Outside In

[Music] 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 [Music] be careful what about that ring around the equator remember you mustn't tear or crease it ah 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 [Music] 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 [Music] 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 [Music] 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 curves turning number [Music] 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 do 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 [Music] 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 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 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 um 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 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 [Music] frowns are like domes curving down [Music] 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 [Music] 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 rules 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 one 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 moran 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 one still not allowing sharp corners

right of course now can the circle be turned into any curve of turning number one 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 one 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 [Music] 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 [Music] 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 [Music] 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 one why can't we align the guides if the turning number isn't one watch what happens when we try to turn a figure eight into a circle [Music] and here both the initial and final curve have turning number zero [Music] using this method or others you can always transform one curve into another with the same turning number this is called the whitney graufstein 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 hmm 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 [Music] 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

eversion we just need to push the middle of the guide strip back through the center of the sphere 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 [Music] here is the whole process with all the guides [Music] the polar caps just move up and down and then rotate into place ah that's why they don't require any springiness exactly [Music] now let's look at two guides in 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 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 uncorrugate 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 pushed 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 [Music] 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 [Music] 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 sphere is now entirely purple wow i think i'm ready to see the whole thing again here goes [Music] [Music] so 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 [Music] [Music] [Music] do [Music] you