along with a friend of mine,

a Klein bottle hat. My question for the Klein bottle hat

requires that I inset a zipper.

Which is why my trusty old Pfaff has a zipper foot.

Why would I want a zipper in a Klein bottle?

I've been told over and over again that if I cut a Klein bottle in half,

I'll get two Möbius loops.

So let me unzip my Klein bottle

into

two pieces of fabric, ja? Here's one half of the Klein bottle.

Here's the other half of the Klein bottle. Check this out. This half Klein bottle has a half a twist.

This edge comes over, around, and then goes under.

This is a Möbius loop. Yeah, it's wider in one part than the other, but it's a Möbius loop.

This is also a Möbius loop. It looks almost the same, with an interesting difference. Check this out.

This also is a Möbius loop, except where this one goes over, this one goes under.

If this is a right-handed Möbius loop, this is a left-handed Möbius loop.

I can zip these two Möbius loops together. This guy zips into here.

And,

with only a little bit of finagling,

come on, finagle. So here's a fabric Klein bottle.

Two Möbius loops, one left-handed, one right-handed,

together, makes a Klein bottle. My head's too big still.

Which is why I've begun making a bigger one of these. So I've got out the

fabric, a zipper foot, have a long zipper, so my next week's project is to make a

bigger Klein bottle hat that will be properly

fitted to me.

And I hope will be like my knitted Klein bottle hats, that, you know, actually work properly.

!!!!

Yeah, if I take a fabric Klein bottle and cut it in half, I'll get two Möbius loops.

What happens if I cut a glass Klein model in half?

Check this out. Come here. Keep keep going, Brady. Come on. Keep going, come on!

[frantic exclamations]

Let's go down.

Okay, so how do you cut a Klein bottle in half? First thing you need is a diamond saw.

Okay, here's a diamond saw, right here. Look along here, there's a diamond blade. Flip it on,

and I can cut practically anything.

I have a tile.

I can cut.

It's a diamond saw, so it cuts through tile like butter. Cuts through glass real easy.

Well, what I'd like to do is take a Klein bottle,

put it in here and cut it.

Problem is, to cut glass as fragile as this, I have to go very slowly.

So,

I can't push it with my hand because I'd be going v-v-v-v.

I need a way to guide this carefully in a straight line.

I built a linear table that will, instead of having nice expensive

linear bearings,

I just used a couple drawer slides, a couple pieces of Plexiglass, a hinge,

and everything's held together with the usual things like this.

I can take a glass Klein bottle, put some Velcro on the bottom and then put a magnet inside of it.

Little magnet right there.

Magnet will be held here. Velcro is held there.

Everything is held together, I hope like this.

Okay, I now have a Klein bottle held in place,

and all I need to do is bring it this way, it'll slice by way of the,

by way of the diamond saw.

So instead of just slicing it by pulling with my hand,

I have a threaded rod, and the threaded rod will be pulled along by way of a drill

which is computer-controlled. I have a little Arduino that will...

[1.21 Jeegawatts!]

Half a Klein bottle. Another half of a Klein bottle.

So, we now see two half Klein bottles, put together,

one complete Klein bottle,

two halves.

Okay, this is fresh from the diamond saw, so it's covered with dust - glass dust. But let's look at this carefully.

I'll start my finger right here,

carry around here,

around, around,

around and back. This has one edge.

Right? One edge. I start here, loop around,

over here, along here, back across

to here. One edge. How about, how many sides does it have? Well, let's start right here.

I can go all the way over to here.

around, and if that wall weren't in the way, I'd be right back to here. It's a Möbius loop. It's got one side,

one edge, a half twist, and if I were to look carefully, I'd notice that this one,

this Möbius loop has the opposite handedness of this Möbius loop.

So, slice a Klein bottle in half,

you end up with two Möbius loops. Glass Möbius, they're glass Möbius loops, admittedly, but it's the real thing.

Okay, I built a machine that'll slice a piece of glass. That'll slice a Klein bottle.

Let's cut a camera in half.

Got an old Russian Zenit 35 millimeter camera.

Ran it through my machine.

Have a sliced camera.

And the lens, of course, is the cool part.

A four-element lens, three of whi-

two, oh, five-element lens, glued together like this, straight out of my slicing machine. Since this is numberphile,

and it's math, is to talk about the wonderful mathematics of

lenses.

Ah, there is wonderful mathematics in a zoom lens!

Tracing rays and figuring out where the focus is. Making sure that the fo- that no intermediate focus lies on the edge of a lens.

Making sure that there's

that all rays from around the edge come to the same focus. I'd love to talk about it.

Maybe some other time.