SPEAKER 1: OK. So, with that, I'm going to introduce Kelly Delp, from Ithaca College. And she will be telling us about flexible [INAUDIBLE], "Playing with Surfaces."
KELLY DELP: Hi. How's my mic? Am I good? Yeah? Great, thanks.
First I want to thank the organizers. It really is an honor to speak here, at this conference, and to share this story about the project that I had with Bill. Starting in 2010, we built a bunch of surfaces. And you can see a lot of them on display in the library across the way.
So the reason we started working together, I think, is because we both really enjoyed making things and building things. So today I'm going to share this story and other stories I learned from Bill, over the time that we worked together, about projects he had with other people, making things. I knew, as a graduate student-- as a graduate student at UCSB, my adviser was Daryl Cooper. And I arrived there, I think, in-- well, I'm not going to say when I arrived there. That'll age me.
Anyways, you can't be a student in geometry and topology long before you learn about Thurston & Thurston's work. The graduate students there were reading Thurston's notes. And, you know, there's all this beautiful mathematics that Bill created.
And yet, when I came here in 2010 and started sitting in on the seminar, somehow I didn't know all these other stories about all this time and energy he put in, making things, and making things that he could share with nonmathematicians to explain mathematical ideas. So that's my goal, today, is to share some of these. the
First story I want to share with you-- oh, and anything that you see today, it's either going to be all Bill or joint with Bill, but nothing I'm going to show you-- it wouldn't exist without Bill-- Bill's influence. And that includes this title.
I get a lot of comments on this title. And I want to say, that for nontopologists, "monkey pants" is even funnier.
You know, somehow, to us, we're like, yeah, monkey pants. But--
--they're like, what? And then I forget to tell them, in some of my pictures the fourth hole is hidden, so they must be really confused. So, I had a lot of fun, going back through old files and emails that I had from Bill. And one of the files I found was this old text file-- said something like "Title for bridges."
So, in 2011, Bill and I were advertised to give a talk at the Bridges math art conference. And it was April, and Bridges says, well, we need a title from you guys. And Bill says, right, let's make a title.
So how do you make a title, if you're doing a project with Bill Thurston? Well, you sit down, he opens up a tech file, and he says, OK, [SNAP] start naming words. So, we started naming words.
And what I thought was interesting is-- just to let you know, in the actual file the small words were nine-point font and the large words about 16. So, as we started to like more words than others, the ones we didn't like became smaller, and the ones that we liked became bigger. And somehow, I don't know who contributed what, but I can tell you 100% it was joint, because he demanded that. You know, like, when we needed names for things, he's like, right-- name things, and I'm not going to-- you know, you have to. Like, tell me what you think.
So I feel like "Playing with Surfaces" might have actually come from me, because he was demanding things. And definitely know that some of these, I'm like, that's too fancy. I think he liked "Curving the Imagination," and I'm like, mmm-- I don't know.
So we ended up with this, and I think it was the right choice. Another thing that I learned from Bill-- and I don't know why I'm surprised by these things, but-- was that he had a long history of making things with people. So, these are pictures from the 1989 Geometry and the Imagination workshop that was held in Minnesota.
And this, here, is Margaret Thurston. And we're very honored to have Margaret Thurston here with us, today, in the front row. And I got to hear stories from her about these projects.
So, this has to be the first mathematical sewing class that ever existed. So some things that I've learned from Bill is that, if you want to make a hyperbolic plane, you want a low angle defect, or things get crazy. So one that works particularly well is 120 degree heptagons. Your angle defect's only about 25 degrees, and you're able to build more before things get out of control and see what happens.
Some other things. Oh, there's, like, something about mathematical constructions. They really invite people to put them on their head. Like, you have to wear them--
I saw someone's small children. They ran into the library, and they immediately picked up something and they're like, this goes here! I don't know why. It is universal, though. You go to the Bridges Conference, and that's what you see.
So, one thing that you see, here-- and I think by "here" I mean this-- I think that white paper is this. We can ask other people here today. Oh, I also want to say, like, these are all stories I learned from Bill. And I'm sure they're going to be incomplete, and some things might be wrong, because my memory is not 100%, maybe not great. So, if anyone here wants to correct me or add details, please do, especially if I get something wrong.
So here we have, at the same workshop-- so, Margaret tells me they recruited her to join this workshop. So Bill told me that his mother was an expert seamstress. And she told me today that she started sewing when she was 10. So a lot of these products really do require quite a bit of skill to do, like, technical skill, to be able to create these things.
So, here's a picture. What they're making now, this was, I think, probably the first physical cloth model of a Klein quartic ever made. So that's what they're working on.
So here's a picture with Conway and Margaret Thurston, working on the Klein quartic. Tony Phillips, Margaret Thurston, and the Klein quartic, still there. Here we see Bill has joined the picture, to discuss the Klein quartic.
And just so I don't leave you-- I didn't have an actual picture of the finished version from this workshop, but Dylan Thurston has since created one. Margaret says she saved her pattern. She never throws out her patterns. She still has them, which is good.
Because, as you can-- oh, you can't really see it very well, here. But the pattern is right here. And it is complicated. What is it, 24 heptagons in the--? Yeah. So there are a lot of them.
So what I hope these pictures illustrate is that you do need technical skill, but also mathematical knowledge is really useful when you're making these things, as well.
So, I saw these pictures. And being a mathematician, lacking skill or knowledge never stopped me from attempting anything. So I was like, I want to make one of these! And they're like, no, this isn't a beginner sewing project, Kelly. You really should start with making a torus that illustrates you need seven colors for any mapping.
So we're all probably familiar with the four-color theorem for the plane. And for any map that you draw on the torus, you need seven colors to make sure no neighborhood is colored the same. So there's your torus. And here's the pattern for the torus. You get a nice one if you use seven hexagons. That'll work.
So I talked to Bill about how I would do this, and he was like-- you know, I showed him my pattern, and he's like, well, I think to get a good torus you really want the longitude to be three times as long as the meridian. That'll give you a nice-looking torus. That's what a torus should look like. We're all happy when those are the dimensions.
And I said, OK, great. So, when you sew things, of course, the seams are on the inside So you very carefully-- and, in my defense, for this story-- I don't know why I tell embarrassing stories about myself, but I do. Keep in mind, this is my first sewing project. I'm learning how to use a sewing machine as I'm doing this. Right?
So I'm sewing on my torus. I sew everything up very carefully, inside out. And I leave a little hole. And at the very last stage, I go to turn the thing inside out.
And that is when I-- yeah, somebody's laughing, right? That is when I learned, when you turn a torus inside out, the meridian and longitude switch places.
So I promise you, if I ever need that in a theorem, I'm going to have that. That's there.
So then I thought, this is awesome! So then I made a square torus that was striped that illustrated this fact. And then I just sat there and played with it and turned it inside and out.
And so this is an-- oh, you can't read the quote! I don't know what's going on with my-- anyways. I think you can probably fill in the gaps. You guys are smart. Figure out what that says.
So this is a quote from the introduction that Bill wrote that appeared in The Eightfold Way. So this beautiful-- many of you have probably seen this sculpture, at MSRI. But I didn't know that it was installed in 1993, when Bill had been director at MSRI for about a year.
And the story he told me that was he told Ferguson about the Klein quartic and some of the beautiful properties that the Klein quartic has. So that was installed while he was director. I think it will be all right. It has to do with the screen size. Most of them won't matter.
So, my story with Bill began in 2010. I was a student in his-- well, I was sitting in on a seminar class. I was visiting Cornell for the semester. And, in February, he says, well, I'm not going to be here for a week in March, because I'm going to be at Paris Fashion Week.
And we're like, what? [LAUGH] What are you talking about? And, being the person I am, like, immediately after class, I was, like, on my computer, seeing who was on-- I'm like, oh my gosh, Bill's going to be at Paris Fashion Week. This is awesome.
So it turned out-- he told us this story-- he'd been collaborating with Dai Fujiwara of Issey Miyake. Dai was a fashion designer who had an engineering background and had read a story about Thurston and his eight geometries of three manifold and thought this would make a great inspiration for Issey Miyake's 2010 spring ready-to-wear line.
So, I'm sure many of us in this room looked-- and I know many of you are familiar with these stories. But, just in case you're not, it's too good not to be told.
So, I'm sure I looked at these, and you're like, how are these the geometries? Like, what-- you know, we're wondering, maybe, I don't know. Circle? Euclidean? Not sure.
One of them definitely looks Euclidean. There. OK. [INAUDIBLE] They're really-- I think they're really connected.
So, there's a picture of them. That jacket was awesome. He wore it for about three weeks, after he came back to the seminar. And, like, you're just like, this is a one-of--
It was an amazingly beautiful jacket. It was a one-of-a-kind jacket. And he's lecturing. Of course, there's chalk everywhere--
--all over the jacket. And you're just like-- oh! So what was the connection? And here's the connection. Bill sent Dai tons of stuff. I love this. I've given this talk a bunch of times, and I get to skip all the "what is a manifold" slides that are usually in here. It's wonderful.
So, one of the things that Bill sent-- well, first he sent tons of ideas to Dai. And he confessed to me, he's like, I think I might have overwhelmed him. So, yeah.
But one of the things he said-- these are actually images drawn by Bill. He liked the shading. He used Adobe Illustrator to draw his pictures. And I think he liked how these turned out.
So, what are these? Each of these is a link in S3. They're each a singular set, in S3. And they all have order 2, and that's the kind of poetic part, I think, that you can-- I didn't know that you could do this, actually, create a representative of a three-orbifold--
OK, let me start over. OK. What are each of these pictures? They each represent a three-dimensional orbifold. The [INAUDIBLE] side is the link that's given. I think they're all links. Are there any knots up there? Yeah, the 5:3. That one's a knot.
And the singular set has an order 2. And they all live inside of S3. And so each one of these is a representative from the particular geometry class that's shown here.
And, just for fun, because-- you know, I'm not going to-- so, if you're bored, you can try and work out why the other ones are what they are. But this is a wonderful picture from Bill's notes on three manifolds of how you get the Borromean link as an orbifold, as a Euclidean or orbifold, with singular set 2. So, I'm sure most of us have stared at this picture, at some point.
Right. So, that was one of the things that Bill sent Dai Fujiwara, was all of these links. And the designers, they loved the links. They grabbed those. And they became an integral part of the fashion line.
And there we can see some of them. And, just because it's a really good picture of-- these scarves look gorgeous. There's the Euclidean three orbifold. So I think this one's Euclidean.
I tried to guess. I had them all written down, and now my slides aren't showing right, so I can't-- I counted the components. And Dylan tells me actually they got some of the links wrong.
Maybe they weren't all right. So maybe what I was trying to do was futile. But there are.
And there are Bill's words. I think he-- I know he had a really great time, in Paris, and enjoyed the experience and thought it was good for mathematics.
So, after Bill came back-- so, of course, this was big news at the Cornell campus. And Fashion Technology heard that Dai Fujiwara had been on campus and hadn't come--
--to the fashion department. They were like-- oh-- so--
They were nervous. So, the first thing we had, we had this great-- so, Bill had been starting to build these assemblages. He was thinking about clothing design, after going to this. And he'd bring these assemblages into class.
I think I left-- I have some of them over in the workshop. This is when I really was impressed with Bill's crafting skills. Because I have spent a lot of time, in my life, building things. Like, for my classes-- you know, the sort of thing that I would do is I would very carefully build paper models of various genus surfaces, up to genus 3, for my classes, to do Euler-characteristic experiments.
And it takes forever. And I would very carefully tape things up, and they would be sort of sad. And they'd last a semester and fall apart.
And so these assemblages-- Bill, I think, had this idea about, like, oh, I want to make some assemblages. And he'd have this idea on Tuesday, and by Thursday he'd bought a laminator and a paper cutter and a riveter and has a whole workshop going. And these things are really permanent and, four years later, still work pretty well. And I was like, wow, I think too small. You know? Like-- you know. So it was very impressive.
So, anyways, these are harder than they look, these assemblages. You can get them to fit any figure, but local changes in angle measure can drastically change the global geometry. So you fix something in one place, and all of a sudden you have to fix it everywhere. And it was actually much more challenging than it might look
He also had pieces-- Harry, who just graduated, or just finished his PhD, I think, last Friday, is wearing one of-- he had a little-- you can measure the curvature of your head with these circular pieces that he had. And I just loved the bear.
So here are some assemblages. One question that you could ask is, what's the-- so, the orange one, if you're looking at, what's the space of flat configuration? Some of them aren't flat at all. But there's at least one-parameter family, I think, of flat configurations. Like, what's the dimension of that?
Oh, this was another thing. And so this puzzle's kind of important, because this is actually-- I think-- is this your puzzle, Liam?
SPEAKER 2: [INAUDIBLE]
That's the one you were talking about, right, the foam puzzle? So this puzzle might have been the start of it. It was a wonderfully meditative puzzle. It's a one-piece puzzle.
So it's this piece of foam. And you can't find these. I've looked. You can't buy them anymore. And it's got one really complicated, like, Jordan curve on it.
And, once you start it, you just keep going. And you put this puzzle together, and it's very meditative.
And so I had this idea-- what's coming next? Oh, right. So, I didn't know, at this time, that Bill had this history of mathematical sewing projects. But I was very excited by the workshop, the fashion workshop, and I was interested in the stuff we were talking about in class.
And I had this idea-- like, wouldn't it be interesting-- I wanted to make a dress. And I was wondering if it would be possible to make it with just one pattern piece. But it had to fit well and look good.
So, like, imagine you have, like, this really complicated curve. And-- because the reason our clothing fits us now is pretty obvious. Like, a lot of times the seams go along the sides of our clothing, because of the symmetry of the human body. Right? So it's very obvious why things fit and where the curvature is.
But what if you had one piece, with an incredibly complicated curve that hid all of the curvature and the reason why clothing was fitting? It would be sort of this magical dress. But, of course, could you do this? Would it develop?
And this question of developing is hard. Like, it might not. Depends upon how much negative curvature you have, maybe. So that was the question.
So, of course, before you start with something that complicated, you start with an easier question. Like, what if you just wanted to make clothing for a sphere? And, to make the problem harder, we're going to use something rigid, like paper. We're going to make paper clothing out of a sphere. How could you do it?
Well, if you had a monosphere, you might just say, well, an octahedron could dress a sphere. It just wouldn't fit very well. Bag-like clothing. It's fine. It's dressed.
So, OK, what's the point, here? So a sphere-- everyone here, I think-- well, not everyone, but a lot of us have probably heard of the Gauss-Bonnet theorem, right? So what's the total curvature in a sphere? Euler characteristics, too. We have 4 pi's worth of positive curvature for a sphere.
So, the combinatorial Gauss-Bonnet theorem tells us that that curvature is concentrated entirely at the vertex. Right? So what's the missing angle at a vertex? 2 pi over 3. If we have an octahedral sphere, a times 2 pi over 3, I hope, is 4 pi.
So all the curvature's concentrated at the vertices. So, to get a smoother, a better-fitting dress for your sphere, the first thing you do is push the curvature out of the vertices, into the seams. And you do that by replacing the triangle edges with circle arcs that will give you right-angled triangles.
So I think the circle arcs here is pi over 6. I think that's the right thing. I shouldn't say things out lout. I shouldn't even make guesses for that, because this is being videotaped.
Anyways, you make right-angles spheres, and you tape that up, and this is what you get. So we taped it up, and we both went, oh! That's not very round. That's not really what we were going for.
And so I think actually Bill was surprised by this. Because he said that, with you, Margaret, they'd made similar things. And with cloth, because cloth isn't rigid, you get really good spheres with patterns like these. But with paper, they're not very satisfying.
So I looked at it, and I said, well, clearly, Bill, we need to make the seams go like this. So I said, make the seams go like this. And I made this motion with my finger.
Oh, you're going to fix it! Maybe. Display. Oh, no, go back-- Display. Oh yeah, yeah, one of those. Do a different one of those. No, the 1,024 by 768, or just pick one. Try a different one.
This is going to mess up my mathematical notebooks, Dylan. It's going to be disaster. [LAUGH]
SPEAKER 3: Can I ask, while we're waiting--
KELLY DELP: Yes!
SPEAKER 3: That is a Reuleaux triangle that you have just constructed?
KELLY DELP: A what triangle?
SPEAKER 3: A Reuleaux triangle.
KELLY DELP: I don't know what a Reuleaux triangle is.
SPEAKER 3: [INAUDIBLE] constant width.
KELLY DELP: Yes. Yeah, it's a circle arc. So, I mean, it's just a triangle, and the edges have been replaced with arcs of circles. OK.
So anyways, I said, we've got to make the seam go like that. And I swear, it was about 45 minutes later, I'm taping together this. And this was incredibly round and satisfying. And I know-- 45 minutes might be a slight exaggeration, but I know that the fat triangle and that guy were taped together in the same afternoon, completely. And they take a long time to tape.
So what happened? How did Bill make that? And this is the answer. So, the trick was, we need a long, meandering curve that's going to cover a greater portion of the area of the sphere but has the same amount of planar curvature as the circle arc. So we want a process to, 1, make a meandering curve, and to, 2, bend the curve the right amount.
So how did Bill think about curves? He thought about the angle function. He thought about the angle of the tangent vector.
So, first thing, define what your angle function should be. And if you're curious-- and I think you can see this. It's not really that important. There's the angle function-- pi times 9/16 cosine x minus 1/9 cosine 3x.
So we all talk about levels of understanding things. I think after meeting Bill, I think my understanding of sine and cosine and composition of functions-- I think I'm about a 2. And I think he's about 50 or something. Because we all know that you can get anything, but--
So, what's going on? So, here's my angle function. So this red arrow, here, just represents the tangent vector for the red point that's traveling around the curve.
So, how did he make the long, meandering curve? He just thought about, well, what direction do I want my curve to happen? And then you just make a unit vector field, to go along with that angle function, and then integrate. You do NDSolve in Mathematica, and then you have your nice curve. And that's what happens.
And you can see, here-- so, what are some properties about the planar curvature? Well, clearly-- so, what is the curvature? The curvature is the rate of change of that tangent vector. And you can see, as it swings, that's where we get the peaks and the valleys.
Oh! So, my curvature I want to think of as sined curvature. Whether I'm turning clockwise or counterclockwise, one's positive, one's negative.
So, some things to point out, here, that you notice about the curve. My claim is that this curve has total curvature zero. And it seems pretty obvious, when it gets done drawing the picture. Maybe I can-- this is tricky. It's not-- pause. Oh, I just slowed it down. Anyways, let's draw it out.
But the way to see that it has-- I love this! This is a great talk for, like, undergrads, because it just uses calculus and the fundamental theorem of calculus.
So you take your starting vector and your ending vector, and you see they're the same. So, as long as your angle function starts and ends at the same place, total curvature zero. So, beautiful. Just pick something periodic that does what you want, and you have it.
So, now, how do you bend? Any guesses? What's, like, one thing to do?
You just add a linear function to your angle. I mean, it doesn't have to be linear, but I like the linear ones best.
So here it is. You just add a linear function, and-- slower. There we go. So now you can see that the total curvature of this curve, you take your start vector and your end vector, and you take the difference, and that's going to be your total curvature
So it's very easy. You just take the linear function that's going to give you total curvature pi over 6. So, we're done with you now. [? Vamos. ?] Oh, no. Got to go this way? Yeah. OK.
And here, you can see that, if you want more bend, you just add-- you just change your linear function, and you can get more bending. And you can see the bending and the spreading of the fingers.
So here's a very good pattern. I think this is, like, version 2 of the sphere. It got good really quickly. And I learned how to tape on the inside, so the finished product looks a little bit better. I can do about, like, 90% to 95% taping on the inside.
And I have to say, it's so incredibly satisfying to tape one of these up, because they start flat, with the fingers spread, and all you have to do is-- you have no choice. Like, you start to tape it up, and you align the seams, and it beautifully curves into a sphere.
And it was so good, that I think I had trouble explaining to people what was going on. So I wanted this intermediate version made, so you could sort of see the shorter curves and you could see more of the octahedron, and it only curved less.
Of course, there's nothing special about the octahedron. You could do this with any platonic solid. Here's a tetrahedron. You only have three triangles meeting at a vertex. So, to get right-angled triangles, you'd need more circle arcs, more bending.
So [INAUDIBLE] just--
I just learned about this this morning! [INAUDIBLE] told me about this. And I haven't read this yet. I've only looked at the pictures. But people have been asking me, like, did you guys talk to Adidas? And I'm like, no, I had no idea.
So I know, in 2010, when the World Cup ball came out, Bill sent me a picture and said, Kelly, did you see this? This kind of looks like our stuff. But I thought this was very interesting.
So it has a practical application--
--I guess-- maybe. Oh, I didn't tell you how-- OK, so, maybe, I want to tell you a little bit more about how this cutter works. So the cutter works like old printers used to work. Like you have--
Like, you really just put this piece of paper down on, like, this tacky, plastic sort of stuff that you feed into this machine. And it has a little tiny knife.
All right. So, the pieces were all designed in Mathematica. Mathematica, it's really easy to take a PDF from Mathematica and put it in, say Adobe Illustrator. Adobe Illustrator-- when you used to be able to buy it, and now you can't-- but Adobe Illustrator has a plug-in that is compatible with the cutter. And it feeds the instructions from your PDF to the cutter. And it runs around, and it comes out, and then you have your piece.
That sounds way easier than it actually is. It's rather fussy. The knives get dull. Sometimes it has to go around twice, to get good cuts. It's hard to peel the pieces off without bending them.
So, I can't stress enough that making things really takes a lot of time. I think to tape up a whole sphere like this well, I mean, it's probably at least two hours. I got faster, but the first time it was about that long.
So anyways, this is what comes out of the machine. You have this thing cut out. And so then you take out your piece, and that's what's left.
And then Bill looks at it. He's like, oh, of course! We have the curves, right there, for negative curvature. So he just goes over to his Adobe Illustrator. If you just look at it from the other side, the fingers are going in. So that's-- you're going to be able to have models of hyperbolic planes and things.
So, you get really nice, smooth models of the hyperbolic plane. I think this is-- yeah, these are 120 heptagons. 1, 2, 3, 4, 5, 6-- yeah. 120-degree heptagons.
And this is actually the picture I have of the monkey pants. And you can see, it's a great picture, but you can't see the fourth hole.
So, we had negative-curvature pieces, positive-curvature pieces. And I said, well, I want to build anything, so we need mixed curvature. How do we make that happen?
And, of course, that can be done. But, before we do that, I think I want to do an experiment. I tried this experiment at [? launch, ?] and I want to try it now. So this is audience participation, here. OK!
So, imagine you're going to have a closed, smooth surface in R3. And if you don't know what one of those is, just think of maybe, say, a human body or the surface of anything-- like, surface of a torus. And you want to pick colors. And you want points of positive curvature, like a shoulder, to be one color and points of negative curvature to be a different color.
So, everybody, right now-- don't say it out loud-- pick a color for positive curvature. OK, everybody have their color? You want things like shoulders to be a certain color.
OK, now close your eyes. OK, you're much better than my students. Raise your hand if you picked red, for positive curvature. OK, look around.
Right? OK? I don't know why this is. But, like, one of the things Bill-- we were always-- you know, he was very concerned about, like, well, what color should this be? And, you know--
So, when you make the matching four-per vertex system-- wait! Somebody who didn't pick red, what color did you pick?
SPEAKER 4: Pink! [INAUDIBLE]
KELLY DELP: Pink is red! Pink is the same as red.
But blue, blue is the exact opposite. How many people picked blue? Just curious. Oh my gosh! That was more than I thought. I think reds win, but, wow. You picked blue?
SPEAKER 5: I picked sky blue.
KELLY DELP: You picked sky blue. That's blue.
SPEAKER 5: Yeah.
KELLY DELP: [LAUGH]
That's just like pink and red. OK, so what's the trick to matching pieces that are compatible? The only trick is, you need the curves to be the same length. And that's easy to get, when you're just defining things with angle functions and parameterized by arc length. You just bend by different things.
So, what's neat is, if you tape two red triangles together, you get positive curvature bending. It's like the sum of the two seams. If you tape red to yellow, here, yellow has no bending, so you get less positive curvature bending. If you tape red to blue, you get just a little bit of bending, because a pentagon has five sides, so there's less negative curvature per side than positive curvature per side.
So there it is. Notice that the total curvature that a triangle contributes is equal to the total negative curvature that a pentagon contributes. So, we started making things. At some point, we moved to foam, because paper was really not forgiving. To tape up the paper, you use just regular--
Like, actually, I'm kind of picky about my tape. I don't like scotch tape, because it's cloudy. I have to find that really thin, clear transparent tape, which is actually kind of hard to find.
So, at some point along the way, we moved to foam. Bill had bought a bigger cutter, and I had come in with this really-- you buy this really thick foam at craft stores. And he had figured out, these cutters aren't made to cut foam, but they will, if you force it.
So the pieces evolved. And I would build things like Archimedes solids, and Bill would build faces that were pretty awesome. But, you know, I really wanted to build a torus. And building a torus was actually surprisingly hard. Oh, I never finished my story why foam was better than paper.
So, with paper, you think something's going to work. And you spend hours, taping up this pattern. And then it doesn't work. It just-- the geometry, you're like, oh, that's not going to work. And then there's nothing you can do but throw it out and start over.
And with the foam, if you used artists' tape, you could at least just go back to a certain point in time and reconstruct, because the artists' tape would come off, whereas the clear tape would-- you're just done.
So, some of the problems with constructing a torus. Why is it hard? So this torus was made with a four-per vertex system. So it was just like the slide that I just showed, not too long ago.
So any smooth surface in R3 has at least 4 pi's worth of positive curvature. And these aren't technically smooth. I guess along the seam, they're not smooth, but they're really good approximations. So they sort of behave like smooth surfaces.
So each triangle contributes pi over 2's worth of positive curvature. So, if I want to make a torus, I need to use at least eight triangles. And, of course, a torus has total curvature zero, so, for every triangle I use I have to use a pentagon.
So the absolute smallest torus I could make would have at least eight pentagons and eight triangles. The smallest number I was able to construct is this sort of star torus, here. And I really feel like it's cheating, because it's actually under stress.
So you have this area problem, where, because the area of a pentagon-- if you have equal side lengths, the area of a pentagon's much bigger. So you want your negative curvature bit to have less area. And that's just not going to happen, with these pieces.
To try and get a better torus-- this is a three-per vertex system that Bill made. That was hanging from a tree in his backyard. There was this great branch on the tree, where things would get photographed. Because they looked good.
So, the three-per vertex system, you can see that torus is much round, but it has a lot of pieces in it. Like, that took a very long time to build. So there's another version of monkey pants.
At some point I, think a lot of people in this room probably got kits of these. We were trying to-- Bill very much wanted these to be developed into a toy. He imagined kids running around with hyperbolic crowns on . And so, yeah.
So these were very similar to the beta version that we sent to a bunch of people. Oh, You can't see that when I do it with a mouse.
So, what are these medallion things like? Well, 1, they're just pretty. But another thing is, to add more [? genus ?] quickly, the idea was that you could add these circles and then attach them to one another and increase the genus in an easy way.
So, as a result of the beta testing, we discovered that they were just a bit too fussy. For these pieces that-- at some point they got the name "zippergons," I think we called them, because it looks like they zip up.
If you had two negative-curvature pieces, they'll sort of stay together. There's a very weak gripping. But, because of the nature of the positive curvature, the fingers spread. And the triangle-to-triangle, they just wouldn't stay together. And even triangle-to-pentagon, they just kind of fall apart.
So we were interested in developing, and particularly Bill was interested in developing, a no-tape system. And he got kind of discouraged. But then he had this ingenious, puzzle-like design.
And this is just really a feat of engineering. I can tell you, this foam is incredibly flimsy. But yet, these seams, they're not as easy as they look.
And I know some of you have been playing with them a little bit. You kind of have to put them in there. And then, after you put it together, you have to massage them in place.
But some of the seams that he came up with were quite good. I mean they'll stay together. And, for instance-- oh, you guys can't see.
Some of the things, like these Archimedean solids, they'll stay together with, like, light tossing. You know, you can toss them around a little bit, and they won't fall apart.
So then we just started playing more. The double pieces got designed. I was really bothered by the fact that we hadn't built what I thought was a good torus.
So one thing you could do is you can make triangles that were twice as big and have half the bending per side. So they bend slower. And they fit together.
And then we started to get some really good tori. And these are also just bigger, because they're hexagonal pieces. And, same principle, the lengths of these pieces--
This is a really nice torus. I unfortunately don't have this one anymore. And I had trouble recutting them. But these are great. The bending along the top seam, which is twice as long, is equal to the negative bending on the inside. So you got really nice round torus.
The soccer ball is interesting. These brown pieces don't have equal bending on each side, because you have brown-to-brown seams, they're-- it was made to be totally smooth. This isn't really a compatible set of pieces. You just had a soccer ball.
My favorite, which I was wearing, here, today. These are really fun, these aliens. It's a great way to make a [? high-genus ?] surface. Right? You start with a five-punctured sphere. You can attach them to each other, attach them up, make lots of different things.
There's a really nice genus-2 surface that was constructed. And-- any questions, thus far? I think I'm going to be done early, but I think that's OK.
So, not to be outdone by [INAUDIBLE]-- which I know is a sentence that's really terrifying to utter. He had this great last slide about Bill's foresight into the future of mathematics. And I just wanted to point out that that wasn't really limited to mathematics.
So this I found in one of my emails. And I was really surprised to find this, because I didn't remember it. And I think what happened was, I saw that, and I'm like, mmm, I don't know if I see earrings, there.
And I was accused of not thinking big enough, at many times. I'm like, I don't know if I see earrings. Which is kind of funny, because these earrings that I'm wearing today-- and I just talked to Henry, and he said he designed these in 2012. And I think it's the exact same picture that was made of foam. And these were 3D-printed out of metal, out of silver. So Bill was also right about the direction of mathematical surfaces and constructions, as well.
So, thanks for your attention.
SPEAKER 1: So, thank you, Kelly. Are there any questions for Kelly? [INAUDIBLE].
AUDIENCE: So when you did the first, you said you very quickly [INAUDIBLE] fingers, or maybe flat or curved?
KELLY DELP: Curved. I-- you know, I don't know-- for me, when I looked at it, I mean, it was exactly like I just said. I said, Bill, make it go like that. And then he did. I don't know what he was--
I mean, it happened very quickly. So I don't know if he was thinking fingers or what. But that was the first curve that came up.
AUDIENCE: You said one of the goals of this was to [INAUDIBLE]. Do you have reactions of, like, nonmathematicians to these gadgets, beyond [INAUDIBLE]?
KELLY DELP: Yeah, they like them, in general. So I've taken them to-- so there's this-- I don't know if that counts as nonmathematicians, but there's this Bridges math art conference, which is actually, I think, a beautiful community. And it really is-- mathematicians go there, artists go there, computer science go there, crafters go there.
And they're definitely-- I thought it was going to be mainly people like me, like, mathematicians with a crafting hobby. But that's not the case. There are actual artists. And they were very positively received there. They were.
I forgot to-- this guy is really pretty. So this is a one-finger model, in terms of a multifinger model. So I think originally we both thought-- which is not surprising to me-- that the number of fingers mattered. Like, more fingers would be better.
But, like, no, it doesn't matter. You can actually get really good results. As long as your curve covers a large portion of the area, that's the only important thing.
So this is actually-- it looked cube-like, but it's actually an octahedral model. But what happened was, a lot of area got concentrated at the vertices, so it smushed out to the [INAUDIBLE], which--
I think this one was the one we were like, oh, yeah, of course. It's not the number of fingers, it's just where the curve goes, relative to sphere. So that was an interesting sort of mistake that looks kind of cool.
AUDIENCE: How many [INAUDIBLE]?
KELLY DELP: In this one that I just held up? This is eight triangles. It's--
KELLY DELP: The what?
AUDIENCE: [INAUDIBLE] the minimal number of, uh--
KELLY DELP: You can definitely-- actually-- four. You can do a tetrahedron. I mean, you could do, actually, one. There's, like-- you can do one with a sphere, really easily. Just think of, like, a flower. Like, actually, just peel an orange. Peel an orange, and then you have--
In fact, I think I-- um-- hand on a sec. Oh, I keep losing my arrow. This is OK, though, because I didn't take up all my time. Ah, there it is. Slide-- don't skip slide. Play
There you go. Just peel an orange. And you can do sphere with one piece, for sure. In fact, as we were working on this project-- I find this really interesting. I'm afraid to mention it, because you might ask me questions about it. And I don't fully understand it. But I'm going to anyways, in the spirit of sharing ideas.
So there's this question, which I think is a really good question. What's the minimal number of Euclidean patches you need to build a particular surface? So it's not too hard to imagine one-patch tori.
But then you have this [INAUDIBLE] what about a hyperbolic surface? What about a closed hyperbolic surface that you want to well approximate with one Euclidean patch? How many patches would you need?
And so one of the things that Thurston talked about in his seminar was, he reckoned you could do it with one. Which is kind of surprising. And that's what these triangles are about.
So, if you have more questions about that, I'd be happy to discuss it with you not in front of everybody.
Because it's kind of cool, but I don't fully understand.
AUDIENCE: Another kind of construction is origami. That's It's been influenced enormously in the last couple of decades by differential geometry and so forth. Have you explored, at all, the connections between what you're doing here and origami?
KELLY DELP: I haven't, no. I know Bill did. So some of the things hanging up in-- he was asking about origami, and have I explored any of the origami connections between this and origami, and, like, no is the answer.
But, around the time we were working on this, Bill was asked to review the video Between the Folds that I-- is that the Demaine brothers? Does anybody know? Anyways, he used the cutter to score paper and make interesting figures that are folded paper and illustrate, you know, starting with flat pieces, and scoring in proper ways, so that, when you sort of just barely fold them, they're negatively curved. And it's interesting, and you can see them over in the library. The one that's hanging down, that was another thing he made.
SPEAKER 1: Are there any more questions for Kelly? OK, let's thank Kelly again.
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Kelly Delp of Ithaca College describes a process, inspired by clothing design, of smoothing an octahedron into a round sphere, June 24, 2014 at the Bill Thurston Legacy Conference.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.