The Script of Outside In (part 2):
Can a circle be turned inside out?

X: That wasn't easy to follow, was it? To figure out what's going on, let's look at something simpler: a circle.

X: We'll build a vertical wall along the circle so that we can color the two sides differently. Can you gradually turn this circle into this other circle, where the purple and gold sides are reversed, without creating sharp corners?

Y: Of course! I can turn a rubber band inside out.

X: Remember, we're really trying to turn the circle inside out. We only built the wall so we could see the different sides.

Y: Oh, yes, the wall has to stay vertical. And it can't have creases, but it can pass through itself. Fine, let me try.

X: Watch out! That was a sharp bend!

X: If we could make sharp bends in the material, we'd be able to turn any curve into any other, by moving each point of the initial curve in a straight line toward a target point in the final curve.

Y: But I can avoid corners altogether by making a loop smaller and smaller...

X: That's an interesting idea, but pulling a loop tight is not really a gradual change. It's like having a corner in disguise, so it's against the rules.

Y: Well, if you can't have corners and you can't pull loops tight, I think it's impossible to turn the circle inside out!

X: Yes! You're right.

Y: Wait a minute. Am I supposed to believe that you can turn a sphere inside out, but not a circle?

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