MV What do you want us to do?
GA can you help me to visualise and to draw in 4D?
GA Yeah.
MV There is something about drawing in 4-D, it’s got some strange appeal.
GA Well I think it’s appeal is that it’s another space to think, visualise and to draw.
MV Some people would say that if it doesn’t exist, they don’t want to know about it.
GA Yeah but I’m interested.
MV So how can we start?
GA could you start by explaining the concept of four dimensions?
MV Yeah well I think that it wouldn’t be that we start with that but actually with 1-d, so for example this thing here this is a 1D square.(drawing) Does that make sense?
GA Yeah it makes sense, (drawing)
yeah that’s 2D.
MV And this is a 2D square.
GA And this is a 3D square.
MV But one way to think about the 2D square is that there is this bit of the surface but the outer shell is made of taking a 1D picture which is, let’s imagine (you know this thing is happening on the plane right?)
GA Yeah that’s a plane…
MV And that’s happening on a line, on an infinite line and these have an infinite plane but let’s consider an infinite line and now just lay down on it the sort of skin of this square. You see the skin of this square is, if the square is 2D its skin is a 1D object which is made out of these four…I suppose they’re cut over here and you can cut it open over there and okay lying down there’s four, one, two, three, four segments, perhaps joined together. And then I can imagine taking this thing and then sort of folding it back into that space it has one dimension more until I close it down and then lock this bit to that bit and then I would have formed the square.
GA So is 4D where the skin is made of 3D?.
MV too quick! We have to really practise it very slowly.
GA Okay yeah. But the idea is that the skin is made of like the way that 2D object the skin is 1D, in 4D the skin is 3D.
MV Yes. So this is a cube in 3D, this is a 3D cube and note that already this picture here is, so to speak, a 2D production because I drew this.
GA Yeah.
MV And it too has a skin and if I open it out it looks a bit like a cross because now it has six faces. And so what I want you to imagine is between these, you see here is a vertex, a corner and then there are these three squares coming into the corner.
GA But that can be drawn as a different type of arrangement, there’s more than one way to draw that.
MV Sure.
GA And this is all isometric is it?
MV Well not really there is essentially only this particular way to do it.
GA Yeah but it could be reflective?
MV Yeah it could be upside down and inside out and yes that’s true.
MV So one important thing to consider this then is what happens at a corner. At a corner I have these three squares hanging, yeah?
GA Is that the vertex?
MV The vertex, call it the vertex yeah?
GA Mm.
MV And there is this gap here yeah? And so imagine this skin to be laid out on the plane and when I see this gap then I somehow can close this gap provided that I then fold this picture along here and here in a dimension one up.
GA Okay.
MV So that then in the dimension one up then I will be able to close that gap somehow.
GA So when you’re saying a dimension one up you mean as in 3D you could just cut it out and you could just do that.
MV I could take this 2D picture, yeah I can cut it out there and then using an extra dimension I could then fold all…you see imagine if you’re going to live in a 2D world you wouldn’t be able to do anything because this is a completely rigid situation but if I’m allowed to go one dimension up then I would be able to fold along these things and then close that gap. And so another aspect that’s interesting about this is how do I picture things happening at a corner? You know if I’m sitting in a corner and I look out from that corner I can picture the corner as a sort of…you see I have this, so you see if I look out from here in the direction of that other corner then this whole face I only see a 1D thing and that’s the same over here, that’s the same over here.
GA On any flat surface?
MV So by sitting at this corner I see a little triangle here and rather…yeah so this is a cube right? So it’s 3D and at the corner I see that the three squares that encroach on that corner somehow the way those three square encroach on that corner is dictated by this triangle because there are three of them and they form it.
GA But would you not also see the bit before the edge because you’re not in the centre of the flat?
MV Well the edge would appear to me just like a point if I look out. If the eye is placed exactly in that…
GA You mean if you’re there you can see that the two wide corners of the triangle are emerging from where you are and then closing again so it’s not a triangle? It’s more like it’s a shape like that unless you’re in the middle and then you’ll see a triangle but if you can see the point before the widest points horizontally then it’s not a triangle.
MV Um I’m not exactly sure how I would see it it’s possible that I would see it more like a circle maybe.
GA Why?
MV No I think I would see a triangle or something because…so what I’m trying to say is what I see is just rays out of this point, rays, so I just see rays out of this point and rays out of this here and rays out of here. If the eye is placed right there…
GA Yeah but you can tell that there is…
MV That’s exactly there.
GA Mm you can tell that the space is coming back towards you again. You can tell that it’s not just going out like that if you’re there.
MV I’m sure you make a lot of sense but I’m not on the same page as you are.
GA Mm, if you try and talk about hyperbolic space, as in the way an ant sees the world or the donut/taurus or something but in terms of understanding how you perceive a flat surface that is a square from that perspective right? So is that the same as standing on a football pitch that is square?
MV We can try to make model of it somehow. In any case perhaps it’s not so important that we agree to get to…so then if I put my eye there my eye is really there, this surface I just see a line.
GA Okay so I see the difference.
MV And then over here I see another line and down there I see another line. So I see three lines but I guess they would all be coming in but yeah exactly. So somehow it’s not clear when…so I do see three lines.
GA I see what you mean but it’s to do with the fact that I was imagining standing on the edge whereas if it’s just your eye that’s on the edge then that’s okay.
MV Right, right, right.
GA So the eye sees that…
MV But in any case…well that’s what I ideally imagine but whether the eye actually sees it or not perhaps is not terribly important.
GA okay.
MV I can use the picture of this triangle to tell me how these three squares are conflating together at that corner.
GA Yeah.
MV And so that was the 3D cube and so what could possibly be a 4D cube? Well, I try to think of is the analogue of this picture well you see to make these…these are 2D pictures so I should be in 3D to make this picture so I should really build some models in 3D to make the analogue of what I do here but then imagine I build this model in 3D and then they would be three dimensional objects and then I would draw 2D pictures exactly like I draw 2D pictures of the cube. Then they would look more or less like this and one way to do it, which is analogous to this way to picture a cube would be to imagine two cubes, this one is a cube and then there is another one right next to it but somehow displaced a bit and I’m not sure how to…and this is not a 2D optimum way to do this particular drawing and then they would be somehow joined in a certain way like that.
GA And why are they not? Why do they have a gap between them?
MV Well just that these two squares, you see I’m just taking two squares and then I’m talking about joining that.
GA Okay.
MV That’s exactly the kind of thing any way. Take two cubes and I sort of join them. Now from this picture it really is not very easy to see what is going on but here is an easier way to see it because you see…no suppose I start rotating this guy and…
GA So if you think about like that joined to that, joined to the…
MV Right, right, right, that’s it yes. But so now you see what happens if I take this guy and I rotate it to that or somehow I move myself right in front of that phase as though I was looking out inside a square room a little bit like what happens in those long shots in the film barrel end then I would see something like that, the other way.
GA Yeah but you’re inside one 3D segment.
MV So then it’s quite easy to see that then that’s what this picture… I see this picture turning into this picture by somehow moving myself, my point of view a little bit and so if I did the same thing to this guy then this picture would turn somehow to this picture where I see an outer cube with a much smaller inner cube and then they would be doing this kind of thing. And so you see…right?
GA I don’t really understand why it’s smaller?
MV Well in the same way as this one is smaller than the one there.
GA Yeah okay.
MV It’s a feature of the perspective there is a vanishing point that makes it smaller but in reality it’s exactly the same.
GA Okay right so is this a mixture of isometric and perspective?
MV Yes. Right so let’s agree that the cube has six faces right?
GA Yes.
MV And so how do we count them? I count them in this picture, one, two, three, four…
GA And five and they are…
MV And the one on the outside. And so here how many? And the faces are square right?
GA Yeah.
MV But now this guy the faces are cubes and can you tell me how many there are?
GA Right hang on okay so on which again? Sorry just ask me that again?
MV You see this guy has six square faces and then how many cubical faces does the 4D cube have?
GA Um hang on. So are you asking me about this one or this one?
MV I’m asking you about the whole 4D cube.
GA Right so that one has got six, okay so on each face there is a cube okay?
MV Yeah.
GA So how many faces does the…so if there’s six and then there’s a cube on each side which means there are six other sides but minus the one that is joining it, minus the one that is the adjoining side, is that right?
MV You are counting the 3D cubes in there now?
GA Yeah.
MV And you think how many 3D cubes do you see in there?
GA six?
MV Here you saw six 2D cubes, in this picture here you saw six 2D cubes.
GA Yeah.
MV How many 3D cubes did you see here?
GA There would be six.
MV There would be eight.
GA Hang on, hang on.
MV There will be eight.
GA Hang on, hang on, hang on.
MV There are six on the outside of that, the little one inside and the big one on the outside.
GA Hang on.
MV You can count it that way as well.
GA But surely… right if there’s… there’s the cube okay?
MV Yeah right.
GA I’m struggling to sort of…like I can understand just you know fine if you’re inside the box okay?
MV Yes.
GA And then you can see the five sides and then you know you’ve got the one side which is your perspective okay?
MV Right, right.
MV Well I think the most similar picture to that would be this one here because you imagine how this is obtained you see if you like what we’re doing another way to look at it is like this right? So what is this picture? So let’s forget whether that’s outside or cut off or whatever it is then we now just see this and if we were to open it up so to speak, to fold it into 2D then we would see this picture here where we then have these gaps and we’ve closed them all by doing that okay?
GA Yeah.
MV Now if we were to do it over here then we’d be thinking of the little cube inside and on each of its faces we put another cube yeah? And then the cube out there and then one out there and then one out there and then we have all these gaps but then we’re using the fourth dimension to fold this whole picture along all these 2D faces and closing it up to this picture so that this face would be closing with that face and this face would be closing to that face.
GA Okay.
MV And then you’d have this picture here where somehow you have now closed them up.
GA And it still makes a cube?
MV It will make a cube and you see you’ve got one in the middle plus six on the wings, down left, right and centre and then the one on the outside. So there won’t be six that’s when you’ve got eight 3D cubes making this picture.
GA Okay right okay that’s fine. Right I understand that because what I was getting confused about was like sides and numbers of cubes.
MV Right, right, right.
GA So I was thinking that there must be six cubes because it’s got six sides but I wasn’t including.
MV The one in the…the little one.
GA That one and that one.
MV Right, right.
GA Okay so I can understand there are eight cubes, because the question is cubes not sides. So hang on let me just draw that because I want to understand. One behind and one in front.
MV Yes.
GA But you see to me right this sort of makes sense right in terms of like a 4D. So if we were to say…what would make sense to me is if you took 50 3D squares or something and you made them into…or whatever number of squares or cubes, okay whatever number of cubes and then you made them into a big cube that seems it would be sensible that that would be 4D because it’s lots of 3Ds building up but then at the end of the day all you get is the 3D.
MV Well the best way to do that is as a 3D model where you really have to imagine a small…you see what’s the analogue of picturing this room in this way you have to think of a 3D cube and then stacking six 3D cubes on top of that but somehow having to deform them a bit so making them into a…I don’t know how to call them, sort of a truncated square pyramid, so that it kind of fits together without leaving any gaps and then imagine having another cube on the outside. So that would be a 3D drawing, you know, so imagine that you are a four dimensional being of course you would draw on a 3D tablet and then the thing we’d just be looking at is what would be the drawing that a four dimensional being would make of his world in the same way as that is a drawing that we make of our world.
GA So do all of these cubes have spaces between them?
MV No because there is absolutely no gap, when it’s all folded together in the fourth dimension there is no gap but if you lay it out in the third dimension then there would be gaps just like there are gaps here.
GA Okay but why is it in a cube?
MV What do you mean?
GA Well I think that’s what…okay right so hang on okay.
MV So imagine that…
GA No hang on, hang on, hang on, right okay. So you’ve got a line right and you understand the line on the plane, right? You’ve got a plane and you understand the plane through the…
MV Imagine the plane is being made by taking a line of length one and moving it off in a new dimension by one step and then as you do that you imagine that line to be dragging some air so to speak then it would make a square as you move that line off in the second dimension by one step. It’s just like taking something like this…
GA When you say step do you mean like…
MV I move it one step that way, one step that way and then it’s going to draw a square.
GA Okay whenever you say step right what you’re saying is parallel lines.
MV Right.
GA So step doesn’t really explain that. How does…
MV When it is another dimension then that’s going along that way.
GA Yeah so how is that step and that is also a step.
MV Yes so maybe I’m heading off in a perpendicular direction and that line is in a new dimension by a unit. And then look at this and I’m taking this square Gemma.
GA Hang on I’m still getting there. So right a 3D object exists in what space? Right the 1D is in 2D.
MV No, no the 1D doesn’t need to be in 2D it will just be in the sole self contained block.
GA Right so 1D can be in 1D, 2D can be in 2D but it can also be in 3D?
MV But it would be not occupying only a 3D space it would be just occupying a much smaller thing. So you’d better thing of it in it’s own 2D space.
GA Okay so 1D is 1D, 2D is 2D, 3D is 3D and 4D is 4D ((laughs)) but is 4D like…because I think it’s quite useful to think about like looking back from one dimension to the other.
MV Yeah except looking back from five dimensions to four dimensions is not going to help very much because in fact you’re trying to look forward rather than look back. It’s good I think what you’re saying is perfectly good and then once you turn yourself a little bit to think about 4D you can try to see 3D yet.
GA 3D?
MV Yeah in the same way as you see the line in space and so on.
GA Right so at the minute I see 3D in 3D?
MV Yes.
GA Okay so that’s all we see is 3D okay. So 4D I think I sort of understand. Why is there 4D?
MV Well I’m going to prove this to you. How do I prove it to you?
GA Why do you think there’s 4D? What’s the purpose of 4D?
MV It hasn’t got any purpose.
GA But in mathematics it’s important.
MV In mathematics yeah absolutely.
GA Why?
MV Why? Because as mathematicians we are used to the fact that there is no problem in N Dimensions in fact in the end or even any number of dimensions or even in dimensions that’s not bothering the mathematician the slightest bit. I even forget, you know, in the formal training of mathematicians I even forget when it’s been completely taken for granted that it’s okay to do things in N dimensions.
GA Okay. So N means no limitation.
MV Or even infinite dimensions.
GA Okay. Infinite dimensions and how can you comprehend infinite dimensions?
MV You don’t you don’t even try to picture it but…
GA So is that the allowed variable within the equation?
MV I’m not even sure how to…
GA Okay lets get back to the 4D. Right I can do 4D with a cube okay.
MV But you want to do something else other than a cube?
GA Yeah so can I do it with a rhombic dodecahedron?
MV No let me give you some other exercises before you do that rhombic okay?
GA Okay. Right how about a cone?
MV Let me… so here’s a octahedron okay? Do you see it?
GA Yeah okay it’s a triangle that way, a triangle that way.
MV At every corner there are four triangles coming in. That’s one way, that’s one way yeah?
GA Yeah.
MV So that would be the picture of the octahedron and now it’s to this picture of the cube, that picture of the cube.
GA Yeah sure but hang on you just said at every corner…is that a corner or is that the vertex? That’s a corner.
MV Yes vertex is the same as corner sorry.
GA Okay there are four triangles, right one, two, three.
MV And the one you’re looking at right in front of you which is this.
GA Yeah okay. So one, two, three, four.
MV Exactly, exactly. Some games have a die that’s made in the shape of an octahedron, there are some games like that but now let me draw you…so now you know that for the purpose of 4D visualisation this kind of picture is easier to work with because then you see I’m just drawing a cube inside a cube. Remember this one here?
GA Okay.
MV So let me draw you the octahedron in that way. And you know that this is in itself sort of interesting.
GA Are all the triangles equal in the octahedron?
MV Yes.
GA Right so is that the platonic solids, all the aspects are equal?
MV Yes.
GA Right okay.
MV And they are of course not only are all the triangles equal they are all equilateral triangles and so it may not be immediately clear that this is also an octahedron that I’ve drawn.
GA Hang on. Okay hang on. But if that is an octahedron would it also have…hang on because that’s like looking into the octahedron but it also has the…
MV I’m looking in from the thing that is on the outside which is the whole triangle on the outside yeah?
GA Yeah so let’s think about if you’re also thinking about it also has a kind of like above as well.
MV I’m not sure I understand what you mean.
GA So in your drawing okay you’re looking into the octahedron?
MV Yes.
GA Okay so you’re looking into it but you’re looking into it within it so you also have this structure behind you?
MV Ah no, no there isn’t another one like that no the structure behind me is just a triangle okay?
GA Right hang on so you’re basically looking…
MV I completely see what you’re drawing there but the thing behind you is just a triangle.
GA That’s fine you’re not in the middle you’re…
MV I’m not completely inside I’m just on the threshold of one of the outer faces yeah but you know what it is you’re making this discussion with me yeah? This is where a lot of this thing is just talking yourself through some kind of story like that and you split your consciousness into various parts and you ask them questions, what do you see when you’re… what is behind you? What is in front of you?
GA Right so you’re using one triangle as a door into the icosahedron?
MV Yes and you see as you talk yourself through it you are making metaphors, you’re making stories, that is a door, that’s how you begin to…that’s the method really.
GA Right so if this is my door…
MV Exactly, exactly.
GA …then I am seeing…
MV You see a little triangle far away which is this triangle here. Can I help you?
GA Mm.
MV So you think of that as being your door yeah?
GA Yeah.
MV So as you walk towards the thing that’s the first thing you meet. So as you stand on the threshold of your door you just have this huge door here, that’s all you can see. As you get really, really close then this face become enormous right and it occupies almost your entire field of vision and then somehow you’re sitting here and you can put an arm here, an arm there and your feet here and your head up here and there’s just this enormous door and that’s all of your world and then you look in and you look in and then what you see I think is this triangle here you see it far, far out, you see, like that’s the door, that triangle there…
GA Hang on right so if that’s the door then, hang on then that’s…
MV No but I would work inside here because you see that’s good. So you see these are in the door.
GA Yeah okay hang on.
MV And you see that little triangle far, far out.
GA Right hang on okay so right this is what I don’t think I understood do you always like describe the doorway as well as the view?
MV Well you do of course you do so that’s your doorway isn’t it.
GA But the thing is what I was thinking is that actually just like imagining what you see through the door but not actually describing the doorway as well. So okay that’s my doorway and then I see this.
MV Absolutely, absolutely.
GA So this far away.
MV And then what’s in red here, maybe you can do it in red also there and you see which triangle do you see far away?
GA This one?
MV Exactly, exactly. And so now…
GA Hang on, hang on.
MV Right that’s not what you see.
GA No.
MV Yes but where do they end those guys? They emanate from the…yes absolutely, absolutely that’s exactly right. Yes.
GA So why…
MV So I don’t think…yeah so you don’t see this.
GA Okay let me try again. So that’s a doorway then inside…is this the wrong way up?
MV No I don’t think so.
GA But I’m just wondering because I’m not going to get equilateral triangles if they’re..
MV Don’t worry they’re not going to be equilateral because there is the perspective effect remember.
GA But yours are…you see that’s not actually a triangle because that’s like that, you see?
MV But you do see, you do see that.
GA But if I do it this way.
MV But you wouldn’t see it that way.
GA Oh no.
MV Because you see that one is the other way, it’s upside down.
GA Oh hang on, oh no, no, no, okay right I get it now. Okay so this is the doorway. This is the triangle in front of it. Now.
MV Yes.
GA Right hang on, hang on. So then it goes like that and then it goes…yeah okay, yeah, yeah.
MV Now you agree with me?
GA Yeah.
MV You get that.
GA Okay.
MV I do that right?
GA Mm.
MV Right now let me draw a 4D analogue of this.
GA Okay hang on. Uh huh.
MV Now imagine you were a four dimensional being so an angel or something and you’re contemplating this object and then you go to the door and this is what you see.
GA So hang on, hang on what’s the difference between what I’m seeing through this door and what I’m seeing through this one. This is one icosahedron where this is…
MV No this is exactly what you saw. Now I see just a triangle, right?
GA Uh huh.
MV Not a triangle a tetrahedron or if you like one could draw it like this, you know, equally well drawing like that. At this point it’s not so easy to do very nice pictures. You could draw a pyramid like that, one of the Egyptian pyramids and then you see now far away in the inside I’m going to see a little inverted pyramid in the inside, yeah?
GA Yeah.
MV And then well you see I’m not doing a great drawing but that’s my inverted pyramid on the inside and then well perhaps we should have looked at it like this to be a bit more except that this is not brilliant looking from this side, we continue thyis way and then…
GA So are you not working with an icosahedron any more you’re doing a tetrahedron.
MV Yeah tetrahedron.
GA No icosahedrons right okay.
MV And then I see a little triangle here, a little tetrahedron there and then another tetrahedron. And then I’d see another tetrahedron on this side and then I’d see another one over there, out there and so then it becomes really tricky to see what else is happening in here, namely exactly how many tetrahedrons does this picture have.
GA Hang on.
MV And remember that here we decided that you counted eight cubes?
GA Yes so then behind you’ve got another one going there and you’ve got another one.
MV You’re pretty much getting the hang of this.
GA And you’ve got another one going there.
MV Yeah, yeah, yeah ((laughs)) that is the coolest thing. You’ll be better than me in no time.
GA ((laughs))
MV ((laughs)) Yeah how many tetrahedron do you see there?
GA Okay well I’m viewing through a tetrahedron?
MV Yes.
GA Yeah okay so I’ve got one, I’ve got two, I’ve got three, four, five, six, seven, eight, hang on, one, two, three, four, five, six, seven, eight, nine, ten.
MV Yeah. That’s correct.
GA Cool. Good. Yeah hang on.
MV You see you’ve got the hang of it but now I’m going to give you a really…
GA No, no, no don’t let me enjoy this part. Hang on because let me just get it right. So is the doorway…let me go through it, so it’s just like a big…
MV Yeah absolutely.
GA Yeah? Okay cool yeah cool that’s really good yeah. Cool yeah okay.
MV So now you answer this question and see whether you can answer this question okay? Now remember that I told you that if I sit here on a cube and I look out from a corner I see this triangle, now if I sit at a corner at the vertex of a hypercube yeah?
GA Yeah.
MV So this the vertex of a hypercube you see and then I see this coming out, this coming out and then this coming out and there are actually three cubes, sorry is it three? There are four cubes encroaching on this corner and so I’m not sure to properly discuss that but there is one cube in there, there is one cube in here, there is…well you see what I’m trying to say is that these guys form, just as that formed a triangle there is a tetrahedron here and then the cubes are encroaching into the inside of that tetrahedron, so the tetrahedron has four faces and there are four cubes coming into a corner, something that we could have seen also from this picture I suppose.
GA Okay.
MV As there being one, two…four. So what I’m saying is now that this tetrahedron now is telling me how the cubes fit into that picture because you see the cubes coming in that particular way…