host Michael Stevens! Christmas time is right around the corner.

You guys know what that means right?

It's time for festive stuff like Santy Clause and funny sweaters and peanut butter and seagulls.

By the way do you guys know why seagulls fly around the sea?

Well because if they flew around the pring, they'd be Pringles.

Okay you know if they flew around a mug they'd be muggles.

Harry Potter.

Quidditch.

Hermione!

And if they flew around the bay they'd be bagels.

Okay okay.

It's time to get serious.

Today I am going to show you how to cut a bagel into two halves that are whole and complete

but yet interlocked by using a cut that follows the surface of a mobius strip.

But before we can mathematically cut this bagel in the festive way I wanna teach you

today we have to ask that question we ask ourselves every morning before breakfast:

How many faces does a sheet of paper have?

Okay.

Look we often think that a sheet of paper has two sides right?

A front and a back.

But does it really?

Is it truly a two-dimensional object?

I don't think so.

Both of the sides of a sheet of paper are actually polygons right?

Rectangles.

Two rectangles, one on the front one on the back separated in three dimensions by the

thickness of the page.

A sheet of paper is actually a flexible polyhedron.

It is an extremely thin rectangular prism which means it has six faces.

This rectangular face on top, the rectangular face on the bottom and then four very very

thin faces around the side just like a die.

Yeah.

Now earlier I prepared some strips of paper.

I couldn't get Christmas colors but I was able to get birthday colors and I'm going

to use these to talk about what happens when we take a rectangular prism and create a hoop.

Alright?

What I wanna begin with is just one strip.

Here's the strip.

Now if I take the strips and I loop it around so that these two opposite faces are joined

I lose both of those faces and the resulting hoop only has a total of four faces.

The south side face, the inside face, and then this top edge which is actually a very

thin face and this bottom edge and they're all completely separate.

If I take some scissors and I snip right there in the middle and then I cut this hoop all

the way around I will separate this right face and this left face and since they are

completely distinct from one another I wind up with two separate hoops okay?

But look what happens when I take a strip and instead of making a hoop just like this

I make the hoop after a 180 degree twist.

Now this is very interesting because now what's happening is that yes, this face and that

face disappear in the join.

However, the twist means that what used to be, let's call this the top face, is now

continuous with the bottom face.

And so if you travel around this bottom face you come back and you connect to the opposite

face, the top.

But we know that the top face connects to the bottom face so now what used to be two

faces has become one.

And let's look at what is here locally, an outside face and an inside face.

They have also connected to each other, to the opposite because of that twist.

If this used to be the outside face, by turning it and joining I now have what used to be

the outside connecting to the inside.

So now there's just one side.

Those two faces have also become one and so if I could cut this shape right down that

thin middle, right down in between along, if I had a very, a very very thin knife then

could separate what is locally here the outside and the inside, I'd wind up with two rings

but I wouldn't because there aren't an outside and an inside.

These two faces here are the same.

To show that let me use two strips.

I'll use a red strip and a green strip and we can imagine that this is actually just

one strip and that I'm going to cut it right down this way down that narrow face and separate

them into two okay?

So imagine that this is just one strip.

I'm going to bring them together into a hoop.

Now normally if there was no twist after the cut I would have myself two hoops right?

But I'm gonna do a twist and this should very clearly show that the inside which is

in this case red is being connected to the outside.

And likewise the outside which is green is being twisted to connect to the inside.

Alright.

Now let me take these.

You have to be very careful that you don't tape too many things together but you also

want the tape to be good enough that you can cut the thing.

Perfect.

Now I'll join these two sides.

ooh that's too big of a piece.

Luckily I have this thinner piece from earlier.

perfect!

Okay.

So here is our twisted hoop which many of you know is called a mobius strip.

This one has a single twist, a 180 degree twist.

Let me now, oh I don't need to cut them!

I want to cut them down that narrow face don't I so I'm pretending that I've done that,

that I've gone all the way around.

But what do I get?

Just one big hoop.

Just one big hoop.

Why should that happen?

Well it's because that twist connected the inside and the outside so that there's only

now one side.

A good way to make this clear is to use some string.

I have two lengths of string here and what I'd like to do is use the string to clearly

whoops I dropped my green string.

I'd really like to illustrate it this way.

Camera person, can you see this?

Wonderful.

Okay so here's a, here's a hoop that's green and here is a hoop that is white.

We can imagine that these are the two sides of the object that we're cutting and perhaps

we're going to cut it right in between and wind up with a separate green hoop and a separate

white hoop.

However, if I take the compound object before cutting and I give it a twist, I'm connecting

as we saw with the paper, sides like this and now I have one continuous loop.

Since green begins and then ends at white and white begins and then ends back at green.

So this is just one big hoop, in fact, I wanna just try this out.

I'm gonna tape the ends together, ohhh!

Okay.

And then I'm gonna tape these two together.

Wonderful.

Okay.

What do we have?

We have one big hoop.

Ha ha hey!

Okay so now let's undo these connections and start again and I wanna do, I wanna do

two, two twists this time.

Two twists.

Okay strings first.

Strings first.

Here's our inside hoop.

Here is our outside hoop.

Great.

Now I hope it's clear that we have a green hoop inside the white, the white hoop's

on the outside.

These are only separate hoops because we already separated them along this line but as an object

to begin with this is just one thing right and we're going to cut it down the middle

and get an inside and an outside, what is green and what is white.

Okay, so now rather than doing a single 180 degree twist let's do a full 360.

So here is the 180, that connects white to green and white to green but another twist

in that same direction connects green back to green and white back to white.

Now look what we have here.

Now the green hoop is a complete separate hoop.

It does not connect to white.

However that second twist got them all intertwined.

Now we have yes, two separate hoops but they are interlocked.

If I stick the greens together and I stick the outside hoop together what do we have?

We have two circles, a green one and a white one that are linked.

This happens with paper as well.

If I take a strip of paper and I make a hoop but before I connect them I do one twist and

then a second twist in that same direction I now have connected faces to themselves.

The outside connects back to the outside.

The inside connects back to the inside but the inside and the outside have crossed over

each other and are now linked so if I cut them in half.

I'm gonna cut it in half this way so instead of cutting what you might in one local region

call an outside and an inside I'm going to cut what in one local region you might

call the right side and the left side.

Watch this.

I will get two separate identical halves but they will be locked together.

Cutting it in half surely we will find ourselves with two pieces.

Nope.

Two interlocked pieces.

Two interlocked pieces.

Now that was done with a cut that twisted 360 degrees.

We can cut a bagel in just the same way.

That's right we're gonna draw on a bagel.

You might not want your kids to see this.

So first of all if I had a huge bagel like the size of a hoola hoop this would be a lot

easier because with a hoola hoop-sized bagel I could stick a knife in and I could go all

the way around just like normal but then before I got back to where I started I'd have a

lot of room to move my knife and rotate it 360 degrees before I got back to where I was.

Introducing the two twists we need for the cut to be the shape of a two twist mobius

strip.

However that kind of 360 cutting is very difficult when you only have a tiny section of a bagel

so what we should do is spread that twist evenly throughout the bagel.

And to make this even easier instead of having to both keep in mind the 360 degree rotation

of the cutting knife in this plane we can call this the xy plane, and on top of that

the knife's rotation around the z axis as I go around the bagel.

Let's remove that second case, that second issue and just rotate the bagel itself.

Perfect.

So let's say that we start here.

Boom.

With the knife horizontally right into the bagel.

Normally we would just go all the way around 360 degrees with the bagel, get back to where

we started and we'd have a top and a bottom half but we want the knife to twist and if

we want 360 degrees to fit within a 360 degree rotation, we need to match it like this so

that a quarter of the way through our rotation of the bagel we have made a quarter of a full

rotation like that.

Okay.

So I'm going to be cutting in like this through that hole and then by the time I've

turned the bagel a quarter turn the knife should be vertical up and down like this right

here.

So I'm going from here to here.

From here to here.

Let's draw that path so it becomes very easy to follow.

From there to there.

Now we're going to have a problem.

The problem is that if I've already done 90 degrees of clockwise rotation and I need

to do 90 more degrees, ooh.

I get myself into this problem.

I want by the time that I reach this part of the bagel for the knife to be horizontal

but I've got the other side of the bagel in the way.

So what's gonna make this easier is just shifting the knife to another orientation.

I'll go up like this and then we need it to go like this.

Ooh!

That's gonna be a problem because our knife can't pass through the other side of the

bagel but what we can do is reposition our knife so think of it this way.

We go in like this.

Then we go up vertical, then we need to turn another 90 degrees clockwise.

But let's do that with the handle beneath this time so we'll turn 90 degrees clockwise.

Alright so on this side we're in horizontally but we've swapped the knife so now since

I wanna draw a line that follows the handle's perspective we're going to draw a line that

goes from here, right, to here.

So from the, did I, yeah, I've got a little cut, where does it come out the other end?

Did I go all the way through?

Yeah there I go.

From this hole up to that hole.

And I'm just gonna draw that path so that I know which side of the bagel I want my handle

to be on.