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Well, thank you for the wonderful introduction, and it's great to be back in Oxford, so so yes, so I'm a mathematician and also a mathematical artist.
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And maybe those seem like a sort of strange combination of things to be doing.
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And hopefully, I'm going to show you how there are more connexions than you might think.
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For that matter, what is it the mathematicians do anyway? Maybe you know what artists do.
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We're not sitting around doing really big sums all of the time, or at least not all of us.
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So let me sort of start off by saying just a little bit about what my mathematics looks like.
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So I'm an apologist. This is more or less typical page of my research work, and you might notice that there's not many numbers here at all.
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It's mostly about sort of surfaces intersecting in space and some sort of complicated configurations and so on.
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So there's obviously a visual component to what I'm doing here.
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And maybe you'd think that's a connexion between mathematics and the arts, but I think there's a stronger connexion,
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which is that both mathematics and artists are sort of allowed to and have to think about fictional worlds.
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So we set up some well that we're going to explore and we see what the consequences are, and we just had nothing to do with the real world.
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It doesn't matter.
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Unlike, say, a scientist or an engineer, which maybe has to have some passing acquaintance with what's happening outside of in the real world.
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So we're allowed to explore these fictional what if questions and then if we come back with something that sort of is useful for the real world?
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Great. If not, it's not a big deal. So let me start talking about my work in in mathematical visualisation and artwork.
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And so I'm pretty much just going to show you a whole bunch of different things.
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And if something doesn't make sense, don't worry, I'll be moving on to something else soon enough.
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And lots of these things that I'm I'm doing a joint work with other people.
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I won't always point out their names, but they'll always be on the side somewhere.
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So, so let's start with this one. This is. Well, what is this?
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This is a 3D print. It's a 3D printed sculpture of something from my work in topology.
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So what you're supposed to imagine here is that you've got sort of a an apple or a ball of dough or something like this.
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And you have an extremely well-trained worm that you've trained to tunnel through this ball and in this sort of knotted fashion.
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And and so what you have left is something called the complement of the Figure eight knot,
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which is one of these is sort of this key core example in in three dimensional geometry entomology.
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So so that's a sort of transition between mathematics and and and this sort of visualisation.
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Let's go back a little bit back to basics.
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You know, if you just want to start doing sort of three dimensional work, you might start with the platonic solids,
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maybe a little bit more complicated than the platonic solids would be the Ahmadi and solids.
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So the platonic solids right you you have these solids made out of triangles or squares or pentagons,
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and it's all the same thing that you see everywhere. It's all made out of triangles or it's all made out of squares,
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and the committee and sides are a little bit more complicated relative to different kinds of face.
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And so maybe you start thinking about Paul here and you say, OK, this is all very nice.
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I've asked my what if questions, is there something I can bring back to the real world? So here's something you could do with some poly hydra.
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So we were the first to produce mass mass produce one hundred and twenty sided dice.
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And so I've got a lot of stuff here and I'm going to be passing it all around. So I guess we'll start here.
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I think you'll you'll be getting all of my things first. So lucky you. So.
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So this beyond the shape of of the dice here, there's sort of an interesting mathematical story with the numbering.
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So how do you put the numbers on a die with 120 sites?
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Well, obviously opposite numbers have to add up to 121, right?
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That's clear, right? Opposite number.
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Why is it that opposite numbers add up to seven on an ordinary cubicle dies?
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So it's sort of an interesting story. What's the one reason?
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It's not really clear,
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but one one reason you could come up with is that an easy way for a die to be unfair is that if it's sort of squished down just a little bit,
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so it's a little bit more like a coin than a than a die. And if if it is, then it's much more likely to land on the big faces and small faces.
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But if you have the opposite sides adding up to the same number for all pairs of opposite sides, then at least it won't roll high or low.
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So it's a sort of dealing with manufacturing defects to get a fair result.
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And so we wanted to do a similar thing here. Obviously, opposite opposite sides should be adding up to a hundred twenty one.
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But that doesn't really cut down the possibilities. Very much so.
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So what we said is, OK, let's suppose that it was you knew that was more likely to land on one of these 10 faces around this vertex.
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You would still like it not to be rolling high or rolling low.
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And the way to make that happen is that you need that the sum of all of the numbers around
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each vertex are would be 10 has to be the same and it is for this arrangement of numbers.
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And it's also true that the sum of the numbers around each vertex are to be six is the same.
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And it's also true that the sum of the numbers around each vertex degree four is the same.
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So that's a pretty difficult problem to make all of that happen.
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There's things like what is an integer linear programming problem? Bob Bosch is at Oberlin College.
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He's an operations research, and he spent about a month of computer time and was very lucky to find this numbering more dice.
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So here's some more dice. These are strangely shaped dice.
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So your ordinary, six sided die is a cube, and that has, it turns out, an excessive amount of symmetry in order to be fair.
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So the thing that you need in order for a die, to be fair,
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is that all the faces of the same and the symmetries of the die take any face to any other face.
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And so this one has less symmetry, but it's just as fact. It also wins people out because it looks like it's melting or something.
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You can do the same thing with 12 sided dice.
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Somehow, this is less impressive than the six sided dice because people don't really know what their ticket will look like anyway,
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so it's hard to tell if they're not the right shape. OK.
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Anyway, onto onto other 3D printed things. So I was in Australia for three years as a post-doc and as a university.
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I called Swinburne University of Technology and they had a whole bunch of money left over at the end of the financial year,
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and so they decided to blow it all on. Some big 3D printed models.
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These are the kind of sort of classic examples from multivariable calculus.
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And you'll notice next to them these much smaller versions, which are just as good at explaining things but much cheaper.
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So I use these in my classes when I teach calculus. And so.
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So this one over here you write down the equation. Something like z equals x squared minus y squared.
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And if you just see the equation, it's difficult to know what it looks like or you know, anything about it visually.
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And this sort of helps helps my students understand what's going on. Let me move on from one sphere to another sphere.
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This is a self-referential sphere. So maybe you can see I've highlighted in red, there's an SE here, a p h e r e.
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So this is a sphere tiled with the words sphere 20 copies of the sphere.
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There's some interesting symmetry here. Here's a bunny tiled with, I think, seventy two copies of the word bunny.
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Let's see if I can point this out. There's a bee here. You and and why.
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So this is. Well, it's this more of a sort of computer science problem than a mathematics problem.
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But anyway, there we go. So I said, I'm with apologist.
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So there's a standard joke about its apologists that they can't tell a coffee
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mug from a doughnut because you can continuously deform one into the other.
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So the real sort of mathematical difficulty here is, OK, I know maybe it's easy enough to decide what shape a doughnut is, right?
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You can write down some sort of equations to describe this shape, and maybe you can use maybe a computer modelling programme to make a coffee mug.
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But how do you deform continuously in some nice way from one to the other?
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So this is joint work with Keenan Crane, who really did most of the work here.
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The words conformal Willmore Flow may make sense to some people in the audience.
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This is what it's about. So moving on to other topological things, so not very common things that technologists think about.
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So I've got this 3D printed, not here.
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This is the trefoil. Not just like the knot you'd make in your shoelaces, except that there's no ends.
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You sort of join it together.
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This note has a very interesting property, which is that it rolls and you have all of these wonderful desks in front of you,
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so you'll be able to try this out for yourself. Why does it roll? What's the sort of mathematical thing going on here?
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So this shape of knot has no try tangent.
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This is the way to say it. So what's the try tangent plane? It's a plane which is tangent to the knot in three different places.
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If it was, if you could sort of hold a plane up to the knot and it was tangent in three places,
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then it would make a little triangle and it would be stable on that triangle. And this one only has it most too.
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So, yes, it knows to, which means that it's always it's always got somewhere that it can.
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It can roll. And as far as I know this, the truffle is the only knot which has this property.
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If anybody happens to know of another knot which has this property, let me know so I can make a rolling version of it.
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Here's another rolling thing at slightly larger scale.
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So, so Li Branswell is in the theatre department at my university, and he's also interested in circus apparatus.
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So there's this class of circus performance where you get inside of a big metal thing and you roll around on the stage,
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and I was asked a question by a performer friend of mine.
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So there's these various things that you can roll around inside of, and they all have the property that, I mean, you might wiggle back and forth,
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but essentially you go in a straight line and then you hit the edge of the stage and you have to sort of go backwards.
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And the question was, can you come up with some design which lets you keep going forwards but somehow curve around and keep on stage?
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And so I'm not a very good acrobat. This guy is, so I'm going to show you what this design actually does.
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So this points in the motion. This is the view from the inside. Maybe you can see on here there's a GoPro attached to it.
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And this is, of course, the view from the outside.
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So the and points in the motion where he can lean one way and switch from rolling in a straight line to sort of shifting from one line to another.
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And so this was an attempt to try and fix this problem, which went somewhere else an attempt anyway.
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Interesting thing and more cars in space.
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So this is a soap film on a frame.
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So this is the mathematical area here is minimal surfaces, surfaces which sort of are pulled tight as as much as possible.
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One of the nice things about working in 3D printing as opposed to, say, computer graphics,
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is you end up with a real world objects and with a real world objects,
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you can interact with other real world things, for example, soap, soap, film, and so you can make these these nine surfaces.
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The not that I showed you earlier, you can also make a nice, minimal surface on the surface with something called a soft surface.
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So a soft surface is the surfaces boundary is a given not.
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And you get this this beautiful shape here.
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They can get a lot more complicated these type of services, but unfortunately it's difficult to get soap films to do the right thing.
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But instead you can 3D print the right thing so.
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So these are sort of various examples of complicated services that would be too complicated to to make the the the the soap film do the right thing.
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But we can calculate with various scary terms like fractional automotive.
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It forms what the shape should be and then we can generate these these things.
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So onto another collaboration, this was a collaboration with Mark O'Mara is a he's a an artist who makes mobile
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so big sort of things that you put up in a hotel lobby usually made out of metal.
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And we looked into making 3D printed versions of these things, and it's sort of nice to be able to.
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You can calculate. OK, so so what does Marco do when he's building this kind of mobile out of metal?
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He starts from the bottom. So you've you've built up this file here, and then you need to decide where does the everything down here?
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Where does it balance? Because that's where you want to put the hook for the next level up.
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And so, so we did something similar, except you can calculate with various things where to put the hook and then you print it out and it works,
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and he doesn't have to spend ages sort of balancing things. Here are some other examples from that same collaboration.
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This is a binary tree. I guess this is more computer science.
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This is a ternary tree when it's split into three at each time, each level as it goes down to Quaternary Tree.
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If you look at the bottom of the ternary tree, you see this happens to you, triangle.
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So to move into fractals, I want to show you some more fractals to explain what the fractal is before I get there.
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So we're going to start with a straight line and then we're going to replace the straight line with a zigzag.
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And then each straight line here, we're going to replace with the same zigzag, right?
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So straight line goes to zigzag. And this one becomes a zigzag.
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I smooth it off a little bit. But it's pretty much going on like this.
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And so usually you might see this. This construction is sort of an animation like I've shown you here.
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And the idea here was to instead of doing an animation through time, you can do an animation through space, which looks something like this.
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You get this beautiful, crinkly surface. Here's a big 3D printed version of this.
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This was at an art show at the Simon Centre for Geometry and Physics at Stony Brook in New York, and we had this set up with this mirror bellows.
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You could see the squiggly thing on the underneath, as well as as seen as sort of the curtain shape.
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And here are some other designs the same sort of idea of it based on different space filling curves.
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These these these constructions are. This is maybe the most familiar or the most famous one is the Hilbert curve turns into this kind of square.
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This one is the dragging curve. Pinsky, our head curve.
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One of the interesting things about working in this sort of field is that, you know, why am I doing this?
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I'm doing this to say, here's this mathematical thing, this abstract thing.
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Let me try and sort of drag it out of the abstract world and make it sort of physical.
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And, you know, and as accurate as a way as possible, realise it.
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But whenever you put something out into the world, people can interpret it however they want and bring some other interpretation.
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So somebody somebody described this one to me as as a skyscraper going for a walk.
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It's sort of here's another one that people sometimes have strong reactions to this sort of ask.
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Monkeys with these weird, distorted limbs.
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But really, this is just a sculpture which is showing a particular kind of symmetry that comes in four dimensional space.
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So there is these sort of there's a ring of four monkeys going around here and there's another ring of four monkeys going around here,
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and this is symmetric if you're able to sort of see what's going on in four dimensions.
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This is based on the geometry of the Hypercube.
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Here's a similar sort of structure built on top of a different four dimensional poly headroom, which is built on the twenty fourth cell.
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Oh yeah, this one's called more fun, more fun than a hypercube of monkeys.
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This one's called more fun than a 24 set of monkeys, and this one's called more fun than a hundred and twenty twelve monkeys.
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There's a series, and there's also an interactive virtual reality version of this.
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This is by heart who you may recognise from her work on YouTube.
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If you go to monkeys, don't happen on the come on your laptops afterwards, not during the lecture.
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Then you will see this, which shows the the symmetry.
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So every few seconds this monkey here is becoming this monkey is becoming this monkey that's doing the symmetry sort of twisting screw motion,
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which is the symmetry that doesn't exist in three dimensional space but does exist in four dimensions.
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If I press various keys and I had any internet, then I would see other things.
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Oh, there it is. Yes, so there's the there's the 24 cell version.
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It turns out with 24 cell is self dual, which means that you can fit another twenty four monkeys in the gaps left between the first set of 24 monkeys.
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And then this is the one hundred and twenty cell version with lots and lots of colourful monkeys.
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And again, it's a different cemetery group in four dimensions, but you can see monkeys become moving to be over the monkey.
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And so on and so forth. OK, so what next?
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I guess what makes a good visualisation, so I'm going to turn down the lights for a second.
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And let's see, does this do it?
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Here we go. No, that doesn't do it at all. How about like that?
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There we go. So. So I'm going to show you one one of, I think, maybe my most successful examples of visualisation here.
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So I've got this 3D printed sphere with a strange pattern on it.
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And the trick here is that if I put the light in exactly the right place.
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Then the curvy grid becomes this beautiful regular, if I could get it in the right place grid.
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Now let me turn the lights back on again and show you a much better photograph of the same thing.
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So, so, so so again, this is using this, this idea, you know, it's a 3D print.
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You can combine it with other real world things and get interesting effects. So.
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So first of all, this sort of a wow effect, right? There was a surprise. How did this curvy thing become this straight grid?
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And then maybe that draws the viewer in, and maybe you get to teach them some mathematics, which is the real point.
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What is the mathematics, by the way? I have to tell you that. So so this is showing something called stereo graphic projection.
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Stereo graphic projection is a map from the sphere to the plane.
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And the way that it works is you trace the light rail from the north pole of the sphere it hits.
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It goes inside the sphere, it hits the sphere somewhere and it continues on down to the plane.
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And the map is just where does it hit on the sphere to where the where does it hit on the plane?
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So I could write down a formula for this, and the formula is not even that complicated,
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but you don't need to know the formula to understand what this map is.
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You can just sort of understand it visually so that the mathematics here is is correct.
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Of course, I mean, there's no point in doing a visualisation if the mathematics isn't correct.
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And there's something else that I really like about this, which is to do with the medium here.
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So, so again, it's a it's a it's a 3D prints and a torch.
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And you know, you could make the same sort of image using computer graphics.
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But if you make it with computer graphics, then people are sort of rightly suspicious, right?
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You can make anything look like anything with with enough computer graphics.
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But this is a piece of plastic in a torch like this is really happening with something actually going on here.
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Last thing to say about this. This was not an easy photograph to take. All right.
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So you might have seen as I was sort of trying to lighten things up. A tiny movement in the in the flashlight produces a big change in the shadow.
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So how did I make this photograph right? I always had my hand here very carefully moving that, and I'm also taking the picture with my other hand.
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What's actually going on here is that the the torch was taped onto a rod and there's a cross beam
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going across the top and there's a clamps down either side and the hand is purely decorative.
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It's just sitting there trying to not move the flashlight and trying to convince you that it isn't a computer render right,
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that this is real because this is a problem that I run into.
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The 3-D printing is so accurate. And of course, the light rays travel in pretty much straight line, so it just looks like a computer.
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So you can do lots of different designs, so all kinds of different things that you can project.
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These were a fun project to illustrate different relationships between the different models of the hyperbolic plane.
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So the 3D print here is this hemisphere.
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And if you sort of do the stereo graphic projection down here, you get the PoincarĂ© disc model, which is maybe the most familiar of these.
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You've seen it in actual prints and so on.
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If you raise the light up very high, so the light rays coming down parallel, you get the Klein model of the hyperbolic plane.
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And if you put the light on the equator of this hemisphere, then you get the upper half plane model of this.
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So we took some of these ideas and ran an exhibition in Edinburgh in 2017, I think it was.
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So we had a globe. It was all very interactive. You could rotate the the globe around to put your favourite country in the middle of the projection.
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This is sort of a full media version of the grid. We had a room full of various timings and this was the sort of finale.
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This is a zoetrope. How many people know what a zoetrope is? Some people zoetrope is sort of the first version of of movies.
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So what's going on here is that there's a disk that's spinning around very quickly and there's a strobe light going on,
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going off every time one of these 30 models I hear and it's spinning once a second.
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So every time one of the models comes to the right place, it flashes the stripes. You always see the right, the right frame.
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And this is some rotating hypercube, very similar to the monkey sculpture that we saw before.
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So. So let's go from right. This is not, you know, the 3D print doesn't actually do this.
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It's an illusion. This is something which looks like it's moving, but it's actually not.
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Let's go to something that should move, but actually doesn't. So maybe you've seen this kind of logo or graphical motif.
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I grew up in Manchester, and I think this this bus shelter ad was about, well, making three different transportation systems work well together.
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But do you think about this for a second if this one's going this way?
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I mean, this was going this way and this one can't move at all.
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And over here, the teachers, the students, the parents come together and nothing is achieved.
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No movement is possible. So here's the challenge. How can you make three gears which are meshing with each other, right?
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They have to. Each power of the gears has to be interacting with each other, and yet they can all move.
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And so here is one of our solutions. There are these three sort of interlinked rings.
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Here's an animation, and I'll pass this one around the the stick through the middle with sort of an afterthought.
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It just makes it easier to to work. But the real thing is the three years he has oh, there's lots of these.
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OK, so he has a sort of linear version of the same idea or three racks rather than gears that that move, you're still working on that.
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So you missed that one story. Here's a version with four.
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This one is very satisfying.
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You have this sort of tetrahedral symmetry, which goes in and out in the current record is five, and the geometries have to keep changing.
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So the question of how to do this and also how to do six yet have any ideas?
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Let me know. This is another example of the medium being important, right, with a 3D printed mechanism.
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You can make it work and you can see that it actually works. If it was a computer animation, you'd be like, Yeah, sure, whatever.
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Pass things around, here's a more mathematical example.
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So if you talk to any geometry or topology, they will tell you that the correct geometry for a Taurus is Euclidean geometry.
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It's flat, and this Taurus really is flat because you can open it out flat on the plane.
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And there it is other sorts of hinged geometry.
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This is a very fun one. This is a sort of hyperbolic doily.
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So this is a model of the hyperbolic plane made out of little hinged triangles.
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I did not prints out three tiny little triangles, then piece them together by hand.
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It had to go in the printer with all of the hinges already in place, and it just comes out of the printer and you shake out.
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The dust and dirt is sort of amazing technology. So there's an interesting mathematical question how do you do that?
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How do you decide where to put the triangles to put it in the printer?
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So this is a sort of simple simulation that shows what actually happened to make the shape so.
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So what we do is, OK, so we need all of these triangles to be, you know, linked to each other and they're not allowed to collide with each other.
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And we start out with them drawing on the PoincarĂ© desk model where we know we can, we can sort of set up where to put them and connect them together.
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So the combinatorics is correct was not corrected at the start is the length.
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As you go further on in the park, create this model. Lengths get shorter.
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And so what we do is we put some springs on each of the edges and shake computationally shake,
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and it eventually converges to some final position where all of the lengths are correct and you
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get this thing here and then another little bit of programming to put sort of hand designs,
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pieces with hinges into the right place. And then this is what we send off to the printer. More sort of moving things,
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this is a puzzle based on a combinatorial version of the vibration 120 cell are getting on to mechanisms that are all exotic, that expand so.
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So this is an interest of mine. A little bit like people know the Hoberman sphere,
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the sphere that sort of you got this little ball and you open it out and you get this big, big ball.
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And this is a similar sort of idea, except that we really wanted to do things in three dimensions.
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So actually, this one I have here alive.
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So rather than right, so the Hoberman sphere, if you think about it, the mechanism is sort of spread out over the two dimensional surface of a sphere.
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And what we wanted to do was make a truly three dimensional version of an expanding mechanism.
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And so there you go. This is based on the crystal structure of diamonds.
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So the way that the carbon atoms are arranged is these sort of, uh,
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I won't hand this around is kind of big, but you can come up afterwards and play with it.
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But the the the way that the the carbon atoms are arranged with each other is sort of beautiful crystal structure,
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and we can exploit that to make some complicated shape like this.
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We can also do not this diamond geometry.
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We can also do a different geometry. So this is cubicle geometry here, I've got a small one here, which I can hand around.
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And so, or again, you go out and explore and maybe you come back with something useful,
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so potentially this has applications in, for example, space architecture.
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You want to some sort of space station or or whatever it is, you want it to be small in the rocket when it goes up.
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But then once it's in space, you want it to expand to a much larger objects.
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So a few other kind of exotic things.
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This is Buckminster Fuller jitterbug.
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So so what's going on here is.
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Well, so when it's small, it's an octahedron. And the four the eight triangles of the octahedron were represented by these sort of y shapes.
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And they're linked together at corners so that they can sort of hinge outwards to a larger figure, which is called a cube octahedron.
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And then it sort of goes back down again. So.
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So Buckminster Fuller was maybe most famous for geodesic domes and things like that, but he did a lot of interesting things.
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Our continued contribution here with adding these gears.
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So it turns out that this structure, mathematically at least, does this kind of motion without the gears.
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But it turns out that the real world sort of doesn't quite behave, and it's sort of jammed sometimes,
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and the gears kind of help it to keep aligned and in the right place.
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And just for fun, here's a larger version, so rather than the previous one one from an octahedron to a Cuban octahedron,
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this goes from a Cuban to eat into a rumba, Cuba to Iran, and just to give a sense of the horrors that await.
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When you try and do this kind of thing here, you'll notice the gearing here is between a square piece and a triangular piece.
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And it turns out that the amount that those pieces rotate is not the same, right?
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In the octahedron, everything was triangle a triangle and everything was rotating the same amount here.
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The the square rotates 90 degrees, whereas the triangle rotates, I think, 120, and they don't rotate a linear rates relative to each other.
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So you have some horrible relationship between the two. Why do I care about all this?
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Well, I need to make gears that rotate at different rates, depending on where they are in their motion.
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And so this is sort of a tricky problem. And here are some more recent work this is.
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So these are two ideas, which in some sense not really connected to each other.
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I mean, they are connected to each other, then fall apart. But there's no axle that goes through these and there's no frame holding it together.
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He has a very similar one, which is showing you that they're really not actually connected together.
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Uh oh. So if you happen to have a screwdriver on you, then you can.
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It's unfortunately just a little bit too too small to fit a pen.
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And this is this is very recent work using similar sorts of ideas.
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So, so there were three guys that are linked together with the red one in the middle.
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I guess you can look. OK. Well, there it keeps changing. Which ones in the middle groups?
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We just play that again. But then you can switch which one is in the middle still holds together,
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but somehow the one that's in the middle, it's actually holding things together is changing.
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So this is sort of a bit like the bombing ring somehow.
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So any teacher will tell you that it's all very well to see somebody do something, but you're not really learning until you do it yourself.
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So I certainly lots of things. What about doing this yourself? Well, maybe not you yet, but my students.
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So I have a 3-D printing lab that's in the mathematics permanent Oklahoma State
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University and I run a class where students learn how to 3D print things.
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So this is the homework for me. I think the fourth week they learn how to use the software and use the 3-D printers.
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And then the homework is, you know, go off and find some cool parametric cabin space and then print one.
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And it's a project based class. So they end up producing beautiful things at the for the midterms and their final fractals and poly hydra.
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And this is to do with sort of pine cone geometry, more things along these lines.
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This has to do with Kepler's laws of planetary motion that these little sections here are filled with water.
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And you can sort of use this to show that they all have the same area, which is Kepler's.
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Third, I think and this is a beautiful example from the last time I saw this.
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So this is showing various areas.
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So this is the different levels of this is showing different times of the Forest Series for the sawtooth wave and the square way.
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And this is just a great, great quote from from George Horn. The more maths you know, the more stuff you can make, which is absolutely true.
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Let me say a little bit about my work in virtual reality. So this is a recent example of non-nuclear and virtual reality.
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This is a three dimensional, hyperbolic virtual reality.
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So I started over here and I walked around the sides of a square in the real world.
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But it turns out that after going round back to the same place in the real world and not back in the same place in the hyperbolic world,
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I have to in fact turn an extra corner to make a right angle Pentagon in hyperbolic space, because that's how hyperbolic space works.
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And this is sort of a good sort of visceral way of of seeing the difference between these things.
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And then I'm going to talk about some other things, but the sounds of so ignore them. So, so taking these sorts of ideas.
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This is using a graphical technique called re-watching.
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Last semester, I was one of the organisers for a programme at a mathematical institute at Brown University in the States.
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And the programme was about illustrating mathematics. So all these sorts of things.
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And with a team of people, we were trying to extend this from three dimensional, hyperbolic geometry to other strange, three dimensional geometries.
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We were particularly going for the eight first in geometry,
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so you may have heard of Bill First and Field's medallist and his famous geometric zation now theorem,
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which says that more or less three dimensional spaces can be classified into into three, sorry, eight different geometries.
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And this is one of them, and it looks very strange. So what is this space?
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Here is the sort of x y plane is a sphere.
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And then the other direction is just a sort of Euclidean translation.
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You get these very weird graphical effects and things get even weirder.
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This is something called nail geometry, which is one of the sort of and the sort of last three really strange geometries that there are.
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And we're still working on this and trying to bring these into again.
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So to make it more accessible and available,
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this you probably won't want to do in virtual reality because the stereo vision just isn't going to work at all.
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So in hyperbolic geometry and three dimensional,
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hyperbolic and three dimensional spherical and a few of the others, stereo vision does work more or less, OK?
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And here it's just completely terrible. It'll just make you sick in no time flat.
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This is very recent work on three dimensional software for investigating three dimensional manifolds called snappy.
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So this has been around. I want to say, for 30 or so years, people researchers have been using the software,
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and now we have the ability to actually visualise what it looks like inside of these spaces that we've been working with.
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This is again, hyperbolic geometry. And this also allows us to does a there's a sense you can deform the space that you're sitting inside of.
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And it gets faintly terrifying when you do.
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So this is again sitting inside of a three dimensional manifolds of the geometry is being squished around in some crazy way,
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which I could get into given another few hours of talking about it.
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But I won't. And one of the fun VR simulation thing, this is another very recent idea.
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Let me quote out of this and actually show you what it is. These things called Camozzi Fractals.
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You can drive through these fractals, which are again based on the hyperbolic geometry of three dimensional manifolds.
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But maybe I won't say much more about what they actually are. You can come and ask me later if you interested.
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And the last topic I want to tell you about is radical video. So, so this is taking video that was taken by a special kind of camera.
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It's a spherical camera. So so here's here's the output from this camera as viewed in the view that comes with the camera.
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So, well, here we are on a tripod, and this camera has lenses on both sides that allow you to see everything around you,
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so it's capturing the entire sphere of data around you. And here we are juggling at Stanford, it turns out.
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So that's what it's supposed to look like when you when you experience it as it's intended,
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but of course, any kind of image or video stored, it's transmitted as a rectangle.
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And so this is what it looks like as it's stored.
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There's sort of the equator around the camera here. This is the South Pole where the tripod is and there's a light up at the North Pole.
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So we're sort of unwrapping it's called the rectangular projection.
377
00:38:44,680 --> 00:38:52,900
So I got interested in these cameras thinking about the transformations of space that you should do so with flat video,
378
00:38:52,900 --> 00:38:58,990
for example, you know how to transform it to zoom into something, right? You just make the picture bigger and then crop.
379
00:38:58,990 --> 00:39:04,630
What are you supposed to do with zooming in spherical video? How do you make it bigger?
380
00:39:04,630 --> 00:39:09,280
You can crop. There is nothing to crop. You see everything. What is Zoom even mean?
381
00:39:09,280 --> 00:39:16,180
And thinking about this led me to think about. Well, it's a sphere of data, and you should be viewing this sphere as the Remon sphere.
382
00:39:16,180 --> 00:39:20,470
And then you should be doing the transformations that are natural to do on the Roman sphere.
383
00:39:20,470 --> 00:39:30,340
Complex numbers. So soon, this video is going to switch to the same thing as before, but modified slightly.
384
00:39:30,340 --> 00:39:38,040
Yeah, yeah. So this video is sort of doubled up.
385
00:39:38,040 --> 00:39:45,360
So what we did here is we took the very briefly we take the image and then we apply the Z squared map z squared.
386
00:39:45,360 --> 00:39:53,610
So I am in the UK and that's it's to to preserves angles, which makes means that people aren't sort of squished in some crazy way.
387
00:39:53,610 --> 00:39:56,280
But it means that there's now two copies of everybody.
388
00:39:56,280 --> 00:40:02,640
So if you show up to joggling and you really want to do a pattern, then of all six people, but only three people are here.
389
00:40:02,640 --> 00:40:13,520
If it has the right symmetry, you can cheat. Here's another version or another VIDEO This is using a particular.
390
00:40:13,520 --> 00:40:17,820
Well, let me just play it and I'll come back in the second version of me.
391
00:40:17,820 --> 00:40:22,800
So I will say my experiment before the start of DR.
392
00:40:22,800 --> 00:40:32,700
So slowly zooming this way along the frame gives you this way, and that is a sort of weird pattern also.
393
00:40:32,700 --> 00:40:39,790
And then what happens? This is the future for the future versions of the year for that and this is the past.
394
00:40:39,790 --> 00:40:47,070
So I want to ask me to explain again what's going on a past version of me.
395
00:40:47,070 --> 00:40:53,340
So I want to say what I can say there.
396
00:40:53,340 --> 00:40:58,740
So, right, so this is a looping video, as you can see.
397
00:40:58,740 --> 00:41:03,840
So this is actually using this kind of zoom that that that is sort of the right thing to do.
398
00:41:03,840 --> 00:41:10,110
Again, it's coming from things called Mobius Transformations and with a bit of,
399
00:41:10,110 --> 00:41:15,960
well, it's not actually green screen, but it's sort of fake green screen. You can get this kind of effect.
400
00:41:15,960 --> 00:41:20,610
It turns out the zoom in in this spherical video feels very much like motion.
401
00:41:20,610 --> 00:41:25,230
It's not quite the same thing, but it feels like like motion. So the camera is not moving at all.
402
00:41:25,230 --> 00:41:31,320
You can see the tripod down here. It's not moving at all. It's just sort of this transformation is giving the impression of movement.
403
00:41:31,320 --> 00:41:40,400
And then I put that the the offset by 30 seconds versions through these these different portals.
404
00:41:40,400 --> 00:41:46,220
OK, so I have one more video to show you.
405
00:41:46,220 --> 00:41:53,320
I'm just going to let this this play, and then maybe I'll come back at the end and say something about this.
406
00:41:53,320 --> 00:42:32,960
This is worth my heart. Oh.
407
00:42:32,960 --> 00:42:43,500
Oh, oh, wow.
408
00:42:43,500 --> 00:43:01,710
And for me.
409
00:43:01,710 --> 00:43:54,320
Oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh oh, oh oh oh.
410
00:43:54,320 --> 00:44:13,560
Ow, ow, ow, ow, ow, ow.
411
00:44:13,560 --> 00:44:22,740
Last thing, let me say a few words about what's going on here. So this is around.
412
00:44:22,740 --> 00:44:32,160
There are three different singers. Well, actually, there's only one singer VI, who is offset in time by 20 seconds from copies of herself.
413
00:44:32,160 --> 00:44:43,260
But we've also offset copies of the room. Well, so we all of that offset the video not by 20 seconds, but by 120 degrees.
414
00:44:43,260 --> 00:44:51,420
The mathematics here in how the transformation of the video happens here, we had tripled space rather than doubled space.
415
00:44:51,420 --> 00:44:54,900
So like the jugglers had a there was a z squared back here.
416
00:44:54,900 --> 00:45:01,560
It's a z cubed. So that sort of explains how we got this image here.
417
00:45:01,560 --> 00:45:08,420
But there's a lot of other strange things going on here. Let's see.
418
00:45:08,420 --> 00:45:14,010
And there are three copies of my website. I've let it go too long.
419
00:45:14,010 --> 00:45:19,260
There were three copies of my. There's only one copy of me. There were moves previously, only two copies of me.
420
00:45:19,260 --> 00:45:26,760
Maybe I'll go back a little bit.
421
00:45:26,760 --> 00:45:49,340
Maybe I'll just let it play out, and then I'll say something you.
422
00:45:49,340 --> 00:45:56,840
OK, so as I was saying, somewhere in the middle. Right.
423
00:45:56,840 --> 00:46:01,040
So where am I? I'm over here and I'm also over here.
424
00:46:01,040 --> 00:46:05,270
Why are there only two of me? First question, second question.
425
00:46:05,270 --> 00:46:11,060
Here's the sheet music. There's only one copy of the sheet music.
426
00:46:11,060 --> 00:46:18,260
So I picked it up. She plays a couple of notes. She passes it to me. I pass it back to her through the piano.
427
00:46:18,260 --> 00:46:25,110
She comes around to put it on the music stand. She plays a couple of notes on the piano.
428
00:46:25,110 --> 00:46:31,200
She picks up a hammer for some reason. Just in time for her to pick up the music again.
429
00:46:31,200 --> 00:46:35,830
So, so there's a lot of sort of interesting choreography going on here. So this video is on YouTube.
430
00:46:35,830 --> 00:46:41,020
Also, there's an explanation video that goes into a lot of detail about what's actually going on.
431
00:46:41,020 --> 00:46:48,690
I encourage you to check that out of your system to see more. I am pretty much done.
432
00:46:48,690 --> 00:46:56,100
I am going to take just a few minutes to say something about my book visualising mathematics with 3D printing.
433
00:46:56,100 --> 00:47:10,110
So the what's the trick with this book is that most of the most of the figures in the book are photographs of 3D printed objects.
434
00:47:10,110 --> 00:47:17,850
And there's a website associated with the book, and you can go, for example, to the chapter on probably Hydra.
435
00:47:17,850 --> 00:47:29,580
And you can click on the page for a figure, and there's a sort of virtual 3D model which you can look around and sort of explore.
436
00:47:29,580 --> 00:47:35,850
And the idea here is there's lots of very interesting, accessible mathematics that people don't really write about.
437
00:47:35,850 --> 00:47:42,270
I think because it's too hard to make the figures that in order to really understand it, you really need three dimensional figures.
438
00:47:42,270 --> 00:47:46,620
And so with this, you can have actual three dimensional figures.
439
00:47:46,620 --> 00:47:53,550
You can buy these things on shape ways. You can download the files for yourself to print on your own 3D printer.
440
00:47:53,550 --> 00:47:56,610
There's also a link to think about if people are familiar with thing of us.
441
00:47:56,610 --> 00:48:04,740
This is a sort of repository for 3D files that if the internet worked for me, I'd be able to go there and show you.
442
00:48:04,740 --> 00:48:11,760
But you can download the files in this sort of community there that's looking at the files and printing them out and saying,
443
00:48:11,760 --> 00:48:18,120
here are the settings to make it work on my on my printer. But that's my talk.
444
00:48:18,120 --> 00:48:22,710
I'm Henry Saige on Twitter and Instagram. And that's my website.
445
00:48:22,710 --> 00:48:43,646
Thanks for your attention.