I wanted to tell you about the maths of my new tattoo, so I got recently a
tattoo this is a hyperbolic helicoid.
And so I was going to explain the math of helicoids 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, I wanted to make sure it was mathematically correct.
So, one of the things about it is that
helicoids are something that showed up in my PhD an awful lot and 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 math tattoos in the future.
So helicoids
are a type of surface that's called a ruled surface. A type of surface that 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 lines.
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 can just continuously
fill it with straight lines. This is the hyperbolic paraboloid. It's also called a high power 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 in architecture because it's very easy to
get straight objects. And then it's just about the way you assemble them that gives you curvature.
They're all still straight. Everything you do about them is straight. So the next one I'm going to show you is a
hyperboloid of
revolution. So what I'm going to do, this has four lines now,
I'm going to continue to add lines in, this makes it a cylinder. And I'm going to 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 to
start moving it around and it will become - this is a hyperboloid of one sheet. So you might
recognize this shape as the cooler for a nuclear power plant?
So that is the the canonical shape for that,
that is this shape here. Turns out that the two surfaces I showed you are
special surfaces because they're not ruled just once but they're doubly ruled so I can take two sets of
straight lines to construct the same surface. - (Brady: It's interesting - because they're made with straight lines)
(it's almost like there's no real curve there, you're just creating the illusion of a curve.) - 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 curve, on the surface.
And that's what makes it a surface and not just a set of curves.
So these ones are doubly ruled so I can turn on the other ruling
so here I have a set of,
sort of crisscross lines, and those will rotate together to form this curvature.
So now to my favorite surface is, is the helicoid. The helicoid is something that - 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
high power surface,
twist it, 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 a helicoid 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 tattoo. So the tattoo is a
hyperbolic helicoid. And it's done in a particular model called the Poincaré disk model and
the first thing we really need to understand is what do straight lines look like in this space.
So, I have all of space is in a disk and this is - the boundary is infinity and
if I 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 point
over here, a straight line that would pass through that point is
going to be something like this. So 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.
We know what straight lines are and I'm going to make a helicoid out of them. So this is what it looks like.
This is the the line I'm gonna rotate about and this is the straight line I'm gonna rotate
So this one this one is the 3d one. So I need to have - this is the plane
but for me to twist it, I need to have a third dimension to twist into.
(Brady: So now it's the same rules of - the lines are defined by the same rules, except we're using a sphere instead of a circle?) - Yes
It should be turning on. Now
I've got lines and they're gonna touch the boundary of a sphere and they have to intersect that in
ninety degree angles. I'm gonna start filling in lines up and down to make my plane. You can imagine
this is a whole plane. - (Brady: So what does that plane look like? Is that plane quite flat? I can't quite tell.)
It, it is flat. This plane right now is
just a cross-section, right through the middle of the sphere.
So it's completely flat both in our version of space and in hyperbolic space.
So this is just the half of the sphere 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 here, I'm gonna add some more lines to make it look a little fuller.
This is the surface that you my tattoo is based on. It is called the hyperbolic helicoid.
So it is a helicoid. It is a ruled surface,
based on twist of a constant rate and this one happens to be in hyperbolic space.
And then the other thing about my tattoo is, so this 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 - - (Brady: And then the plane goes along for the ride does it?) - And the plane goes along for the ride, yeah.
It's taking the curve and pulling the North Pole
around, down towards the equator. Zero is up top and pi over 2 is at the equator. So that's what I've done here.
So my tattoo, I believe, is at pi over 2. So it's a quarter of the way around the circle.
(Brady: I want to see your tattoo again now)
(So I'm looking at the South Pole, and the North - so the North Pole is not right above the South Pole?)
(It's kind of bent is it?) - Yeah, so I've again pulled this,
the centerline is one of these straight lines. So
it is also a curve that meets the, the
sphere in right angles.
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,
I guess one is at the South Pole and the other it's at the equator.
(Brady: Ok, but sort of towards me, it's not - ) - Yeah, I rotated it,
so that you got to see both poles.
So
helicoids,
they're not just a ruled surface. They're also a minimal surface.
And those are surfaces like - so films and things like that. And
that's what I've spent a lot of years of my life studying. In particular there's a theorem that says
minimal surfaces locally are either bits of helicoids or pairs of pants. And pairs of pants is
actually a technical term. And so I've spent a lot of time looking at how to decompose surfaces into helicoids.
There's a conference we always go to which is called Bridges. We went there and they do math art,
so they had set up a gallery and they said well, I mean you can go do over whatever you want,
just leave your shoulder here and we'll show it with the rest of art pieces!
Now a tattoo is getting very interactive with your mathematics, and you might not want to go that far.
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Which of the following shapes could not be a cross section of the cylinder?
I've got to be D, surely. Ah
again! Oh, yeah I see I could have liked put up there