so into something Different right this one isn't new it is jarred a little bit, but basically you've got this disc Cube here And I can go backwards you see the colors change Now you've seen these I think everybody's played with them, or most people have played with them Now what gets a little bit lost with all these designs here and the different pictures that are usually on top of them is something really amazing and I've just highlighted this by kind of getting rid of all the pictures here, so I've just Made the whole surface here white okay, so what you see here is white and now I'm going to basically turn it right and When I turn it, I've generally now actually finished with turning what's happened? Is that the white has turned into the black so what's really going on here? Is that the cube turns inside out? so it's made up of eight little cubes and every one of the little cubes three of the faces the three faces that you see here are colored black and then the other three faces of colored white and The Cube the big Cube is hinged together in such a way that you can turn this whole thing inside out very nice And so that's what's really happening, okay? Now there's actually a couple of different ways in which you can hinge these things together, and there's some nice mathematics there But that's not what I want to talk about today There's actually one step up from this and I want to show you that one here This thing here is is one of the highlights of my collection. It's a real miracle It's actually it's actually a cube like this, so it's colored one color outside one color inside in this case. It's Silver and gold okay, so let us take it out So it's a silver on the outside and now I'm just going to turn it into the gold so here we go Now it's gold alright again. It's a bit stiff and now but it can actually do a few more things okay, so let's just flex it again and You can already see [there's] something weird happening here oops So the whole thing kind of comes apart into two Parts and actually when you when you fold those up you get a star here, and you get a star there and even better so you can you can turn these guys here inside out and then this star here turns into a cube And That one here turns into a cube Too so now we've got two cubes where there was only one originally. I mean, how good is that right? Okay, and actually. I mean these are of course hollow cubes, so if you actually look inside You can see well those stars that we see here They actually fit right in there And so I mean I could just kind of slide the star in here only problem is I can't really close it at this point in time So for that, I have to fit the two Two pieces here together in the right way. I'll just have a look at it like this. It's exactly the same At this point in time. I actually can't slide them together. So what I have to do is I have to turn this thing Like that now this looks slightly different right? So there's walls here. There's no walls there and at this point in time I can take this and Recombine it and then it's just a normal thing again Okay, so what's going on here? Let's just get this star out again. Okay, so the stars out. So there's the stars out What just don't have a really really close look at this this star okay? So this star? when you actually have a look around it, there's lots of these stars you can see what so I've got one that's Here, and I've got another one. So here got one here. I've got a lot more on on this side here So, what's this star? Well this star is something very special. It's got a name in Mathematics. It's called stellated rhombic dodecahedron well, that's quite a mouthful. So what is it actually what makes this this cube work, okay? Okay, so let's have a really really close look at At a cube So what I've got here is six pieces That that you can combine into a cube so basically one side of the cube is this and Then what you see here is a kind of a pyramid going in to the middle of the cube, okay? and there's six sides of the cube and so from big six sides of the cube there's these six pyramids going into the middle and then when you combine these these parts here you actually build a cube so So that's a complete Cube Now, I have to get it apart again, and so now they're back okay, and so the Secret behind this stellated Rhombic Dodecahedron well it's a star let's call it a star is that we can combine these six pieces into two halves of the cube in a special way So let's just do this again so I can put two of these guys together Now I can put that one into there and then I've actually got half of the cube here And I can put the other two or the other three together in exactly the same way, and I've got now two parts of the Cube and these two parts of the Cube now fit together like This so if I slide them together the cube will be complete, okay? And now if we actually have a really really close look I can kind [of] put them together like this And you can already see kind of part of the star here merging, right? I've actually 3D printed quite a few of these things so I can show you so we put two of them together There's eight of them You know the original big cube consists of these little eight little ones And so every single one of the little ones gets split into half right, and then it's just Hinge together in such an ingenious way that you know the whole thing actually comes apart into these two different stars Okay, so there's the star here So that's that's how you get it well while we're at it I might as well kind of explaining a couple of side Miracles here, okay? So we're going to do this up a couple of side Miracles um So let's just get rid of the star and do something slightly different and then get back to the star so there's actually a second way of combining three of these guys into two halves, so if I just remove like the yellow part here and attach it [like] that now if I remove the White part here and attach it like this we can get kind of two different parts here we can also slide them together, so if I kind of go like this, and I go like that I can actually slide the two parts together and that will be a different way of splitting the cube up into two parts based on the same pyramids alright so based on the same pyramids and well, this this is actually something that Well, maybe some people will recognize you know some people will recognize So I've got a good another version of it here again 3D printed, and I'm just going to slide them together, okay? So let's just do this so if you put them together like this you Know and just press they come together and maybe now it's a bit clearer or what I'm talking about Have you seen this sort of setup before which is that we know? well actually This is just kind of a square version of of what you see in a tennis ball, right, right? So you know when when you kind of just follow this curve around. It's very much like in the tennis ball now I've actually cut this thing open and there you see the two different parts here two different parts and well I've also done something else. Which I think is kind of neat which is I've actually taken a tennis ball like this. Which is hollow and then kind of filled in the bits in the middle so that when you slide the two parts together you get a complete ball here, so you get a complete ball and That is actually something so that's what this looks like so that's how you have to extend things to the middle so that Things fit together nicely so you kind of fit them together like this And then you get something very very nice, and what you see there is actually yin and yang So the tennis ball curve, you know, it's basically just a 3D version of yin and yang So that was my first kind of side miracle here You know that's that's also something that you basically can't see anywhere else We made this up right so it's the 3D yin, and yang here And it's it's all part of this you know geometry that we're talking about here Okay, so now let's get back to our star effect our star and I will just want to show you a few other things that this the star can can do okay, so Star based on this this way of splitting up the cube okay, so that's important So let's just do something else with the star the star actually comes up Quite a bit if you just keep your eyes open you'll actually find a star in all sorts of places so for example Where do you think I found this one Giuseppe? mmM close The dog toy Department, and now I found it somewhere else Just recently in the reject chop here I got this to see itsy bitsy a puzzle eraser pack, and there's a star, and I've actually got a wooden version of this It's actually quite famous, and I think quite a few people who have seen have seen this before so it's this sort of wooden puzzle here, right and when you kind of go like this it just falls apart and it falls apart into six identical pieces okay so six identical pieces Now when you try to combine them what you do is you basically do it one at a time What everybody tries to do it one at a time? So now we are at this stage here with this stage, and we have to fit the last guy in okay so we have to fit the last guy in and that's actually pretty much impossible at the stage one and so The way to get this thing together is actually based on these This way of splitting the cube into two parts Okay, so the thing to see is that one of the pyramids here is actually visible Right here in the center of one of those pieces okay, so we can see [it] right [here] these guys here They combine exactly into the cube as these six pieces here, okay? So the way, we're going to do it is actually to arrange the big pieces just like one of those hearts So that we see one of those halves in the middle. Okay? Let's just do this so we kind of go like this we go like that and Now what we've got is like a little propeller okay, so we've got a propeller and in the middle We've got one of those [halves] and now what we do is We create a second propeller like this, so we kind of go like this. It's a bit tricky and like that and We've created a second propeller now this propeller is actually not the one we want Because the propeller is kind of both propellers are going the same direction Right so what we want is two propellers that are going in opposite directions So let's just take this piece out and put it in on the other side, right? So now the propellers are going in different directions, okay? Now what you can do is you can just line up these two? halves that you see in the middle the two halves of the cube like that and Slide the whole thing together, and that's it so if you've got Like lots of these cubes Identical you can just put them right next to you Go, just like you know like build a brick wall, and you kind of build a whole wall So you can fill the whole space or all of space, right? Seamlessly you can do the same with these stars? So if you have a lot of these stars you can fit them together seamlessly To fill all of space, and it comes a little bit. I mean, it's it's it's almost obvious when you look at This guy here, and we just kind of put two halves together So I'm going to put the two halves together right, so What you can see here, I hope is Well, you see these two boxes, okay? Now what we can obviously do is we can kind of fill the space here if I've had and I love two boxes I could Put them here on top okay, and then I could just use these boxes and fill the hole of space, right? But when you can see at the same time is that when we've kind of put the boxes together there's like holes here which can be filled with one sort of star and Where you can see here in the middle. There's another star forming already right? So when we put all the boxes together We'll actually have like lots and lots of solid stars already with gaps that are stars themselves And then if you kind of put stars in you've got a whole space filled up, right so so that works and actually We're kind [of] going back to this puzzle here. I bought all the puzzles in the shop and actually got myself like lots of these things [I] actually combined all the colors together like lots of these and With these I can actually kind of start start making one of those space fillings Which is a lot of neat, so I'll just start with this so here's one of those halves, right? there's another one of those halves so it's kind of fit them together like that, and I can actually start building a tower like this, you can see that the tower is already solid in the middle and so the tower is Solid in the middle, and then maybe we do a different tower, so we'll do another tower here so this Tower and that tower and Now I'll just see how I can fit those two guys together. So there you can see they fit really nicely together And if I had another tower here that would just go straight in here so like these these things that are jutting out here They fit into these these gaps here. I don't they just keep on going right? Let's just put one in okay So there so that goes in here [alright], and there we've got another one and that goes right there Alright, and you can kind of just continue like this and fill all of the space and obviously nobody knows about these things (Now they do!) Now they do, so we've got like a main miracle right and we've got all these side miracles And they're all based on really really you know nice Little facts about the Cube that nobody knows about I leave you with another one of those things to chance that does something amazing This is special kind of ball. Okay. It's a special kind of ball when I do this It turns inside out, and I do that it turns inside out again This guy here although. It looks like a ball is actually based on a cube
These are some neat toys, but please understand that this has nothing to do with Banach-Tarski, not even remotely or "sort of".
Where can you buy these cubes that he is using?
Is Banach-Tarski now a verb? :)