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The Script of Outside In (part 1):
What does it mean to turn the sphere inside out?

Y: Hey, I read somewhere that mathematicians can turn a sphere inside out.

X: Yes, that's true.

Y: What's the big deal? Just poke a hole in it and pull it through.

X: Sure, but the point is to do it without making a hole.

Y: But then it seems impossible!

X: You're right, you cannot do it with an ordinary sphere like a basketball. You have to understand the rules of the game: this sphere is made of an abstract elastic material that can stretch and bend and pass through itself. [MW]

But you cannot rip or puncture this material without destroying it, and you cannot crease it or bend it sharply. [MW]

Y: If the surface can pass through itself, what's the problem?

X: Do you think allowing self-intersections makes it easy? Try it.

Y: I'll push the two halves right through each other.

X: Be careful. What about that ring around the equator? Remember, you mustn't tear or crease it.

Y: Argh---let me try again.

X: That's no good either---you're pinching it infinitely tight.

Y: But then there's no way! It's impossible! You have to crease or pinch it to turn it inside out.

X: It is surprising. But watch this:

Y: ... Is this it? Is this a sphere turning inside out?

X: You bet!

Next: Can a circle be turned inside out?
Up: The Script of Outside In

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The Script of Outside In (part 1): What does it mean to turn the sphere inside out?

The Script of Outside In (part 1):
What does it mean to turn the sphere inside out?