Live from Vsauce studios in Los Angeles, California, this is Michael Stevens Living Live with your
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.
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.
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.
Remover this is equivalent to another 90 degree rotation of the knife.
I’m going to do 90 degrees and then from here I’ll do 90 degrees perfect.
And then from here I need to do another 90 degrees clockwise so I’ll go from here to
here.
I can do that without having to swap without having to flip my knife.
We’ll go up here so when we've rotated the bagel 3/4 of the way I wanna be vertical again.
Perfect.
So let me just connect these two lines.
That entry point and that entry point and we need to rotate another, a final 90 degrees
clockwise.
And again this is gonna be tough if the handle’s up here because I’m gonna hit the other
side of the bagel so let’s go ahead and reverse and at that point I’m gonna put
my knife here and I’m gonna move up 90 degrees.
Starting like this, up 90 degrees back to where I started so I’ll draw that path from
the handle’s perspective from this hole up to the starting point.
Okay so now we’ve got some lines on our bagels and these are the lines that we’re
going to follow.
If you can keep track of all of this in your mind, if you’ve practiced a little bit it
won't be too hard to do this without drawing on a bagel and I'm gonna try that next but
this is just to show you why we have to flip the knife and that we truly are doing something
that is equivalent to rotating the knife 360.
Yes we might flip it at some points but we still are moving through 360 degrees with
our knife so here we go.
Where should I start?
Forget which one was the actual starting point.
Doesn’t matter though does it.
Okay so I’m gonna start up here and then my knife is going to follow that line.
Nice fresh bagels work the best.
If they’re too crumby it takes too much force to cut them and they can kinda fall
apart.
You want them to be a bit chewy and soft so they don’t split.
Alright so there's the other and and I just, see, I’m getting a little bit of splitting
but that’s fine.
So here I am, alright.
Knife handle is straight up and I wanna flip the knife for my next 90 degree rotation of
the knife and I wanna follow this line.
Okay.
So let’s watch myself do that.
I’m gonna hold the bagel like this though because I want to keep the same frame of reference
as best I can.
Uh oh.
I’m kind of tearing it.
Be very careful.
It’s very easy to hurt yourself doing this.
Do it at your own risk.
Okay now it’s a little bit scrunched up but that's okay.
It’s still going to be delicious and mathematical.
At this point I don’t need to flip my knife.
I can keep following this line.
Here’s our third 90 degree turn.
Yeah.
Good.
That sawing motion is helping me a lot.
I should be doing more of that.
Okay we’re vertical again.
Now for the home stretch I flip the knife, put it in like this.
It’s still a vertical line.
The flipping is just caused by the limitations of our tools.
From here I wanna pull the knife while cutting up along this path.
Ready?
Here we go.
Curling up, whoa!
Getting that knife pretty close to my hand.
Please guys be careful.
Take your time.
Alright.
I’ve gone all the way through.
This cut is a two strip mobius, this strip is a two twist mobius strip and as you can
see our bagel can be divided into two intact identical rings that are interlocked.
Smear a little spread on that, serve it to your guests and they will say this is really
hard to eat but I love it.
Merry Christmas and as always, thanks for watching.
As promised here I am cutting a bagel with no lines.
I’m going to first rotate the knife 90 degrees and then there’s another 90 and another
90 for a total of 270 here’s the final 90 for the full 360 and what do you know, when
you pull the bagel apart we have two halves that are identical to one another.
They’re not even mirror images or anything but they're interlocked with no seems or gaps.
It will surely impress and it’s certainly fun to lean how to do but again guys be careful
please.
Knives are sharp and bagels well, let’s just say they've got strong personalities.
Wait what's that?
You wanna see more footage of me cutting bagels?
Well good because I do too.
Here’s some footage we shot earlier that didn’t get into the episode where I cut
a bagel with a single twist mobius strip being the shape of the cute so right now I’ve
cut the bagel half way around and my knife is rotated just 90 degrees clockwise.
I’m going to rotate the knife another 90 degrees before I return to where I began so
there’s a total of just 180 degrees of rotation of the knife which means the surface of the
cute is equivalent to a single twist mobius strip so when I’m finished I won’t get
two separate halves that are interlocked.
I won’t even get two separate things.
I will just have a well it’s kind of strange how you would describe this.
It’s basically a bagel that's twice as far around but only has one side so when I finish
which I’m about to do here, watch how I pull the bagel apart and of course I can’t
pull it apart but I have gone all the way around and you can spread whatever you want,
cream cheese, peanut butter, jam, all the way around without ever having to switch your
knife to a different bagel half.
See look at that!
Beautiful.
This is also a very fun way to cut bagels.
It’s a little bit simpler too because you don’t have to rotate the knife as much.
So go out there and have some fun and as always, thanks for bageling…naw kidding just thanks
for watching.
That made no sense.
Bageling?