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.
And maybe those seem like a sort of strange combination of things to be doing.
And hopefully, I'm going to show you how there are more connexions than you might think.
For that matter, what is it the mathematicians do anyway? Maybe you know what artists do.
We're not sitting around doing really big sums all of the time, or at least not all of us.
So let me sort of start off by saying just a little bit about what my mathematics looks like.
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.
It's mostly about sort of surfaces intersecting in space and some sort of complicated configurations and so on.
So there's obviously a visual component to what I'm doing here.
And maybe you'd think that's a connexion between mathematics and the arts, but I think there's a stronger connexion,
which is that both mathematics and artists are sort of allowed to and have to think about fictional worlds.
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.
It doesn't matter.
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.
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?
Great. If not, it's not a big deal. So let me start talking about my work in in mathematical visualisation and artwork.
And so I'm pretty much just going to show you a whole bunch of different things.
And if something doesn't make sense, don't worry, I'll be moving on to something else soon enough.
And lots of these things that I'm I'm doing a joint work with other people.
I won't always point out their names, but they'll always be on the side somewhere.
So, so let's start with this one. This is. Well, what is this?
This is a 3D print. It's a 3D printed sculpture of something from my work in topology.
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.
And you have an extremely well-trained worm that you've trained to tunnel through this ball and in this sort of knotted fashion.
And and so what you have left is something called the complement of the Figure eight knot,
which is one of these is sort of this key core example in in three dimensional geometry entomology.
So so that's a sort of transition between mathematics and and and this sort of visualisation.
Let's go back a little bit back to basics.
You know, if you just want to start doing sort of three dimensional work, you might start with the platonic solids,
maybe a little bit more complicated than the platonic solids would be the Ahmadi and solids.
So the platonic solids right you you have these solids made out of triangles or squares or pentagons,
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,
and the committee and sides are a little bit more complicated relative to different kinds of face.
And so maybe you start thinking about Paul here and you say, OK, this is all very nice.
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.
So we were the first to produce mass mass produce one hundred and twenty sided dice.
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.
I think you'll you'll be getting all of my things first. So lucky you. So.
So this beyond the shape of of the dice here, there's sort of an interesting mathematical story with the numbering.
So how do you put the numbers on a die with 120 sites?
Well, obviously opposite numbers have to add up to 121, right?
That's clear, right? Opposite number.
Why is it that opposite numbers add up to seven on an ordinary cubicle dies?
So it's sort of an interesting story. What's the one reason?
It's not really clear,
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,
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.
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.
So it's a sort of dealing with manufacturing defects to get a fair result.
And so we wanted to do a similar thing here. Obviously, opposite opposite sides should be adding up to a hundred twenty one.
But that doesn't really cut down the possibilities. Very much so.
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.
You would still like it not to be rolling high or rolling low.
And the way to make that happen is that you need that the sum of all of the numbers around
each vertex are would be 10 has to be the same and it is for this arrangement of numbers.
And it's also true that the sum of the numbers around each vertex are to be six is the same.
And it's also true that the sum of the numbers around each vertex degree four is the same.
So that's a pretty difficult problem to make all of that happen.
There's things like what is an integer linear programming problem? Bob Bosch is at Oberlin College.
He's an operations research, and he spent about a month of computer time and was very lucky to find this numbering more dice.
So here's some more dice. These are strangely shaped dice.
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.
So the thing that you need in order for a die, to be fair,
is that all the faces of the same and the symmetries of the die take any face to any other face.
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.
You can do the same thing with 12 sided dice.
Somehow, this is less impressive than the six sided dice because people don't really know what their ticket will look like anyway,
so it's hard to tell if they're not the right shape. OK.
Anyway, onto onto other 3D printed things. So I was in Australia for three years as a post-doc and as a university.
I called Swinburne University of Technology and they had a whole bunch of money left over at the end of the financial year,
and so they decided to blow it all on. Some big 3D printed models.
These are the kind of sort of classic examples from multivariable calculus.
And you'll notice next to them these much smaller versions, which are just as good at explaining things but much cheaper.
So I use these in my classes when I teach calculus. And so.
So this one over here you write down the equation. Something like z equals x squared minus y squared.
And if you just see the equation, it's difficult to know what it looks like or you know, anything about it visually.
And this sort of helps helps my students understand what's going on. Let me move on from one sphere to another sphere.
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.
So this is a sphere tiled with the words sphere 20 copies of the sphere.
There's some interesting symmetry here. Here's a bunny tiled with, I think, seventy two copies of the word bunny.
Let's see if I can point this out. There's a bee here. You and and why.
So this is. Well, it's this more of a sort of computer science problem than a mathematics problem.
But anyway, there we go. So I said, I'm with apologist.
So there's a standard joke about its apologists that they can't tell a coffee
mug from a doughnut because you can continuously deform one into the other.
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?
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.
But how do you deform continuously in some nice way from one to the other?
So this is joint work with Keenan Crane, who really did most of the work here.
The words conformal Willmore Flow may make sense to some people in the audience.
This is what it's about. So moving on to other topological things, so not very common things that technologists think about.
So I've got this 3D printed, not here.
This is the trefoil. Not just like the knot you'd make in your shoelaces, except that there's no ends.
You sort of join it together.
This note has a very interesting property, which is that it rolls and you have all of these wonderful desks in front of you,
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?
So this shape of knot has no try tangent.
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.
If it was, if you could sort of hold a plane up to the knot and it was tangent in three places,
then it would make a little triangle and it would be stable on that triangle. And this one only has it most too.
So, yes, it knows to, which means that it's always it's always got somewhere that it can.
It can roll. And as far as I know this, the truffle is the only knot which has this property.
If anybody happens to know of another knot which has this property, let me know so I can make a rolling version of it.
Here's another rolling thing at slightly larger scale.
So, so Li Branswell is in the theatre department at my university, and he's also interested in circus apparatus.
So there's this class of circus performance where you get inside of a big metal thing and you roll around on the stage,
and I was asked a question by a performer friend of mine.
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,
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.
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?
And so I'm not a very good acrobat. This guy is, so I'm going to show you what this design actually does.
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.
And this is, of course, the view from the outside.
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.
And so this was an attempt to try and fix this problem, which went somewhere else an attempt anyway.
Interesting thing and more cars in space.
So this is a soap film on a frame.
So this is the mathematical area here is minimal surfaces, surfaces which sort of are pulled tight as as much as possible.
One of the nice things about working in 3D printing as opposed to, say, computer graphics,
is you end up with a real world objects and with a real world objects,
you can interact with other real world things, for example, soap, soap, film, and so you can make these these nine surfaces.
The not that I showed you earlier, you can also make a nice, minimal surface on the surface with something called a soft surface.
So a soft surface is the surfaces boundary is a given not.
And you get this this beautiful shape here.
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.
But instead you can 3D print the right thing so.
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.
But we can calculate with various scary terms like fractional automotive.
It forms what the shape should be and then we can generate these these things.
So onto another collaboration, this was a collaboration with Mark O'Mara is a he's a an artist who makes mobile
so big sort of things that you put up in a hotel lobby usually made out of metal.
And we looked into making 3D printed versions of these things, and it's sort of nice to be able to.
You can calculate. OK, so so what does Marco do when he's building this kind of mobile out of metal?
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?
Where does it balance? Because that's where you want to put the hook for the next level up.
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,
and he doesn't have to spend ages sort of balancing things. Here are some other examples from that same collaboration.
This is a binary tree. I guess this is more computer science.
This is a ternary tree when it's split into three at each time, each level as it goes down to Quaternary Tree.
If you look at the bottom of the ternary tree, you see this happens to you, triangle.
So to move into fractals, I want to show you some more fractals to explain what the fractal is before I get there.
So we're going to start with a straight line and then we're going to replace the straight line with a zigzag.
And then each straight line here, we're going to replace with the same zigzag, right?
So straight line goes to zigzag. And this one becomes a zigzag.
I smooth it off a little bit. But it's pretty much going on like this.
And so usually you might see this. This construction is sort of an animation like I've shown you here.
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.
You get this beautiful, crinkly surface. Here's a big 3D printed version of this.
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.
You could see the squiggly thing on the underneath, as well as as seen as sort of the curtain shape.
And here are some other designs the same sort of idea of it based on different space filling curves.
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.
This one is the dragging curve. Pinsky, our head curve.
One of the interesting things about working in this sort of field is that, you know, why am I doing this?
I'm doing this to say, here's this mathematical thing, this abstract thing.
Let me try and sort of drag it out of the abstract world and make it sort of physical.
And, you know, and as accurate as a way as possible, realise it.
But whenever you put something out into the world, people can interpret it however they want and bring some other interpretation.
So somebody somebody described this one to me as as a skyscraper going for a walk.
It's sort of here's another one that people sometimes have strong reactions to this sort of ask.
Monkeys with these weird, distorted limbs.
But really, this is just a sculpture which is showing a particular kind of symmetry that comes in four dimensional space.
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,
and this is symmetric if you're able to sort of see what's going on in four dimensions.
This is based on the geometry of the Hypercube.
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.
Oh yeah, this one's called more fun, more fun than a hypercube of monkeys.
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.
There's a series, and there's also an interactive virtual reality version of this.
This is by heart who you may recognise from her work on YouTube.
If you go to monkeys, don't happen on the come on your laptops afterwards, not during the lecture.
Then you will see this, which shows the the symmetry.
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,
which is the symmetry that doesn't exist in three dimensional space but does exist in four dimensions.
If I press various keys and I had any internet, then I would see other things.
Oh, there it is. Yes, so there's the there's the 24 cell version.
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.
And then this is the one hundred and twenty cell version with lots and lots of colourful monkeys.
And again, it's a different cemetery group in four dimensions, but you can see monkeys become moving to be over the monkey.
And so on and so forth. OK, so what next?
I guess what makes a good visualisation, so I'm going to turn down the lights for a second.
And let's see, does this do it?
Here we go. No, that doesn't do it at all. How about like that?
There we go. So. So I'm going to show you one one of, I think, maybe my most successful examples of visualisation here.
So I've got this 3D printed sphere with a strange pattern on it.
And the trick here is that if I put the light in exactly the right place.
Then the curvy grid becomes this beautiful regular, if I could get it in the right place grid.
Now let me turn the lights back on again and show you a much better photograph of the same thing.
So, so, so so again, this is using this, this idea, you know, it's a 3D print.
You can combine it with other real world things and get interesting effects. So.
So first of all, this sort of a wow effect, right? There was a surprise. How did this curvy thing become this straight grid?
And then maybe that draws the viewer in, and maybe you get to teach them some mathematics, which is the real point.
What is the mathematics, by the way? I have to tell you that. So so this is showing something called stereo graphic projection.
Stereo graphic projection is a map from the sphere to the plane.
And the way that it works is you trace the light rail from the north pole of the sphere it hits.
It goes inside the sphere, it hits the sphere somewhere and it continues on down to the plane.
And the map is just where does it hit on the sphere to where the where does it hit on the plane?
So I could write down a formula for this, and the formula is not even that complicated,
but you don't need to know the formula to understand what this map is.
You can just sort of understand it visually so that the mathematics here is is correct.
Of course, I mean, there's no point in doing a visualisation if the mathematics isn't correct.
And there's something else that I really like about this, which is to do with the medium here.
So, so again, it's a it's a it's a 3D prints and a torch.
And you know, you could make the same sort of image using computer graphics.
But if you make it with computer graphics, then people are sort of rightly suspicious, right?
You can make anything look like anything with with enough computer graphics.
But this is a piece of plastic in a torch like this is really happening with something actually going on here.
Last thing to say about this. This was not an easy photograph to take. All right.
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.
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.
What's actually going on here is that the the torch was taped onto a rod and there's a cross beam
going across the top and there's a clamps down either side and the hand is purely decorative.
It's just sitting there trying to not move the flashlight and trying to convince you that it isn't a computer render right,
that this is real because this is a problem that I run into.
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.
So you can do lots of different designs, so all kinds of different things that you can project.
These were a fun project to illustrate different relationships between the different models of the hyperbolic plane.
So the 3D print here is this hemisphere.
And if you sort of do the stereo graphic projection down here, you get the Poincaré disc model, which is maybe the most familiar of these.
You've seen it in actual prints and so on.
If you raise the light up very high, so the light rays coming down parallel, you get the Klein model of the hyperbolic plane.
And if you put the light on the equator of this hemisphere, then you get the upper half plane model of this.
So we took some of these ideas and ran an exhibition in Edinburgh in 2017, I think it was.
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.
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.
This is a zoetrope. How many people know what a zoetrope is? Some people zoetrope is sort of the first version of of movies.
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,
going off every time one of these 30 models I hear and it's spinning once a second.
So every time one of the models comes to the right place, it flashes the stripes. You always see the right, the right frame.
And this is some rotating hypercube, very similar to the monkey sculpture that we saw before.
So. So let's go from right. This is not, you know, the 3D print doesn't actually do this.
It's an illusion. This is something which looks like it's moving, but it's actually not.
Let's go to something that should move, but actually doesn't. So maybe you've seen this kind of logo or graphical motif.
I grew up in Manchester, and I think this this bus shelter ad was about, well, making three different transportation systems work well together.
But do you think about this for a second if this one's going this way?
I mean, this was going this way and this one can't move at all.
And over here, the teachers, the students, the parents come together and nothing is achieved.
No movement is possible. So here's the challenge. How can you make three gears which are meshing with each other, right?
They have to. Each power of the gears has to be interacting with each other, and yet they can all move.
And so here is one of our solutions. There are these three sort of interlinked rings.
Here's an animation, and I'll pass this one around the the stick through the middle with sort of an afterthought.
It just makes it easier to to work. But the real thing is the three years he has oh, there's lots of these.
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.
So you missed that one story. Here's a version with four.
This one is very satisfying.
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.
So the question of how to do this and also how to do six yet have any ideas?
Let me know. This is another example of the medium being important, right, with a 3D printed mechanism.
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.
Pass things around, here's a more mathematical example.
So if you talk to any geometry or topology, they will tell you that the correct geometry for a Taurus is Euclidean geometry.
It's flat, and this Taurus really is flat because you can open it out flat on the plane.
And there it is other sorts of hinged geometry.
This is a very fun one. This is a sort of hyperbolic doily.
So this is a model of the hyperbolic plane made out of little hinged triangles.
I did not prints out three tiny little triangles, then piece them together by hand.
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.
The dust and dirt is sort of amazing technology. So there's an interesting mathematical question how do you do that?
How do you decide where to put the triangles to put it in the printer?
So this is a sort of simple simulation that shows what actually happened to make the shape so.
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.
And we start out with them drawing on the Poincaré desk model where we know we can, we can sort of set up where to put them and connect them together.
So the combinatorics is correct was not corrected at the start is the length.
As you go further on in the park, create this model. Lengths get shorter.
And so what we do is we put some springs on each of the edges and shake computationally shake,
and it eventually converges to some final position where all of the lengths are correct and you
get this thing here and then another little bit of programming to put sort of hand designs,
pieces with hinges into the right place. And then this is what we send off to the printer. More sort of moving things,
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.
So this is an interest of mine. A little bit like people know the Hoberman sphere,
the sphere that sort of you got this little ball and you open it out and you get this big, big ball.
And this is a similar sort of idea, except that we really wanted to do things in three dimensions.
So actually, this one I have here alive.
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.
And what we wanted to do was make a truly three dimensional version of an expanding mechanism.
And so there you go. This is based on the crystal structure of diamonds.
So the way that the carbon atoms are arranged is these sort of, uh,
I won't hand this around is kind of big, but you can come up afterwards and play with it.
But the the the way that the the carbon atoms are arranged with each other is sort of beautiful crystal structure,
and we can exploit that to make some complicated shape like this.
We can also do not this diamond geometry.
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.
And so, or again, you go out and explore and maybe you come back with something useful,
so potentially this has applications in, for example, space architecture.
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.
But then once it's in space, you want it to expand to a much larger objects.
So a few other kind of exotic things.
This is Buckminster Fuller jitterbug.
So so what's going on here is.
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.
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.
And then it sort of goes back down again. So.
So Buckminster Fuller was maybe most famous for geodesic domes and things like that, but he did a lot of interesting things.
Our continued contribution here with adding these gears.
So it turns out that this structure, mathematically at least, does this kind of motion without the gears.
But it turns out that the real world sort of doesn't quite behave, and it's sort of jammed sometimes,
and the gears kind of help it to keep aligned and in the right place.
And just for fun, here's a larger version, so rather than the previous one one from an octahedron to a Cuban octahedron,
this goes from a Cuban to eat into a rumba, Cuba to Iran, and just to give a sense of the horrors that await.
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.
And it turns out that the amount that those pieces rotate is not the same, right?
In the octahedron, everything was triangle a triangle and everything was rotating the same amount here.
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.
So you have some horrible relationship between the two. Why do I care about all this?
Well, I need to make gears that rotate at different rates, depending on where they are in their motion.
And so this is sort of a tricky problem. And here are some more recent work this is.
So these are two ideas, which in some sense not really connected to each other.
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.
He has a very similar one, which is showing you that they're really not actually connected together.
Uh oh. So if you happen to have a screwdriver on you, then you can.
It's unfortunately just a little bit too too small to fit a pen.
And this is this is very recent work using similar sorts of ideas.
So, so there were three guys that are linked together with the red one in the middle.
I guess you can look. OK. Well, there it keeps changing. Which ones in the middle groups?
We just play that again. But then you can switch which one is in the middle still holds together,
but somehow the one that's in the middle, it's actually holding things together is changing.
So this is sort of a bit like the bombing ring somehow.
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.
So I certainly lots of things. What about doing this yourself? Well, maybe not you yet, but my students.
So I have a 3-D printing lab that's in the mathematics permanent Oklahoma State
University and I run a class where students learn how to 3D print things.
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.
And then the homework is, you know, go off and find some cool parametric cabin space and then print one.
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.
And this is to do with sort of pine cone geometry, more things along these lines.
This has to do with Kepler's laws of planetary motion that these little sections here are filled with water.
And you can sort of use this to show that they all have the same area, which is Kepler's.
Third, I think and this is a beautiful example from the last time I saw this.
So this is showing various areas.
So this is the different levels of this is showing different times of the Forest Series for the sawtooth wave and the square way.
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.
Let me say a little bit about my work in virtual reality. So this is a recent example of non-nuclear and virtual reality.
This is a three dimensional, hyperbolic virtual reality.
So I started over here and I walked around the sides of a square in the real world.
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,
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.
And this is sort of a good sort of visceral way of of seeing the difference between these things.
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.
This is using a graphical technique called re-watching.
Last semester, I was one of the organisers for a programme at a mathematical institute at Brown University in the States.
And the programme was about illustrating mathematics. So all these sorts of things.
And with a team of people, we were trying to extend this from three dimensional, hyperbolic geometry to other strange, three dimensional geometries.
We were particularly going for the eight first in geometry,
so you may have heard of Bill First and Field's medallist and his famous geometric zation now theorem,
which says that more or less three dimensional spaces can be classified into into three, sorry, eight different geometries.
And this is one of them, and it looks very strange. So what is this space?
Here is the sort of x y plane is a sphere.
And then the other direction is just a sort of Euclidean translation.
You get these very weird graphical effects and things get even weirder.
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.
And we're still working on this and trying to bring these into again.
So to make it more accessible and available,
this you probably won't want to do in virtual reality because the stereo vision just isn't going to work at all.
So in hyperbolic geometry and three dimensional,
hyperbolic and three dimensional spherical and a few of the others, stereo vision does work more or less, OK?
And here it's just completely terrible. It'll just make you sick in no time flat.
This is very recent work on three dimensional software for investigating three dimensional manifolds called snappy.
So this has been around. I want to say, for 30 or so years, people researchers have been using the software,
and now we have the ability to actually visualise what it looks like inside of these spaces that we've been working with.
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.
And it gets faintly terrifying when you do.
So this is again sitting inside of a three dimensional manifolds of the geometry is being squished around in some crazy way,
which I could get into given another few hours of talking about it.
But I won't. And one of the fun VR simulation thing, this is another very recent idea.
Let me quote out of this and actually show you what it is. These things called Camozzi Fractals.
You can drive through these fractals, which are again based on the hyperbolic geometry of three dimensional manifolds.
But maybe I won't say much more about what they actually are. You can come and ask me later if you interested.
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.
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.
So, well, here we are on a tripod, and this camera has lenses on both sides that allow you to see everything around you,
so it's capturing the entire sphere of data around you. And here we are juggling at Stanford, it turns out.
So that's what it's supposed to look like when you when you experience it as it's intended,
but of course, any kind of image or video stored, it's transmitted as a rectangle.
And so this is what it looks like as it's stored.
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.
So we're sort of unwrapping it's called the rectangular projection.
So I got interested in these cameras thinking about the transformations of space that you should do so with flat video,
for example, you know how to transform it to zoom into something, right? You just make the picture bigger and then crop.
What are you supposed to do with zooming in spherical video? How do you make it bigger?
You can crop. There is nothing to crop. You see everything. What is Zoom even mean?
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.
And then you should be doing the transformations that are natural to do on the Roman sphere.
Complex numbers. So soon, this video is going to switch to the same thing as before, but modified slightly.
Yeah, yeah. So this video is sort of doubled up.
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.
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.
But it means that there's now two copies of everybody.
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.
If it has the right symmetry, you can cheat. Here's another version or another VIDEO This is using a particular.
Well, let me just play it and I'll come back in the second version of me.
So I will say my experiment before the start of DR.
So slowly zooming this way along the frame gives you this way, and that is a sort of weird pattern also.
And then what happens? This is the future for the future versions of the year for that and this is the past.
So I want to ask me to explain again what's going on a past version of me.
So I want to say what I can say there.
So, right, so this is a looping video, as you can see.
So this is actually using this kind of zoom that that that is sort of the right thing to do.
Again, it's coming from things called Mobius Transformations and with a bit of,
well, it's not actually green screen, but it's sort of fake green screen. You can get this kind of effect.
It turns out the zoom in in this spherical video feels very much like motion.
It's not quite the same thing, but it feels like like motion. So the camera is not moving at all.
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.
And then I put that the the offset by 30 seconds versions through these these different portals.
OK, so I have one more video to show you.
I'm just going to let this this play, and then maybe I'll come back at the end and say something about this.
This is worth my heart. Oh.
Oh, oh, wow.
And for me.
Oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh, oh oh, oh oh oh.
Ow, ow, ow, ow, ow, ow.
Last thing, let me say a few words about what's going on here. So this is around.
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.
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.
The mathematics here in how the transformation of the video happens here, we had tripled space rather than doubled space.
So like the jugglers had a there was a z squared back here.
It's a z cubed. So that sort of explains how we got this image here.
But there's a lot of other strange things going on here. Let's see.
And there are three copies of my website. I've let it go too long.
There were three copies of my. There's only one copy of me. There were moves previously, only two copies of me.
Maybe I'll go back a little bit.
Maybe I'll just let it play out, and then I'll say something you.
OK, so as I was saying, somewhere in the middle. Right.
So where am I? I'm over here and I'm also over here.
Why are there only two of me? First question, second question.
Here's the sheet music. There's only one copy of the sheet music.
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.
She comes around to put it on the music stand. She plays a couple of notes on the piano.
She picks up a hammer for some reason. Just in time for her to pick up the music again.
So, so there's a lot of sort of interesting choreography going on here. So this video is on YouTube.
Also, there's an explanation video that goes into a lot of detail about what's actually going on.
I encourage you to check that out of your system to see more. I am pretty much done.
I am going to take just a few minutes to say something about my book visualising mathematics with 3D printing.
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.
And there's a website associated with the book, and you can go, for example, to the chapter on probably Hydra.
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.
And the idea here is there's lots of very interesting, accessible mathematics that people don't really write about.
I think because it's too hard to make the figures that in order to really understand it, you really need three dimensional figures.
And so with this, you can have actual three dimensional figures.
You can buy these things on shape ways. You can download the files for yourself to print on your own 3D printer.
There's also a link to think about if people are familiar with thing of us.
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.
But you can download the files in this sort of community there that's looking at the files and printing them out and saying,
here are the settings to make it work on my on my printer. But that's my talk.
I'm Henry Saige on Twitter and Instagram. And that's my website.
Thanks for your attention.