Hey, I read somewhere that mathematicians can turn a sphere inside out.
Yes, that's true.
What's the big deal?
Just poke a hole in it and pull it through.
Sure, but the point is to do it without making a hole.
But then it seems impossible.
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.
But you cannot rip or puncture this material without destroying it.
And you cannot crease it or bend it sharply.
If the surface can pass through itself, what's the problem?
Do you think allowing self-intersections makes it easy?
Try it.
I'll push the two halves right through each other.
Be careful.
What about that ring around the equator?
Remember, you mustn't tear or crease it.
Let me try again.
That's no good either.
You're pinching it infinitely tight.
But then there's no way.
It's impossible.
You'd have to crease or pinch it to turn it inside out.
It is surprising.
But watch this.
Is this it?
Is this a sphere turning inside out?
You bet.
That wasn't easy to follow, was it?
To figure out what's going on, let's look at something simpler.
A circle.
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?
Of course.
I can turn a rubber band inside out.
Remember, we're really trying to turn the circle inside out.
We only built the wall so we could see the different sides.
Oh, yes.
The wall has to stay vertical, and it can't have creases, but it can pass through itself.
Fine.
Let me try.
Watch out.
That was a sharp bend.
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.
But I can avoid corners altogether by making a loop smaller and smaller.
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.
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.
Yes, you're right.
Wait a minute.
Am I supposed to believe that you can turn a sphere inside out, but not a circle?
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.
And what's that?
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.
Hey, and if there's no net turning, the turning number is zero.
Right.
I had a hard time following the net turning for this winding track.
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.
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.
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.
I see. The turning number measures happiness.
If you insist.
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.
So a curve can only turn into another curve with the same turning number.
Right. The turning number is the fundamental property I mentioned before.
Now what's the turning number for the two circles?
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.
Good.
So the reason you cannot turn a circle inside out is that that would change the turning number.
But wait. Doesn't the same argument prove that 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.
Not quite. Your analogy is good.
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.
But how does that change anything? Spheres don't have saddles.
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.
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.
Okay. I'm willing to believe that turning numbers don't prevent the sphere from turning inside out as they do the circle.
But that doesn't mean you can actually do it.
We'll get to that. I know it's hard to see.
Steve's Mail proved it was possible in theory in 1957, but it took seven years before Arnold Shapiro found a practical way to do it.
Since the problem remained hard to visualize, more methods were invented later by Bernard Morin and several others.
I'll show you Bill Thurston's method, invented in 1974.
Let's go back to curves for a bit. Remember that this circle can only be changed into curves of turning number 1.
Still not allowing sharp corners, right?
Of course. Now can the circle be turned into any curve of turning number 1? Say this one?
Let's see. I'll try to go backwards from this curve to the circle.
I think I got it. There.
Excellent. Now try this one.
I'll undo this loop first and push this fold back. Now here.
Here we go.
Very good. And this one?
Whoa. You're not going to ask me to do every single curve of turning number 1, are you?
Of course not. What we need is a general method. Do you remember the simple way to transform one curve to another when sharp bends are allowed?
Yes. You just go straight from one to the other.
That's the one.
When the curves have the same turning number, this method can be adapted to work without sharp bends. The trick is to add waves to the curve.
Can we do it on a simpler one?
Sure.
We start by marking small pieces of the curve that will serve as guides for the transformation.
We'll concentrate on these segments now.
We move the centers of the guide segments straight toward their final destinations on the circle, without any rotation.
Next, we rotate the guides so that they are lined up with the circle.
Okay. What about the parts in between?
That's where the waviness comes in.
We make the connecting segments between adjacent guides bulge out into corrugations.
This allows the segments to move freely around each other as long as they remain more or less parallel.
Oh, I see. The guides can move around without creating sharp bends.
Correct.
Here is the transformation of the whole curve.
The original curve in blue develops sharp corners, but the wavy curve is springy enough to remain smooth throughout.
We have to keep adjacent guides roughly parallel as we rotate them to align with the circle.
This is possible as long as the turning number of the original curve is 1.
Why can't we align the guides if the turning number isn't 1?
Watch what happens when we try to turn a figure 8 into a circle.
And here, both the initial and final curve have turning number 0.
Using this method, or others, you can always transform one curve into another with the same turning number.
This is called the Whitney-Graustein theorem.
And what does this have to do with the sphere?
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.
This is going to get complicated.
Then for now, let's look at a single guide strip along with the caps.
Start by pushing the two caps past each other.
Before, when I pushed the poles through, it made a crease.
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.
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.
Right.
Then you can straighten out the belt by turning each end half a turn in opposite directions.
To finish the aversion, we just need to push the middle of the guide strip back through the center of the sphere.
Hmm. Can I see how two guide strips interact?
Sure.
You can see that there are two places where the strips intersect near the central axis.
And the gold sides that started facing out are now facing in.
Here is the whole process with all the guides.
The polar caps just move up and down and then rotate into place.
Ah, that's why they don't require any springiness.
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 aversion.
The whole sphere is made from 16 rotated copies of this piece.
That looks pretty complicated.
Yes, but the corrugation is just following the twisting of the guide strips that you saw before.
Can I see that from pole to pole?
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.
Let me see the whole thing.
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 uncorrigate.
I still don't understand. Is there some other way to look at this?
Okay. 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.
Wait a minute. This ribbon looks just like the wall under the monorail, and it's turning inside out.
You had finally convinced me that that was impossible.
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.
Wow. I think I'm ready to see the whole thing again.
Let's go.
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.