I that can turn a out. that's What's big Just a in it and pull it through. but point is to do it without making a But it right, you cannot do it with an ordinary a You to of this is of an abstract that can and and pass through But you cannot rip or this without it, and you cannot it or it sharply. If can pass through what's Do you think allowing it Try it. I'll push two right through What about that ring around you mustn't or it. try again. That's no good pinching it tight. But no way! It's You to or pinch it to turn it out. It is surprising. But watch this ... Is this it? Is this a turning out? You That wasn't to follow, was it? To out what's going on, look at a build a wall along so that can color two Can you gradually turn this into this and gold without sharp Of I can turn a band out. trying to turn out. only built wall so could Oh, wall has to stay And it can't but it can pass through try. Watch out! That was a sharp If could sharp in to turn any into any by moving point of initial in a straight toward a point in final But I can avoid by making a loop and That's an but pulling a loop tight is not a gradual It's having a in so it's against if you can't and you can't pull loops tight, I think it's to turn out! right. Wait a Am I to that you can turn a out, but not a is about that would to if you to turn a out---and that cannot our motions. And what's that? I'll a monorail atop wall. Now about monorail traffic is that car only forward, and it always wall on its right. a diagram to monitor car's On this track, car is always turning As it around it full turn toward On a track, car might turning and right, but amount of turning circuit is always of full turns in or of full turns it toward is turning For a is turning toward right than toward turning is and if is no turning, turning is Right. I had a hard following turning for this winding track. That's natural. But way to right find spots in a particular looking from south, that would monorail is going toward our wall on. At of points track is curving away from us. from it looks a at car would turning At track looks a frown, curving toward us; at points car would turning right. of full turns car a and it a frown. Starting at two... four... two... and finish with turning is of minus of frowns. I turning If you insist... Now thing about turning is that it a according to our Frowns and can or but only in pairs that out. of minus of frowns So a can only turn into with turning Right: turning is I Now what's turning for two Hmm... This has and no frowns, so turning is And if gold is frown and no It turning all on it's Good. So you cannot turn a out... is that what would turning But you can't turn a out? This has a and this has a frown. So turning Not Your analogy is good, but to it must look at a and all points it is horizontal and gold is on top. draw horizontal to points to bowls, curving up; frowns curving down. But points is horizontal that bowls nor and look from and frowns from a bowl or a horizontal form rings. a form an X. But how that anything? don't Ah, but point is how Look: a and a can and out. a bowl and a can out. But bowls and of sign, normally don't unchanging for is this: add and bowls, and subtract This is 1 for no which is out! go back to for a bit. that this can only into of turning 1? Still not allowing sharp right? Of Now can into any of turning 1, say this I'll try to go backwards from this to I think I got it. Now try this I'll undo this loop first... and push this fold back... Now go. good! And this Whoa! not going to ask to do of turning you? Of not. What is a Do you way to transform to sharp You just go straight from to That's turning this can to work without sharp trick is to add to Can do it on a start by marking small of that will as for transformation. on now. of straight toward final on without any rotation. so that up with OK, what about parts in That's in. out into corrugations. This allows to around as long as or Oh, I can around without sharp is transformation of original in sharp but wavy is springy to smooth throughout. to roughly as to align with This is as long as turning of original is Why can't align if turning isn't Watch what try to turn a into a ... And both initial and final turning Using this or you can always transform into with turning This is And what this to do with A lot. Think of as a stack of into a and off by caps and Just as our by dividing into by into strips that with wavy strips. out at top and bottom, so as to match caps. Hmm, this is going to for now, look at a strip, along with caps. Start by pushing two caps past I through, it a Stop has a loop in Now turn two caps in want to loop in to twisting at Oh I know---it's a If you put a loop in and pull tight, loop turns into twisting! Right. you can out by turning half a turn in To finish just to push of strip back through of Hmm. Can I how two strips You can that two strips axis. And gold that facing out now facing in. is with all polar caps just up and down and into Ah, that's why don't any Now look at two and corrugation from a to This chunk is building block of is from of this That looks but corrugation is just following twisting of strips that you saw Can I that from to corrugation so that motion not any or just in that saw thing! strips and push caps past twist caps to undo loops, and push across Finally, I still don't Is way to look at this? OK, into thin horizontal ribbons. look at ribbon at a You can north push down into south. A ribbon is position to and corrugations Ribbons to so split to what's going on. On right, tracks ribbon from so its not This highlights that in on position of ribbon in At ribbon just twists and up or down. Wait a This ribbon looks just wall monorail, and it's turning out. You'd finally that that was I'll play that again. that our walls and had to stay But ribbon can twist around in it's part of a way to is to build up of at a important This is corrugation Now just caps through This is of twisting can activity at At of twisting corrugations in of pushing horizontally through of Finally, show uncorrugation is now Wow. I think I'm to thing again. You right: you can turn a out without poking or it, though you can't do it for a This is should a about this stuff!
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noodles 1 year ago -
Hey, read somewhere mathematicians sphere inside Yes, true. the deal? poke hole Sure, the hole. then seems impossible! You're sphere like basketball. have understand the rules the game: sphere made elastic material stretch bend itself. puncture material destroying crease bend the surface itself, the problem? self-intersections makes easy? the halves each other. Be careful. the equator? Remember, tear crease Argh---let me either---you're infinitely then there's impossible! have crease inside sphere inside bet! easy figure let's something simpler: circle. We'll vertical the circle we the sides differently. circle other circle, where the purple sides are reversed, creating corners? course! rubber inside Remember, we're really the circle inside We the we see the different sides. yes, the vertical. have creases, itself. Fine, let me bend! we make bends the material, we'd be able curve other, each the curve line target the curve. corners altogether smaller smaller... interesting idea, really change. like corner disguise, the rules. Well, have corners impossible the circle inside Yes! You're minute. supposed believe sphere inside circle? Yes! There something fundamental curves have change were circle inside something change under allowed explain. Imagine the the rule the travels keeps the purple We'll use the direction. the left. goes the circle once, makes one the left. more complicated the sometimes be left sometimes the net after one complete some number one direction the other. The number makes the left called the curve's number. curve where there more the the left, the number negative. Hey, there net the number zero! time the net there's another get the answer: the where you're traveling direction, like due east. Let's see, since we're the be wherever the right---where we see the purple face Exactly. some these the Viewed here, like smile; these places, the be left. others the like these the be The net number increases when the passes smile, decreases when passes zero... one... three... three... we three. The number the number smiles the number see... the number measures happiness! the nice the number remains the same when curve changes rules. smiles appear disappear, balance The number smiles the number never changes. curve another curve the same number? the number the fundamental property mentioned before. the number the circles? one one smile the number one. the outside---one smiles---minus one! makes sense---on one curve you're left the time, the other the opposite. the reason circle inside change the number! wait---doesn't the same argument prove sphere inside sphere three-dimensional smile, one three-dimensional they have different numbers! quite. make complete, we general surface consider the where We'll stripes make these easier locate. Smiles are like are like domes, there are other where the surface are neither domes. They are saddles, like smiles one direction another. Near dome, the stripes Near saddle, they does change Spheres have saddles! the these features interact. dome saddle come together cancel Likewise, saddle cancel domes, like electrical charges the same get near each other. The number surfaces, then, domes saddles. number the sphere matter face Let's curves Remember circle be changed curves number corners, course. the circle be turned curve number one? Let's see: curve the circle... There. Excellent. one. here... Here we Very one? You're me every single curve number one, are course we need general method. remember the simple one curve another when bends are allowed? Yes. one the other. the one. When the curves have the same number, method be adapted bends. The waves the curve. we simpler one? Sure. We pieces the curve serve guides the We'll concentrate these segments We move the centers the guide segments their destinations the circle, Next, we rotate the guides they are lined the circle. the between? where the waviness comes We make the connecting segments between adjacent guides bulge the segments move freely each other, they remain more less parallel. see---the guides move creating bends. Correct. Here the the whole curve. The curve, blue, develops corners, the curve enough remain We have keep adjacent guides parallel we rotate them the circle. possible the number the curve one. we the guides the number one? happens when we figure eight circle. here the curve have number zero. method, others, one curve another the same number. called the Whitney-Graustein theorem. does have the sphere? the sphere circles, deformed barrel shape closed above below. we made curves more pliable them guide segments connected waves, we divide the barrel guide alternate The waviness dies the the get complicated... Then, let's single guide the the each other. Before, when pushed the poles made crease! before the crease, when the guide the middle. we the opposite directions, because we convert the the middle the ends. like belt! the middle the ends the Then straighten the belt each end opposite directions. the eversion, we need the middle the guide the center the sphere. see guide interact? Sure. see there are places where the intersect near the central the sides started are Here the whole process the guides. The move then rotate place. they require springiness. Exactly. let's guides the between them, pole the equator. the fundamental the eversion: the whole sphere made sixteen rotated copies piece. pretty complicated. Yes, the the the guide before. see pole pole? Yes. The provides flexibility between the guides their does create pinches creases, like the waves the curve we before. Let me see the whole We corrugate the connecting between the guides, the each other. We the the middle the equator the sphere. we uncorrugate. understand. there some other we'll divide the sphere We'll one time. see the pole the near the pole rather tame: the guide segments keep their relative one another, the never get very deep. closer the equator are wilder, we'll the screen see the the camera the above, apparent size does change. overhead view the symmetries are hidden the side view the left, where we see the the space. the equator, the doesn't move minute. like the under the inside convinced me impossible! Remember represented circles vertical. here the space, because sphere. Another understand the eversion progressively the surface the sphere few stages. the phase... we've pushed the each other... the middle the phase: we see the complex the equator... the end the phase, the have nearly become figure eights... Here we're the middle the center the sphere... we the phase. The sphere entirely purple. ready see the whole Here goes! were sphere inside holes creasing even circle. great---somebody make movie