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.

One out.

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