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Pretty much everyone loves eating pizza,

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but it can be a messy business.

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Pizza is soft and bendable, so how can you stop all that cheese from falling off?

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You might know some of the tricks:

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you can use two hands,

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not so classy,

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or you can use a paper plate

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and allow only the tip of the pizza to peak out.

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There is one other trick, though:

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holding the crust, you can sort of fold the slice down the middle.

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Now the tip of the pizza isn't falling over,

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and you can eat it without getting tomato sauce all over yourself

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or accidentally biting off some of that paper plate.

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But, why should the tip stay up just because you bent the crust?

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To understand this, you need to know two things:

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a little bit about the math of curved shapes,

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and a little bit about the physics of thin sheets.

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First, the math.

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Suppose I have a flat sheet made out of rubber.

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It's really thin and bendable, so it's easy to roll into a cylinder.

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I don't need to stretch the sheet at all, just bend it.

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This property where one shape can be transformed into another

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without stretching or crumbling is called isometry.

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A mathematician would say that a flat sheet is isometric to a cylinder.

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But, not all shapes are isometric.

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If I try to turn my flat sheet into part of a sphere,

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there is no way I can do it.

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You can check this for yourself by trying to fit

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a flat sheet of paper onto a soccer ball

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without stretching or crumbling the paper.

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It's just not possible.

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So a mathematician would say

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that a flat sheet and a sphere aren't isometric.

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There is one more familiar shape that isn't isometric

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to any of the shapes we have seen so far:

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a potato chip.

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Potato chip shapes aren't isometric to flat sheets.

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If you want to get a flat piece of rubber into the shape of a potato chip,

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you need to stretch it,

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not just bend it, but stretch it as well.

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So, that's the math.

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Not so hard, right?

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Now for the physics.

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It can be summed up in one sentence:

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thin sheets are easy to bend but hard to stretch.

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This is really important.

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Thin sheets are easy to bend but hard to stretch.

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Remember when we rolled our flat sheet of rubber into a cylinder?

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That wasn't hard, right?

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But imagine how hard you would you have pull on the sheet

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to increase its area by 10%.

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It would be pretty difficult.

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The point is that bending a thin sheet takes a relatively small amount of force,

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but stretching or crumbling a thin sheet is much harder.

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Now, finally, we get to talk about pizza.

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Suppose you go down to the pizzeria and buy yourself a slice.

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You pick it up from the crust, first, without doing the fold.

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Because of gravity, the slice bends downwards.

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Pizza is pretty thin, after all,

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and we know that thin sheets are easy to bend.

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You can't get it into your mouth,

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cheese and tomato sauce are dripping everywhere,

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it's a big mess.

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So you fold the crust.

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When you do that, you force the pizza into something like a taco shape.

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That's not hard to do.

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After all, this shape is isometric to the original pizza, which was flat.

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But imagine what would happen if the pizza were to droop down

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while you are bending it.

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Now it looks like a droopy taco.

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And what does a droopy taco look like?

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A potato chip!

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But we know that potato chips are not isometric to flat pieces of rubber,

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or flat pizzas,

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and that means that in order to get into the shape it's in now,

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the slice of pizza had to stretch.

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Since the pizza is thin, this takes a lot of force,

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compared to the amount of force it takes

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to bend the pizza in the first place.

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So, what's the conclusion?

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When you fold the pizza at the crust,

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you make it into a shape where a lot of force is needed to bend the tip down.

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Often gravity isn't strong enough to provide this force.

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That was kind of a lot of information,

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so let's do a quick backwards recap.

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When the pizza is folded at the crust,

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gravity isn't strong enough to bend the tip.

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Why?

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Because stretching a pizza is hard,

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and to bend the tip downwards,

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the pizza would have to stretch.

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Why?

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Because the shape that the pizza would be in,

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the droopy taco shape,

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is not isometric to the original flat pizza.

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Why?

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Because of math.

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As the pizza example shows,

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we can learn a lot by looking at the mathematical properties of different shapes.

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And it's especially nice when those shapes happen to be pizza slices.