CLIFF STONE: So, how come
we eat pizza the way we do?
You know, as well as I do:
If I just hold it like this, it flops over.
You know, as well as I do,
that if I curl it like this:
I can eat.
So, why do we eat pizza like this?
Gauss.
Gauss tells us why.
What does Gauss has to do with it?
Gauss came up with this absolutely nifty theorem
called the Theorema Egregium.
Also, in English, called the
Remarkable Theorem.
Curvature is an intrinsic property of surfaces.
Curvature? We all kind of know a curve is.
What's curvature?
This piece of paper has no
curvature at all. Right?
It's flat going this way, flat going this way.
We'll come back to that.
What about a sphere? A sphere clearly has
curvature going outward this way,
and an outward going curvature this way.
If both curves are outward,
Gauss says: multiply them together
This is positive, this is positive. Positive multiplied by positive means
Oh, this has a positive curvature!
So this is a positive curvature right here.
Everywhere on a sphere
it's positive curvature.
What about a banana? For a banana I see along here,
along here,
it's curving out. Along this way its
curving out,
so this is positive curvature.
But watch this, watch this!
Over here, oh, it's curving outward along
here, we remember that. But along here
it's curving inward. So this one
is a negative Gaussian curvature,
this one is a positive. A positive times
a negative means that at this point
I have a negative curvature. So, someplace,
along a line here, along a line here
the curvature of a banana changes from
being positive Gaussian curvature to
negative Gaussian curvature.
For a torus, along here,
it's curving outward, that's cool. Along
this way it's curving outward outward
Outward times outward is positive. So, that's positive.
But over on the inside, over here,
it's curving outward there, but inward
here!
So this means this is negative curvature.
So it's negative curvature around here,
Positive curvature, so some place around this line in a torus,
and along some place around here
It goes from negative gaussian curvature to positive. That makes sense to me.
What Gauss said in his remarkable theorem:
This curvature is intrinsic to the surface.
So, if I have something that starts out
with a certain integrated Gaussian total curvature
then, no matter how I stretch
it and turn it around and move it,
it's going to stay the same.
I know that this piece of paper starts
out flat.
If I bend it... I'll draw a line here.
I'll bend it sort of along that line.
Ok, along there, in this direction
there's negative curvature .
But along here,
So it's minus, going along there,
but this has still zero curvature.
So, it's a straight line,
it's a straight
line going that way.
But going this way, it's negative curvature.
Flipping on the other side...
Look over here! I still have
no curvature in that line.
And a positive curvature over here.
Well what's the Gaussian curvature
right at this point, right here?
It's some positive number times zero.
Ah! Gauss tells us that Gaussian curvature,
the multiplication of this direction curvature
and this direction curvature is
intrinsic to the surface.
That means that if I bend it this way
and I get positive curvature
at this point, since the surface started out
with no curvature, flat, no curvature at all,
then if I add positive curvature here,
I'd better multiply that by something
that has zero curvature,
which is a straight line.
So, if I have curvature in this direction
I'd better have no curvature there.
What about a pizza?
Come here, Brady! Come, come here!
If I have a flat pizza
and I lift it up like this
it's going to bend down!
If it bends this way,
then... Oh! The Gaussian curvature is positive here
but I have zero going this way.
But I can take advantage of that by saying:
I will make this have no
curvature along here and
negative Gaussian curvature,
negative curvature there,
so at each of these points, the
curvature stays the same.
If it tries to flop this way while I'm curving it this way,
the pizza would be violating
Gauss's remarkable theorem.
Pizzas don't like to do that.
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CLIFF STONE: In other words, the reason
why there is a correct way to eat pizza
and this won't flop over
is that if that flopped over like this,
I would be forcing a point right here to
change its Gaussian curvature.
Not just an intrinsic part of a piece of paper,
but an intrinsic part of a piece of pizza.