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• So we're going to talk about space-filling curves.

• So... already that sounds sort of, counter-intuitive, like a curve,

• is like a one, like its a very skinny one-dimensional thing but space is like...

• ...two-dimensional or something so how can you have a curve?

• That fills up everything in space? So...

• We're going to build a curve starting with something very simple and then we're going to sort of change it.

• So this is three sides of square.

• So at the moment this is really not very, space-filling, right?

• Its just three lines, so we need to kind of make it more squiggly and we'll get there.

• So lets draw a grid, on this, so...

• Like, four squares, and so my curve is like going through the centers of those squares.

• So what we're going to do to make this, uh, red curve more squiggly...

• ...is...

• ...divide each of these squares into four.

• So I can chop it up...

• We wanna make a more squiggly version of this but we're going to use this same shape.

• So here's how you do it; you sort of have to rotate it a little bit to make it work so...

• There's the same shape but I've rotated it.

• And then we'll do one like that,

• another one like that,

• and one like this.

• And then, we need to connect them up, so let's,

• connect up like this,

• like this, like this.

• And you can see this is, maybe slightly more space-filling?

• Like, its getting more places than this one was,

• And its sort of made of the same stuff.

• So, well, lets do another one, why not?

• And then in each one of these squares I need to put this thing here, so let me...

• ...make a...

• ...a grid.

• And then I just have to turn things the right way again.

• So I wanna take this thing,

• and rotate it and put it here.

• So its gonna start like that...

• ...and its gonna end like that.

• And in the middle its going to do this...

• And its gonna do this...

• And let's connect them up with red, 'cos...

• OK so we got that shape,

• and then this one over here,

• let's do this one next 'cos its easy 'cos its just, its just the reflection of this one...

• OK so connect them up, boop, boop, boop.

• And then these, I'm gonna go to this now, so this is gonna be like that...

• (whispered) Connect up, connect up, connect up.

• OK, and then again, we, connect up in the same way,

• we do across like that,

• and like that,

• and like that.

• And, we've got something that's...

• more squiggly, more, sort of, space-filling.

• And you can see how you would just, like, continue doing this.

• So, so this is the Hilbert curve.

• So this was invented by, or discovered,

• by David Hilbert in 1891.

• Really these are like steps in the construction of it.

• The Hilbert Curve is when you do this infinitely many times.

• (Brady, off camera) You're using a sharpie which has thickness to it,

• Right.

• (Brady) but surely a theoretical line has no thickness...

• Yeah, zero thickness.

• (Brady) ...So if something has zero thickness how can it fill any space?

• Well, right, every step...

• ...the thing that you have has no thickness, its filling nothing,

• but somehow when you go infinitely far...

• ...well actually what happens is that...

• ...in a, rigorous way which, I'm not going to get into,

• but, actually what happens when you go infinitely far,

• this, curve,

• hits every point of the square.

• So somehow, at the limit, when you get infinitely far,

• boom, you've got everything.

• This construction is not just used for this sort of infinite thing, this is also...

• ...um, you know, a, uh, way to, sort of pack,

• sort of two-dimensional data, in a one-dimensional order, right?

• So, s'pose you have, a bunch of two-dimensional data,

• that you need to store for some reason,

• and you need to store it in some linear order,

• but you also want this bit of data,

• and this bit of data,

• they're quite close to each other in two dimensions and you would like them to be close to each other in the ordering.

• And the Hilbert Curve kind of gives you a way of,

• uh, often two points that are close together in two dimensions are also close together in the linear order.

• Right, whereas if you did, like,

• you know, just sort of, if you just ordered it by rows,

• then that's sort of, less likely to be true.

• Anyway, so its important in, um, computer science...

• I think there's even some theories that the way DNA is packed up when its wrapped up in a chromosome is sort of like this.

• But anyway, so what I was interested in was,

• this, sequence of curves, like the sequence of polygons, how you get there.

• And so you can think of this, like as an animation,

• going from one to the next, to the next, and so on through time,

• or, you could think of this, as,

• space rather than time, so, well...

• So there we go, this is a, 3D-printed sculpture that shows this construction.

• At the very top,

• you've got the original curve,

• and then...

• Well so you can see there's like a band here, that's just a sort of, intermediate band but then the next band down,

• So two down,

• is the next step of the sequence.

• It's probably a little hard to see, but

• it starts here, and then it goes across, and then up,

• and then back, and then it goes up...

• It does that curve that we had before.

• And then, it just keeps going down,

• so another two steps, and another two steps.

• So there's the bottom, that's, that's quite far along,

• in the, let's see how many, so the first step, second step, third step, this is the fifth step,

• of that sequence.

• And...

• I mean,

• in principle I could just keep going, right?

• But, it gets very thin, and difficult to print.

• So this is one of many space-filling curves, or fractal curves, so...

• Uh, so there's one called the, the Dragon Curve.

• (Brady) Oh I like that one.

• Yeah, oh right you did one on this.

• (Inset video) Lots of them, way more than you can fold a bit of paper,

• you get this, rather stunning, pattern...

• (Inset video) And it keeps going, but in the middle it gets very intricate.

• So how does the Dragon Curve one? So here's the first one you just have a straight line...

• ...and then, the next level down, there's,

• just a, like a, right angle corner.

• And then the next one down, this, this one, every band is another step of the, the Dragon Curve sequence.

• And then at the bottom you've got your, Dragon Curve.

• So this is like...

• One, two, three, four, five, six, seven, eight...

• Nine? I think this is nine iterations of the Dragon Curve.

• But right, when you ex-, sort of extrude it through space, you get this, this, cool, uh, this cool shape.

• Somebody, somebody once described this to me as a sky-scraper going for a walk.

• It's sort of a nice image.

• Recognize this shape? That would be the, uh, Sierpinski Triangle,

• except there's a curve called the Sierpinski Arrowhead Curve.

• Let's turn it around...

• So it starts with, again this is smoothed off a little bit,

• but it starts with three sides of a hexagon...

• ...and then...

• ... well it gets more squiggly, um, but yeah you end up with...

• ... the, uh, something that looks like the Sierpinksi Triangle.

• I wanted to show you the Clone Troopers.

• Uh, [laughs], so,

• Right, so, you make all of this,

• like this is just math, right, I mean this is just a simple sequence.

• But then, once you've made it, and you see the faces...

• The eyes and the noses, and the, and the, mouth here of...

• ...um, at least, I don't know, may-maybe you've blacked all...

• ...blanked your mind of all references of the prequels, but,

• Clone Troopers is what this said to me.

• So in the other room there's one based on something called the Terdragon Curve.

• It's like the Dragon Curve but there's a three in it,

• and there's a slightly different process that does it,

• and yeah, this is a, really big one, that we...

• ...dyed, in a very large, trash can.

• So, final things. This is sort of a three-dimensional version,

• of, the two-dimensional Hilbert Curves.

• So there's sort of, three-dimensional versions of a lot of these...

• This is the third step, right, so rather than,

• showing the whole sequence, sort of as an extruded animation,

• this is just the third step, of that sequence.

• And you get this very squiggly curve that is filling three-dimensional space.

• Or a three-dimensional cube.