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# The Script of Outside In (part 3): The turning number (for curves)

X: Yes! There is something fundamental about curves that would have to change if you were to turn a circle inside out---and that something cannot change under our allowed motions.

Y: And what's that?

X: I'll explain. Imagine a monorail atop the wall. Now the rule about monorail traffic is that the car only travels forward, and it always keeps the purple wall on its right.

We'll use a diagram to monitor the car's direction. On this track, the car is always turning left. As it goes around the circle once, it makes one full turn toward the left.

On a more complicated track, the car might sometimes be turning left and sometimes right, but the net amount of turning after one complete circuit is always some number of full turns in one direction or the other. The number of full turns it makes toward the left is called the curve's turning number.

For a curve where there is more turning toward the right than toward the left, the turning number is negative.

Y: Hey, and if there is no net turning, the turning number is zero!

X: Right.

Y: I had a hard time following the net turning for this winding track.

X: That's natural. But there's another way to get the right answer: find the spots where you're traveling in a particular direction, like due east.

Y: Let's see, since we're looking from the south, that would be wherever the monorail is going toward our right---where we see the purple wall face on.

X: Exactly. At some of these points the track is curving away from us. Viewed from here, it looks like a smile; at these places, the car would be turning left. At others the track looks like a frown, curving toward us; at these points the car would be turning right. The net number of full turns increases when the car passes a smile, and decreases when it passes a frown. Starting at zero... one... two... three... four... three... two... and we finish with three. The turning number is the number of smiles minus the number of frowns.

Y: I see... the turning number measures happiness!

X: If you insist...

X: Now the nice thing about the turning number is that it remains the same when a curve changes according to our rules. Frowns and smiles can appear or disappear, but only in pairs that balance out. The number of smiles minus the number of frowns never changes.

Y: So a curve can only turn into another curve with the same turning number?

X: Right: the turning number is the fundamental property I mentioned before. Now what's the turning number for the two circles?

Y: Hmm... This one has one smile and no frowns, so the turning number is one. And if the gold is outside---one frown and no smiles---minus one! It makes sense---on one curve you're turning left all the time, on the other it's the opposite.

X: Good. So the reason you cannot turn a circle inside out...

Y: ... is that what would change the turning number!

Next: The turning number (for surfaces)
Up: The Script of Outside In
Prev: Can a circle be turned inside out?