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• 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.

• 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.

CLIFF STONE: So, how come we eat pizza the way we do?

B1 中級

# 私たちはピザを食べる驚くべき方法 - Numberphile (The Remarkable Way We Eat Pizza - Numberphile)

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林宜悉 に公開 2021 年 01 月 14 日