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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.

  • 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.

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

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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
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