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  • I wanted to tell you about the maths of my new tattoo.

  • So I got recently Ton Thio, This is Ah, hyperbolic hell a coid.

  • And so I was going to explain the math of Sheila coins and a little bit of hyperbolic space.

  • So I actually went into the tattoo parlor with the line work and the shading, and I had an artist who was kind enough to actually transfer that onto my body.

  • I wanted to make sure it was mathematically correct.

  • So one of the things about it is that he, like coins, are something that showed up in my PhD an awful lot on I'm an aficionado of hyperbolic space.

  • Anything negatively curved, So I combined them and made a tattoo.

  • This is my first tattoo, but may get more mouth tattoos in the future.

  • So he'll a coins are a type of surface that's called a rolled surface, a type of surface, but is made entirely out of straight lines.

  • So I've got straight lines that are displayed here, and I can change the number of lines on the surface, and I've got other parameters that I will be changing throughout.

  • So one thing you'll see is there's this opacity here.

  • So this is showing the full surface so I can turn.

  • Turn that on so you can see a bit of surface and see how it's being filled up with one's.

  • This here is going to be the simplest rolled surface you can possibly have is the plane.

  • It's just created with a whole bunch of straight lines, so I could just continuously fill it with straight lines.

  • This is the hyperbolic Paraiba loin, also called, Ah, hype our surface.

  • I'm gonna basically sort of twist up the ends of these.

  • And so this is my parameter here I call height.

  • So as I stretch it, I end up with this surface here.

  • This is something you see in architecture a lot.

  • This is in a lot of roofs.

  • Actually, you see a lot of ruled surfaces and architecture because it's very easy to get straight objects on.

  • Then it's just about the way you assemble them that gives you curvature.

  • They're all still straight.

  • Everything you do about them in straight.

  • So that's what I'm going to show you is Ah hyperba Lloyd of revolution.

  • So what I'm gonna do This has four lines.

  • Now I'm gonna continue to add lines in this, makes it a cylinder, and I'm gonna turn on my surface so there's a bit of surface back here.

  • So what I'm gonna do is each of these lines I'm gonna twist it right about here, and that is going Thio.

  • Start moving it around and it will become This is ah hyperba Lloyd of one sheet.

  • So you might recognize this shape as the cooler for nuclear power plant.

  • So that is the economical shape for that.

  • That is this shape here.

  • Turns out that two surfaces I showed you our specials surfaces because they're not ruled just once, but they're doubly rolled, so I can take two sets of straight lines to construct the same surface.

  • It's interesting because they're made with straight lines.

  • It's almost like there's no real curve that you're just creating the illusion of a Yeah, it is a bit like that.

  • But if I were to make them infinitesimally small and start filling them in one by one by one, I would sweep out every single points on the curb on the surface.

  • And that's what makes it a surface, and not just us out of carbs, so these ones are doubly rolled so I can turn on the other ruling.

  • So here I have aside, off of crisscross lines and those will rotate together to form this curvature.

  • Now, to my favorite surface is is the hell accorded the hell?

  • A coId is something that you imagine starting with a plane and then you start filling it in with lines like this.

  • But this one I'm going to instead of like we did with the hype, our surface twisted.

  • This one twists at a constant rate that one twisted, not at a constant rate.

  • And here we're gonna end up with a surface that looks quite a bit like a wind chime.

  • So he liquid is a surface that's constructed of straight lines that rotate about a central line at a constant rate.

  • So let's get to the tattoos.

  • So the tattoo is Ah, hyperbolic hell a coid.

  • And it's done in a particular model called the Planck Array desk model.

  • And the first thing we really need to understand is what do straight lines look like in the space?

  • So I have all of space is in a disk, and this is the boundary is infinity on defy Have a point at the very center.

  • A straight line is a straight line that passes through the center.

  • So this is a straight line.

  • If I had, say, another points over here, a straight line that would pass through that point is going to be something like this.

  • This is an arc of a circle, and it intersects the boundary in right angles.

  • So this is what straight lines are in this particular model.

  • When know what straight lines are?

  • I'm going t o make a hell a coid out of, um so this is what it looks like.

  • This is the line I'm gonna rotate about.

  • And this is the straight line.

  • I'm gonna rotate.

  • So this one, this one is the three D.

  • Once I need tohave.

  • This is the plane.

  • But for me to twist it, I need to have 1/3 dimension to twist into.

  • So now it's the same rules of the lines are defined by the same roses that were using a sphere instead of a circle.

  • Yes, it should be turning on now I've gone lines and they're gonna touch the boundary of a sphere and they have to intercept that in 90 degree angles.

  • I'm gonna start feeling in lines up and down to make my plane.

  • You can imagine this is the whole plane.

  • So what does that plane look?

  • Look, that playing quite flash.

  • I can't tow it is what this plane right now is just a cross section right through the middle of this year.

  • So it's completely flat.

  • Both in our version of space and in hyperbolic space of This is just the half of this fair right here.

  • So now that I've got a plane, that's pretty full, and I'm gonna start twisting around that line.

  • So once it twists quite a lot, you'll start seeing something that looks reminiscent of my tattoo.

  • So once it gets to hear, I think I'm gonna add some more lines to make it look a little fuller.

  • This is the surface that my tattoo is based on.

  • It is called the hyperbolic Kill A coid.

  • So it is a hell.

  • A coid.

  • It is a rolled surface based on twist of a constant rate on this one happens to be in hyperbolic space.

  • And then the other thing about my tattoo a service is straight, but I can actually change what the center line does.

  • So I kept calling it a curve before I started with a straight line, but I don't need to keep it a straight line.

  • I can let it twist around, too.

  • So that and then the plane goes along for the ride on the plane goes along for the ride.

  • Yeah, taking the curve and pulling the North Pole around down towards the equator.

  • Zero is up top and pie over to is at the equator.

  • So that's what I've done here.

  • So my tattoo, what I believe is that pie over to So it's 1/4 of the way around the circle.

  • I want to see your tattoo again.

  • So I'm looking at the South Pole and the North, so the North Pole's not right above the South Pole.

  • It's kind of bent is a Yeah, so I've again pulled this.

  • The center line is one of these straight lines, so it is also a curve that meets the, um, the sphere in red uncles.

  • So what I've done is I've pulled it down so that the north Pole of this and the South Pole of this are, um I guess one is at the South Pole and the others at the equator.

  • Okay, but sort of towards maids.

  • It's not e.

  • I rotated.

  • It s O that you got to see both poles, so he like coins.

  • They're not just a ruled surface there.

  • Also a minimal surface on those air surfaces, like so films on things like that.

  • And that's what I've spent a lot of years of my life studying in particular.

  • There's a serum that says minimal surfaces locally are either bits of hell a coid Zohra pairs of pants in pairs of pants is actually a technical term.

  • And so I've spent a lot of time looking at how to decompose surfaces into hell codes.

  • There's, ah, conference we always go to which is called Bridges.

  • We went there and they dio Mathare so I had set up a gallery and they said, Well, I mean, you can go to ever whatever you want.

  • Just leave your shoulder here and we'll show it with There has to be art pieces.

  • Now a tattoo is getting very interactive with the mathematics, and you might not want to go that far, But if you do want to interact with mathematics.

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I wanted to tell you about the maths of my new tattoo.

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