X: A lot. Think of the sphere as a stack of circles, deformed into a barrel shape and closed off by caps above and below. Just as we made our curves more pliable by dividing them into guide segments connected by waves, we divide the barrel into guide strips that alternate with wavy strips. The waviness dies out at the top and bottom, so as to match the caps.
Y: Hmm, this is going to get complicated...
X: Then, for now, let's look at a single guide strip, along with the caps. Start by pushing the two caps past each other.
Y: Before, when I pushed the poles through, it made a crease!
X: Stop before the crease, when the guide has a loop in the middle. Now we turn the two caps in opposite directions, because we want to convert the loop in the middle to twisting at the ends.
Y: Oh I know---it's like a belt! If you put a loop in the middle and pull the ends tight, the loop turns into twisting!
X: Right. Then you can straighten out the belt by turning each end half a turn in opposite directions. To finish the eversion, we just need to push the middle of the guide strip back through the center of the sphere.
Y: Hmm. Can I see how two guide strips interact?
X: Sure. You can see that there are two places where the strips intersect near the central axis.
Y: And the gold sides that started facing out are now facing in.
X: Here is the whole process with all the guides.
Y: The polar caps just move up and down and then rotate into place. Ah, that's why they don't require any springiness.
X: Exactly. Now let's look at two guides and the corrugation between them, from a pole to the equator. This chunk is the fundamental building block of the eversion: the whole sphere is made from sixteen rotated copies of this piece.
Y: That looks pretty complicated.
X: Yes, but the corrugation is just following the twisting of the guide strips that you saw before.
Y: Can I see that from pole to pole?
X: Yes. The corrugation provides flexibility between the guides so that their motion does not create any pinches or creases, just like the waves in the curve that we saw before.
Y: Let me see the whole thing!
X: We corrugate the connecting strips between the guides, and push the caps past each other. We twist the caps to undo the middle loops, and push the equator across the sphere. Finally, we uncorrugate.
Y: I still don't understand. Is there some other way to look at this?
X: OK, we'll divide the sphere into thin horizontal ribbons. We'll look at one ribbon at a time. You can see the north pole push down into the south. A ribbon near the pole is rather tame: the guide segments keep their position relative to one another, and the corrugations never get very deep. Ribbons closer to the equator are wilder, so we'll split the screen to see what's going on. On the right, the camera tracks the ribbon from above, so its apparent size does not change. This overhead view highlights the symmetries that are hidden in the side view on the left, where we see the position of the ribbon in space.
At the equator, the ribbon just twists and doesn't move up or down.
Y: Wait a minute. This ribbon looks just like the wall under the monorail, and it's turning inside out. You'd finally convinced me that that was impossible!
X: I'll play that again. Remember that our walls represented circles and had to stay vertical. But here the ribbon can twist around in space, because it's part of a sphere. Another way to understand the eversion is to progressively build up the surface of the sphere at a few important stages. This is the corrugation phase... Now we've just pushed the caps through each other... This is the middle of the twisting phase: we can see the complex activity at the equator...
At the end of the twisting phase, the corrugations have nearly become figure eights... Here we're in the middle of pushing horizontally through the center of the sphere... Finally, we show the uncorrugation phase. The sphere is now entirely purple.
Y: Wow. I think I'm ready to see the whole thing again.
X: Here goes!
Y: You were right: you can turn a sphere inside out without poking holes or creasing it, even though you can't do it for a circle. This is great---somebody should make a movie about this stuff!
Created: May 17, 1995 --- Last modified: Jun 14 1996
Copyright © 1995-1996 by The Geometry Center All rights reserved.