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# The Script of Outside In (part 4): The turning number (for surfaces)

Y: But wait---doesn't the same argument prove you can't turn a sphere inside out? This sphere has a three-dimensional smile, and this one has a three-dimensional frown. So they have different turning numbers!

X: Not quite. Your analogy is good, but to make it complete, we must look at a general surface and consider all the points where it is horizontal and gold is on top. We'll draw horizontal stripes to make these points easier to locate. Smiles are like bowls, curving up; frowns are like domes, curving down. But there are other points where the surface is horizontal that are neither bowls nor domes. They are saddles, and look like smiles from one direction and frowns from another. Near a bowl or a dome, the horizontal stripes form rings. Near a saddle, they form an X.

Y: But how does that change anything? Spheres don't have saddles!

X: Ah, but the point is how these features interact. Look: a dome and a saddle can come together and cancel out. Likewise, a bowl and a saddle can cancel out. But bowls and domes, like electrical charges of the same sign, normally don't get near each other.

X: The unchanging number for surfaces, then, is this: add domes and bowls, and subtract saddles. This number is 1 for the sphere no matter which face is out!

Next: How to turn any curve into any other of the same
Up: The Script of Outside In
Prev: The turning number (for curves)