Embedded - 313: Where the Paper Knows It Needs to Fold

Episode Date: December 13, 2019

Robert J. Lang spoke with us about origami, art, math, and lasers. Robert has many origami books, here is a subset: Origami Design Secrets: Mathematical Methods for an Ancient Art (the one we talked about most, has the hummingbird crease pattern) Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami (his new one, a textbook!) Origami in Action: Paper Toys That Fly, Flap, Gobble, and Inflate (not a theory book, just fun folds) Origami Sea Life (not mentioned but probably the book Elecia will be getting next) Robert’s website langorigami.com is full of neat goodies: Gallery Origami design software including a pointer to the Origamizer by Tomohiro Tachi Crease patterns! Suggested other books: Tomoko Fuse's Origami Boxes: Beautiful Paper Gift Boxes Origami to Astonish and Amuse by Jeremy Shafer Origamido has a number of books. Robert uses Origamido paper but it is unobtanium to most people. Unless you are in Maine.   (Note: book links are affiliate links, we get a little kickback if you buy from there.)

Welcome to Embedded. I am Alicia White. I'm here with Christopher White. Our guest this week is Robert J. Lange, one of the foremost origami artists and theorists in the world. We'll be talking about art, math, engineering, and origami. Hi, Robert. Thanks for joining us. Howdy. Could you tell us about yourself? Sure. I am an origami artist and consultant. My passion for origami has extended my entire life.

I started folding at the age of six. When I was in high school, I also started getting pretty interested in mathematics, and that led me to a scientific career, studying engineering and physics and lasers and the like. And after a while, I had the idea of putting the two together, math and origami, and each has enriched the other. And so now I've been a full-time origami artist and sometime consultant on math and technological things relating to origami for about 20 years. I'm a little jealous. I think the listeners have heard me talk about origami, but not on the level you do it. We'd like to do lightning round where we ask you short questions and we want short answers. And if we're behaving ourselves, we won't say why and how. And what about step 47? What's your favorite equation? What's my favorite equation? It varies over time,

but my favorite right now is the equation that gives the fold angle as a function of length along a curved fold. It's a beautiful equation, and it comes from a solution to a nonlinear differential equation. Around a curved fold? Yes. Yes. All right, well, we can just talk about methods for solving nonlinear differential equations now for the rest of the day. No. Yes, that's definitely going to bring in a big audience. It's fine. It's all about me, really. I mean, they know it's the spread sink fold, which is where you flatten a middle point by spreading the edges. And there's just something very satisfying as you spread the edges and that point disappears into smooth flatness. I'm so glad you didn't say closed sink. I use a lot of those. I inflict them on my fans,

but they're hard. So that makes it difficult to make them a favorite. This next question is not mine. It's definitely Alicia's. How long should practice origami pieces be allowed to sit on the table before being disposed of follow-up how full of your house as long as you like them um or actually this depends on whether you have roommates or not if you're answering only to yourself hey keep it there forever uh but you know if a roommate's got to look at this crumpled or semi-crumpled mess, then you might show a little consideration and get rid of it after a little while. What's your favorite paper? Origami dough paper. It's paper that's made by an origami artist and paper maker, master at both Michael LaFosse and his partner Richard Alexander. And it's thin and strong, takes a crease wonderfully, has a beautiful texture. Just everything about it is fun to fold. All right, one more. What is your favorite laser wavelength?

Favorite laser wavelength? Boy, that's tough because I've played around with so many. But I'm going to have to say 980 nanometers. That's the wavelength of pump lasers. It was my company's main product. It was what made my company successful when I was doing lasers. And it was what allowed me to become an origami artist full-time professionally. So I give it a lot of points. All right. So as you mentioned, you did start out in tech with semiconductor lasers and

optoelectronics? That's right. Which everyone knows what semiconductor lasers and optoelectronics are, right? Well, I do. All right. Maybe we should have you explain that before we get to my origami questions. So just really quickly, a laser is a device that produces very pure, concentrated light. They can be made from many materials, but you can make them from certain kinds of semiconductors like microchips. and that's the kind that I worked with. And optoelectronics is just a field of everything that mixes optics, light, with electronics, electricity. And the cool thing about semiconductor lasers is for a long time you had to have, you know, a big apparatus, right, with flashbulb, pump things into these giant lasing mediums. Cooling things. Getting them down to diodes and things was a big apparatus, right, with flash bulb, pump things into these giant lasing mediums. Cooling things.

Getting them down to diodes and things was a big, big advance, right? It was huge. But, and the kind of funny thing is the first semiconductor lasers actually came along just a few years after the first lasers of any sort back in the 1960s. But they weren't practical devices long-lived robust and so forth until getting somewhere into the late 1970s early 1980s and of course and now they've taken over they're everywhere taken over the world and they're in laser printers. They're even in now some of the smartphones used to do facial scanning and the like. So they're quite ubiquitous now. What was the shift? I mean, what made it possible to go from non-robust to everywhere? It was a lot of small advances. Probably one of the largest,

one of the most significant, was the development of a thing called the double heterostructure, which basically boiled down to people realized if they made particular sandwiches of different kind of semiconductor lasers, that when you powered them up, you could get all of the electrons and holes, the electrical carriers, to fall into this very, very thin slice of semiconductor where they'd recombine and produce light. And that made them hugely more efficient than they were to begin with. And then after that, it was just a lot of manufacturing iteration over the years to find out the reasons that made them die quickly and eliminate those reasons,

make the materials purer and the crystal structure have fewer defects and all these things piled up to the point that they eventually became very robust and long-lived indeed. One of the interesting things I remember when I was an undergraduate, one of my professors who was one of the inventors of one of these big ion lasers, the big tubes with high-voltage power supplies. And I remember him talking about how even though people could make these semiconductor lasers, there was just no way they'd be able to get the price down to match the price of the helium neon, the glass tube lasers that were in grocery store scanners,

because those could be made for as little as $30. And of course, now the semiconductor lasers are made for pennies. Okay, so you were doing this, your patents and lasers and optoelectronics and all of the fun stuff. And then you were doing origami on the side? Or was it just sort of an after-work, play-with-your-hands hobby? Well, it was on the side or was it just sort of a after work play with your hands hobby? Well, it was on the side. I would say it was a much more than a hobby. It was a real passion. So, but yes, it was after work nights and weekends. Um, I'd been doing origami my whole life started by folding designs from instructions by other people, but after a few years,

started creating my own designs. And that was what really hooked me. The creation of new things that had never existed before. And so I did that, and I wrote up instructions for my designs, and I wrote several books, what I call recipe books, collections of folding instructions for some of the things that I'd come up with over the years. So this was a real passion, but it wasn't a job. It was, I guess, a hobby. I mean, it was a hobby that made money. So, you know, it's more than a hobby then. Yeah, although books don't make a lot. Origami books don't make a lot of money. So it brought in a little extra cash on the side, but it wasn't a big moneymaker. And that wasn't my motivation. Actually, it's never been my motivation for doing origami. It's wonderful that now it can be my full-time job. But even then, I wasn't writing

books for money. I wrote books because I had something to say. And what I had to say was, here's my creations. Here's how to fold them. Here's how you can fold things like this too. Was it something that you, for a long time, had thought, wow, it would be great if I could make this my primary career or did it just happily slowly occur? It was never in my wildest dreams. In fact, when I was in grad school, my office mate asked me, he saw how passionate I was about origami. He said, hey, Wilfred, if you had a choice between doing origami and doing lasers, which would you do? And I said at the time, oh, I would do origami. But it never remotely occurred to me that it would really become a full-time occupation.

And it wasn't really until I was sort of into it full-time, after I left my laser job, that I realized, hey, this origami can be a full-time occupation. The main motivator for leaving lasers was to write my book about how to design, Origami Design Secrets, because I've been thinking about that book for about 10 years. I wanted to write it, but I had come to the conclusion that to do it justice, I needed to work on it full time and have nothing else interrupting or competing with the brain cells that I needed to write that book. And so that was the big motivator for quitting my job, go write that book. But then once I started doing that, I also started following up on all the leads and opportunities that I had let go by when I had a full-time job. And after a few years found that I had enough opportunities, enough work in the laser and the origami field that I could just keep doing that forever.

And so kind of got the best of both worlds. I got to do what I loved and get paid. And I think anyone who can do both of those is pretty darn lucky. What do you get paid to do? I mean, there's the books, but as you said, they don't make a lot of money. In aggregate, over time, they make some. But do people pay you to fold beautiful things and you sell them as art? Or do people pay you to design new things or to do performance art? I don't know. What do people pay you to do? We're doing origami art for advertising where people wanted origami imagery to advertise a product.

If your product is not naturally photogenic, you think, how can I make it photogenic? Let's say, you know, I'm selling shares of stock and well, let's or some banking product. Let's show you the things you can you could do with all the money you make from using banking product. Let's show you the things you can do with all the money you make from using our product by folding them from that money, that sort of thing. So that started off. But there's always been private art commissions

where, yes, someone pays me to create a beautiful piece of artwork that they're going to display in their home or something like that. In recent years, it's been a lot of consulting, both for private companies who are developing a product that somehow involves folding, and especially in recent years for government-sponsored research taking place at universities and national laboratories where the object of the research somehow involves folding. So like solar arrays for satellites? Yeah, solar arrays, telescopes, even manufacturing processes that involve, for example,

folding a membrane around a composite material as part of the manufacturing process of that composite, creating honeycomb structures for internal strengthening of panels in aerodynamic structures, where you'd like to create that honeycomb by folding sheet metal or folding carbon fiber composites in some way. So in these applications, people generally are not interested in a beautiful art object. They don't want a bug. They want something functional, something more geometric. But the folding and the mathematics of folding doesn't know or care what you're using it for. So we can use math to create beautiful animals. We can use it to create extremely stiff, strong mechanical structures.

You mentioned your book, and now you've mentioned math. And I know with my origami stuff, usually when I think of an origami book, it's about 50 pages with a few different models and creases. And they say, you know, turn it, repeat this fold on eight different points. And I just want to make sure that people understand that your book's a little different. At 750 pages, it has quite a bit of math and not like wimpy math, but actual math. No, no math is wimpy. All math is beautiful. All math is beautiful. But like, let's see, page 100-ish, you introduce the idea of a bookworm, an origami bookworm that travels along the creases and measures how long the creases are in order to figure out if there's any wasted paper that could be used to create other details. One of the things I tried to do in my design book

and tried to do more in the second edition was to actually minimize the amount of explicit explicit mathematics. And so instead of writing an equation that says, you know, let f of x be the metric geodesic between points p sub 1 and q sub 2, I tried to actually use more intuitive analogies like a bookworm that's crawling from one point to another. Because even though formal mathematics, equations, and algebra is incredibly powerful, not many people know that, and a lot of people are actually put off by it. So I tried very hard in the design book to use as little math as I could. And going from the first to second edition, I actually took out a chapter of explicit math to instead put in more less obviously mathematical notions. So the thing is, math is much more than arithmetic and algebra and equations. It's more generally the study of

patterns and relationships. And so I the study of patterns and relationships. And so I tried to describe patterns and relationships in ways that people who didn't have much or even any formal mathematics could still appreciate. And that is very effective. I can look through the molecules and the trees and the circle packing and say, oh, wow, that's some nice math. But it's true that there aren't equations. It's a matter of starting to understand how the geometry works out. Yeah. So, towards the end of the book, the most powerful method for designing what I call polygon packing, there is a formal mathematical description, but it's much easier to understand. It's just like fitting shapes together, like putting together a

jigsaw puzzle. You have squares and hexagons and rectangles and T-bars, and you just need to put them together so that they fit. And people understand that and can do that even without much math. So origami is really just an optimization problem. I wouldn't say it is an optimization problem, but there are optimization problems sprinkled throughout origami. And that actually establishes some pretty neat connections to pretty sophisticated math that most people don't ever need to deal with or care about, but kind of mathematical nerds really like. Because optimization problems very often include some of the hardest unsolved problems in math and computer science. I've talked on a previous show about curved crease origami, and we were talking to some geophysicists, and we kind of geeked out about how that is seen. The same principles are seen in geologic formations. But I haven't found good ways to mix curved crease origami with straight folds.

Yeah. And that's not just because you haven't found it. I don't think a good description of that exists. And so what I'm working on right now, even just before we started this interview, is a book on the mathematics of curved folding that tries to bring together all the relevant equations and the like. But kind of presented in a framework that grows out of straight crease polyhedral folding. And in fact, I originally intended what I'm working on now to be the final chapter of the tessellation book, because there are tessellations that use curved folding. But as I was writing this chapter, and it got up to about 150 pages, just that chapter, and it became evident it really needed another 100 or 150 pages to do it justice. I said, let's pull it out of that book and make it its own book. And so that's what's going to happen. And I'm working on that now with it's also going to be a collaboration working with Eric Domain at MIT and some of his students

and colleagues. Eric Domain wrote the Huffman curved folding essay that was part of the reading in the previous podcast about this. And I also liked Mitani, one of the Japanese artists who does a lot of curved folding. He has software. And in fact, there are many pieces of software for different styles of folding. Do you use any of these? Yeah. Um, I use, uh, uh, mostly I use the software that I've written, um, which I've posted on my website and, and various other people use. Uh, so the tools I've written are, there this one called TreeMaker, which I actually started and published back in the 90s. And it kind of kicked kicked open the door of using software to design non-trivial origami.

I also wrote a program called Reference Finder that helps find folding sequences for particular points and distances. And I've written a software package called Tessalatica that for about the last 10 years has been my main tool for analyzing problems with both straight fold and curve fold design. It's kind of a package of programming language subroutines. So it doesn't have a terribly friendly user interface. But once one gets up the learning curve, it's quite powerful. I used it for the thousand or so figures in the tessellation book, and I'll be using it for many of the figures in this curved folding book.

What are some of the figures you're going to be making with the curved folding book? Oh, well, there are figures showing the relationships between vectors, different vector frames on surfaces. There's a set of three vectors called the Frenet frame. There's a set called the Darbu frame. So these figures are mostly mathematical plots, but we'll also be computing and rendering some art pieces. And we'll be doing sort of reconstructing mathematically some of the Huffman work, reconstructing and these creating computer models and renders of some of the curve folding designs I've done. And so kind of a mix of technical illustration of mathematical concepts and some imagery, computer generated of art pieces, as well as photographs of actual

curved fold art pieces. With curved fold, you often need the curves to be very precise. Do you use a laser cutter to do the pre-creases? I do. And in fact, that was one of my main motivations for getting a laser cutter and exploring it when I first started doing that about 10 or 15 years ago, that I could compute a curve folding pattern and then print it out and then try to fold by hand along the printed lines. But one of the surprising things about curved folding is it's just unbelievably sensitive to small errors in the curved. I mean, just the slightest deviation from smoothness creates very visible buckles and imperfections in the curved surface. And it was frustrating to not be able to make the physical object as pristine and precise

as my computer model said it ought to be. So then I started using a laser cutter to score the folds in and that gave the precision that I needed. Some of your straight folds have gotten very complicated and what's it called package folding where you have to make all the creases and then you can only put it together after you've made all the creases together. Is that the right word? Yeah, the concept you're describing is, we call it a collapse, where you pre-crease all of the creases,

so you pinch them so that they're going the right direction, but you don't fully fold any crease you leave the paper mostly flat but but covered in these little pre-creases there that where the paper knows it needs to fold a valley fold over here and mountain fold over there and so forth and then you have to bring all those creases together at once that's what's called the collapse, because you're kind of collapsing the square down to this much smaller, very complicated shape. And a surprising development, a side effect of the development of mathematical methods of design, is that most mathematical designs don't have a step-by-step folding sequence. It's actually very rare in the larger mathematical landscape of possible folds. And it's surprising because, well, every origami book, you know, for 100 years showed a step-by-step sequence. And people just naturally thought, well, every origami design should have a step-by-step sequence

because everything we'd ever seen did. But it turned out that the people who were designing origami, because they were developing their designs by step-by-step folding, whether intuitive or experimental or so forth, it was always step-by-step what they were doing. So they were only discovering designs that have a step-by-step sequence. Once we started doing mathematical designs, we could design what the finished piece had to look like, but there was absolutely no guarantee, and usually not the case, that there was a step-by-step

way of getting from the beginning to that last step. And so most of the time, the only way to make those patterns happen is a collapse. Pre-crease everything and then bring all the folds together at once. Do you use your laser for doing the pre-creases on the collapsing ones too? Oh yeah, yeah. Basically anything that needs to be pre-creased, I would much prefer to use the laser cutter to score the folds, to define the folds in the paper before I do the collapse. And so whether it's curved or polyhedral or flat folded, if there's precreasing involved and the finished appearance can tolerate the marks that the laser

cutter will leave behind, then I'll use it. There are some patterns I'll still fold by hand because I don't want to see the visible marks left behind by the laser cutter. Chris is laughing at me because this morning I asked him what he would use a laser cutter for. I was trying to convince him that we should get one, and it was his idea. They are marvelous toys. I highly recommend them. You mentioned the order of folds and how important it has been historically, but with this collapsing method, less so. I find as a beginner folder that sometimes it's very

irritating to discover the fold order is not optimal. I had one this morning where it made me do a blintz fold, which meant I need to know the center of the paper, but it didn't let me fold the paper in half. And then later it made me fold paper in half. And I was like, okay, there was an easier way to do this. Yeah, that's just sloppy. I hope it wasn't one of my designs. No, it wasn't. I mean, when we write a book, when we write a recipe book that has step-by-step folds, I think this is pretty true of most origami authors, that we do try to find sequences that are efficient and even interesting. So where you do in the early steps, you create all the reference points that you need later on, and you make extra folds if you need a reference later on. You don't make reference folds that would leave marks in visible parts of the paper where you don't want them. So we do work on coming up with a sequence.

But I guess there's both different levels of success at the goals of making the sequence useful. And it's also subjective. To one person, the sequence might be very, very long and tedious because you're making all sorts of little references along the way, and someone else might say, no, I can skip all that. I just know how to make the fold that I need at the very end of things. It is tough. I mean, do you do all the little little tweaks as you go along or do you save them all to the end? So you have a, if like you have an elephant, do you curve its trunk when you're working on its trunk or do you wait until the end? Because then at least you end up with a basic elephant

before you start messing with things. Exactly. How did you, how did you go from following plans to creating models? I mean, you mentioned the mathematics of it, but did you start out by just tweaking legs and changing things a little bit? Yeah, the math came much later. I started trying to change things almost from day one. Basically, as soon as I succeeded folding, I think probably the very first thing I said, started saying, well, how could I, can I change this? Can I make it different? Can I, you know, if it's, I made the jumping frog, let's change the position of the legs. You know, can I make it stand upright? Can I make it look like it's halfway through a jump rather than sitting on a lily pad?

And then there were things like there's a pretty famous model called the talking crow. You make it by turning a cootie catcher inside out. And so you get a head with a mouth and front legs, or wings, but no body. And I thought, well, can I make an animal that has that kind of a head, but has a whole body? How could I do that? Well, let's try blincing the paper one extra time to create some extra paper, and then maybe I can pull out some paper andinkering and seeing if I try something different, can I get something new? And with origami, the barrier to doing that is very low. You can just try it. If you mess up, grab a new sheet of paper and try something different.

So origami was a good hobby for an innate tinkerer like me. That's sort of where I am right now. Okay, well, maybe I'll blintz it first or maybe I'll fold the edges around, see if I can get a little extra oomph here and there. And I'm not really into the leap over into starting from a blank piece of paper and designing something. And it took me a long time to get to that point, you know, but as I evolved, the kind of the changes I would make got bigger and bigger. So, for example, I might now say, well, what if I make half of the paper into a bird base and half the paper into a frog base? What does that get me? And in fact, yes, a frog with wings keeps him from whomping his butt when he lands. So making bigger and bigger changes.

But also, then I started to recognize deeper underlying patterns. Like the idea, if I had one point and I wanted to get two points, there seemed to be a relationship between how long I could make the two points related to how far apart the tips of those points had to be on the unfolded paper. That's this bookworm idea that I describe in the design book. And in fact, to a large extent, the narrative flow of my design book, where I go from simple ideas to more and more complex ideas, is really a mirror of the 20-odd year evolution that I went through as I went from very simple design tweaks all the way up to full-fledged from scratch design. Going through the design book, do I need to actually fold all of the models in order to understand the material? Or can I get past this whole hummingbird and baby section and just skip over to the math? It depends on your innate talent. So this is all optional.

So, you know, in principle, you could completely jump to the last chapter. But a lot of people would have trouble with that. So I recommend to people that they work their way through sequentially because even if you even if you look at it like the hummingbird and the baby and you say oh yeah i could fold that so i'll skip it but just the process of folding and thinking about why why are you doing what you're doing what is each step getting you in terms of the final result? That practice will build the synaptic connections in your brain that will help you in the later stages as the designing gets more sophisticated. I have found that to be true, that if I skip the folding part and just try to go on it doesn't

make as much sense but once it makes sense to my hands it makes a lot more sense to my brain yeah and that's i think that's one of the neat things about origami as a way of learning mathematical concepts is that it gives you the opportunity of making this tangible connection with your handling, with being able to take the shape and rotate it and visualize, look at it from different angles. And so build a stronger mental model of what's going on. And so that's why I encourage in the design book people to actual fold things rather than making just a dry mathematical treatise. Here's the equations, you know, QED. So that's the Origami Design Secrets, Mathematical Methods for an Ancient Art.

But you do have a new book, which you mentioned briefly, Twists, Tiling, and Tessellations, Mathematical Methods for Geometric Origami. Yeah. And that one is very definitely a math book. And there's no way of avoiding formal math. So to do this kind of geometric origami, you pretty much have to approach it using formal mathematics. So what I tried to do was present the mathematics that people need, but just the math that they need, And give it in context of visible shapes. So you can see what the equations are describing. And also classify the math for one's educational level. Because there is a lot of origami design that could be done with just early high school math.

Even a lot of tessellations, amazingly beautiful things. You do need some math, but only a little. And so I labeled every chapter with one, two, or three stars, depending on the level of math that's needed. So people will know what's reasonable to ask of them. I mean, if you, if you've had early high school math, um, the chapters that are one star, I think you ought to be able to take those on. Yeah. I mean, there are, you will learn new things. It's not necessarily going to be super easy, but it's not totally out of the question for you. Whereas a two-star

chapter, which requires more like trigonometry, someone who doesn't know trig, they're not going to have a chance of doing that work. So I say skip those chapters if you're a one-star person. But if you do have trig, then there's a broader family of designs that you can create. And finally, the third level is undergraduate engineering math. And that's the most advanced math in that book. But people who have that as basic background would be able to set up design and one would hope fold pretty much everything that's described within the book. I don't have this book, but I'm looking at its table of contents in Amazon. And chapter 1.6 is

vector formulations of vertices including translation rotation and reflection and that's a three-star section so it's not that chapter one is all one star but each chapter has subsections and they may have one two or three star subsections so i think that is the three-star subsection of chapter one. But if you're a one-star math reader, you skip that, and you go on to chapter two, and you're back to one star again. This is very much a textbook. Yes. Where are they teaching origami as a class? And where can she sign up? I mean, I think Domaine is at MIT,

but is anybody else teaching origami as a class? Not that I know of. So, one of my colleagues has been after me to teach an origami course at my local university. And my excuse for putting them off was that I needed a textbook to teach out of. And now that I've written that book, I kind of lost that excuse. So I will probably be in the next year or two teaching a class from that book. Other than people interested in origami, who would be the people who take the class? Undergraduate mechanical engineers? Yeah, it's people who are interested in folding.

And folding shows up in a lot more than just the origami art. So the fields where they arise are both mechanical engineering, computer science, because in computer science you work a lot, it's computational geometry in particular, Eric de Mainsfield, you work with manipulations of surfaces computationally. Architecture is another area because architects like to make shelters and facades and forms that are made from sheet-like material. Now we're talking about folding materials like wood and metal and concrete. But this is all, in fact, doable. And architects are quite interested in this. The counterpart to Eric Demaine in Japan, a Japanese professor of computer science and architecture, is Tomohiro Tachi.

He's a professor in both fields, and he is, like Eric, a leading expert on the math and computational science of origami. Now I have a new name to look up. How exciting. Oh, definitely. Particular thing to look up for what Tomohiro is probably most famous for is Origamizer. So that'll be the thing to look up, which is a program that takes a description of any surface and gives you the flat folding pattern that folds into that surface. Yeah, but the flat folding pattern, it's too hard. Oh, it's incredibly hard. In fact, and there's a wonderful, there's a video online of Tomohiro folding the Stanford bunny, basically a bunny rabbit from one of his patterns. And it's, you know, it's slow motions, or it's the opposite of slow motion, it's time lapse. So you see him moving very,

very fast and the paper slowly takes the form of this rabbit. But the most impressive thing about the video is watching the shadows of the sun move from one side of the room to the other over the course of this video, because it was about eight or 10 hours of folding. What are some of the things people do wrong when they start folding? I mean, impatience is one of them, but what are the common mistakes in beginners? Probably the most common mistake is not being aware of everything that's going on as they're making a fold. And it's a very natural human thing, sort of a tunnel vision. If the instructions say, bring point A to point B to make the fold,

you're really looking at points A and B, and as you're flattening the fold, you're just sort of seeing that happen in the periphery of your vision, but not really focusing on it. And what happens is that as people, they bring point A to point B, and then they flatten it, but the fold is forming a little bit off, not quite in the right place. Maybe the left side is at a slightly different angle than the right, so a buckle is forced to form between them. And that imprecision then will throw off subsequent folds. So eventually they get to a point where their paper doesn't match the instructions and they can't go any further.

And then they say, I hate this. I'm never going to do it again. But it all boils down to trying to keep track of everything that's going on as you're folding. And so impatience plays into that. If a person is rushing, they're just not able to stop and spend the time to look at how the fold is forming as they're flattening it. And so when I teach people, I'm very often urging them to slow down, you know, start to make the fold, but pause. And as you're flattening it, watch as the fold falls into place to make sure that it's really forming in exactly the right place. And that the left side and the right side are going to line up and so forth and so on. I recently did a cat. And I really, really, really wanted to have a good cat because it was for an electronics thing and I put electronics in it so it would purr if you pet it. But it gave me motivation

to want to get to a good cat. And I had to fold probably 50 of them. I got so that that's what I would just do. I would just fold the cats as I was working on it because I was having such a hard time making the ears look right to me. And it really changed my opinions of how to do origami. The repetition made it so much easier to see these slight tweaks that if I did them in the beginning, it messed it all up. Yeah. And cats and people are really hard. I think it's because both have personality, and we're exquisitely sensitive to personality.

So you picked a subject that is particularly difficult to do well. Do you have a good cat? I mean, I should know, but I look at your bugs more than your animals. I've done a few cats. I have one on my website that I'm sort of happy with, but I don't feel like it's a it's a good cat but it's not a great cat i ended up with one of montral's cats and it's a it's an origami cat you're not going to say oh that's a cat it's you're going to say oh that's an interesting origami cat yeah yeah john john has a very distinctive way of stylizing, of abstracting his subjects.

He's not striving for sort of a literal interpretation in that very often the proportions of his models are pretty different. But there's a very definite kind of look to it um part of that is is what he's striving for is is sort of a beautiful and interesting structure and uh and and his structures are always really really interesting um but uh but yeah it's it's i I kind of think sometimes John's look is more traditionally Japanese than some of the traditional Japanese. It's like computer graphics. Many of his models look like they are made of polygons. And I like it. It's a neat style but many of yours look like

they're full high def 1080p bugs and by bugs i don't mean software bugs but cicadas and and why do you like bugs so much they're just cool um you know what can i say i've always loved nature so for one thing they were accessible um because uh growing up um i would entertain myself by going hiking in the woods and looking at uh you know looking at the plants and animals and and bugs were always accessible. They're a little bit scary. Not so much to actually be scary, but there's just that little frisson of scariness that them attractive to sort of overcome the creepiness factor. And then they're challenging. As an origami subject, especially back in the 70s and 80s, the hardest thing to design were considered to be bugs, insects, spiders, arthropods, because they have long, skinny legs and lots of them. And the first mathematical algorithms for design were developed in part to go after that class of subject,

but also turned out to be especially good for that type of subject. Circle packing is really well suited to bugs and insects and arthropods. It's less suited to blobby things, you know, like elephants or geometric shapes like cubes and rectangles and buildings and cars. But it's really, really good at insects and spiders and the like. Can you describe circle packing? Yeah, the basic idea is that every appendage, like an arm or a leg, is represented on the unfolded paper as a circular region. And you can, I think, get the idea of why that works by considering an

umbrella. That when an umbrella is closed, it's a long skinny shape. That's the leg of an insect. But when you open an umbrella, when you unfold it, it becomes a great big circle. And so you can imagine if I want to make a shape that has, let's say, six long skinny legs, each of those legs is like a little umbrella. And when I open it, each is going to become a circle. And so I need to figure out where on the paper all those umbrella circles go in such a way that they don't overlap, that I'm not asking the paper to be in two different legs, that every bit of paper is either in one leg or another leg or is somehow in between. And that's the basic idea of circle packing is simply packing those umbrella-like circles into a square in such a way that you make them all as

big as possible so you don't have a lot of leftover paper, but they still all have to fit in the square without overlapping. That's a wonderful description. I'm really impressed because for me, it would have been two and a half hours later. I would have still been saying, well, and then. So I appreciate that. Okay, I wanted to go on to a couple of questions that I have about origami.

As opposed to what the previous hour has been? This is where we get into the private lesson, eh? Wet folding. Yeah. I know you do wet folding. And I know that the first time I tried it, I quickly realized that wet did not mean soaking. Yeah.

But how does wet folding work? Do I crease it before wetting or after? And how wet is wet? So the first thing to know is there is actually two distinct styles of wet folding. There's what you do with thick papers and what you do with thin. And they're completely different or very different. The classical wet folding is thick paper wet folding. That's kind of the technique that Yoshizawa developed. And there, you dampen the whole paper before you start folding. But the craft of doing it, the trick, is to get the paper to the right level of dampness. So as you,

it sounds like you discover, if it's too wet, it doesn't work. The paper is floppy, it rips easily. As you fold it, the fibers come up and get fuzzy, and it's just a disaster. I mean, the fuzziness can look really cool if you do it right. Yeah, but if you do it right, it's a very narrow range. So generally, you don't want that. And so the goal is to get the paper to a state that, when I teach classes on this, I describe it as leathery. The paper, if you hold up a sheet of, let's say, me-tints, pastel paper, and you shake it, it rattles. And then if you dampen it, and then shake it, it changes the sound. The rattling goes away. And so when I teach this class, I actually teach people to listen for the level of dampness in their paper, to dampen it to the point where the

rattling has gone away, but it still has some resistance to being folded. So you want the paper to have some resistance to being folded, but not to be so stiff that a fold cracks the paper. And getting the paper to that level of wetness just requires practice and experience. You have to try it, and eventually you recognize when you're borderline too wet or borderline too dry, and then you know how to keep re-dampening it as you're folding it to hold it at the right level of dampness as you're folding the shape. Do you use a hairdryer to make it dry faster so that you can put in curves and personality, or do you tape it? I do use tape most of the time because I live in California and California is so dry. I don't need to accelerate the drying.

So I'll just fold it and then set, I'll use bits of tape to hold it in place and then set it aside for a few minutes and it'll start drying out. And then when it gets to that dry enough level, I can either re-dampen parts that I want to continue folding or let it dry completely. A couple times though, if I've been under a time pressure, like I recently did a video for Disney where they filmed the whole process and I didn't have unlimited time. And so Disney, uh, where they filmed the whole process and we, I didn't have, I didn't have unlimited time. And so then, yeah, I did use a hairdryer to speed on, speed up the drying process. I don't like to, because a hairdryer, uh, will, uh, will dry it in homogeneously, dry it unevenly. So some parts will dry faster than others,

and then it can warp in the drying when that happens. So thin paper wetting, that's pretty dry, actually. That's barely wet. Yeah, so for wet folding with thin papers, we basically don't wet the paper at all for most of the folding sequence. And it's only when we get down to the final shaping aspects that we'll dampen it, and then only selectively. So I'll use a paintbrush and a little glass of water, and I'll just dab little bits of water inside the layers of, let's say, a leg or an antler that I'm trying to compress.

So I'll dampen just that part, compress it, throw some strips of tape on to hold it into shape, and then let it dry and then go on to the next leg or another part of the shaping of the body. That's probably what I need to do with the hummingbird head, isn't it? Yeah, that would be a good way of doing it. Certainly, when I have wet-folded my hummingbirds, I don't dampen all the paper from the beginning. I use sufficiently thin paper that I'll just do the selective dampening in the final shaping. Do you have a favorite size of paper to start with? Usually as big as I can get, as big as I can get in high quality paper.

Since for most of my kind of bugs and insects and the like, I use origami dough paper. And so my favorite size is the standard size sheet of origami dough paper which is about a 50 centimeter square um oh really that's kind of exciting that's big yeah um i i think because pre-cut origami paper comes in 15 and 25 centimeters, there is this presumption sort of that that's the size one should use. But generally, things will get easier as the paper gets larger, up to about a meter in size. And beyond a meter, it starts to get hard again, just because the paper is very unwieldy. All right. That makes sense. And for those of you who are not metric, 15 centimeters is six inches, I think.

Yeah. So we could re-record that and say standard origami paper is six or 10 inches. It's fine. I've been using three-inch paper because I throw away so much of it, and I think that is not great. Well, for modular origami, three inches is great because it makes these lovely, delicate, little, intricate, intricate little balls and the like. But for representational origami, I think three inches is little on the small side. I told you. I know. But I had green and red paper, so is little on the small side. I told you. I know. But I had green

and red paper, so I could do these hummingbirds with this tiny piece of paper with these intricate folds. You can't learn that way. Well, you can get six inch green and red paper from Origami USA. So you can at least go up to up to that size for your hummingbirds. You don't do or your books don't have a lot of modular origami and does that count as origami what about scissors what about um all of the what about when is it cheating what are the rules yeah yeah when is it cheating so since there is no governing body of origami, there is no, and there are no sacred holy books, except for mine, perhaps. In fact, there are no sacred holy books. So there is no objective answer to what the rules are or what's cheating so really everyone is free to uh to decide what what rules they want to play by

and we're free to decide what we want to call origami because yes the word origami is japanese but it was only in use for about a hundred years and and during those hundred years it only applied to the traditional designs. It certainly was not applied to the tens of thousands of modern designs. So you have to say, well, when does something qualify as origami? And do you accept a very broad or a very narrow definition? I accept a broad definition. I say origami is an art form in which the primary means of creating the shape is folding. That's all. I mean, that's what it means in Japanese, fold paper. Yeah, and I don't even restrict it to paper. I say, so if you are folding polymer, tortillas, metal, wood, those I think are all forms of origami if folding is primary. So that means I don't stipulate whether it has to be a square or some other shape, whether you can use cuts, whether you can use more than one sheet of paper. So by my definition, modular origami is definitely a form of origami when folding is primary.

There's a certain style of modular origami that uses really simple little triangles and puts hundreds of them together to make shapes like pineapples. And I kind of feel like that's that's borderline if um i'm not going to try to put it on one side of the line or another personally it's not it's not interesting to me so i i won't go down that path but if someone says wants to call it origami and some people do i won't say no you're allowed to, because it really is a personal choice. So in terms of tools, you mentioned scissors. Certainly, even the most ardent traditionalist who's using square paper was using cutting tools to cut the paper to square.

And if you want to cut it to square, you want to cut it to some other shape. If you want to put slits in the paper, which is very traditionally Japanese in the Senbazuru or Akata folding of 10,000 cranes, that uses scissors. So sure, scissors are allowed if they allow you to create the art. Other tools, bone folders, tweezers, laser cutters, even the little pair of optical tools that I sit on my nose so that my 50-year-old eyes can see fine, intricate folds. Yeah, those are tools, too. You have a book on action models, which are things that move. Let's see, the book is Origami in Action, Paper Toys that Fly, Flap, Gobble, and Inflate. Does math apply there? How do you figure out how to make things move as well as design them in the origami software. So math definitely applies there, although it didn't to anything that was in that book at that time.

I wrote that book in the, I think, in the late 1990s. And so the designs were some of my own designs I knew of by other people that I especially liked. And I don't think anyone used explicit math to design those objects. But the mechanisms that make things move are very much things that can be described well by mathematics. And there is, over the last 10 years, there's been an explosion in the field of mechanical engineering of the mathematics of origami mechanisms. How you design and describe folding mechanisms that either come from origami or are inspired by origami or are explicitly origami. And the mechanical engineers are studying and describing them, not because they want to make

monkeys that clap their hands, but because they want to make solar arrays that deploy with 100 moving parts but only need a single actuator to make the whole thing expand or shrink. Or they want to make a shelter, you know, a building that expands and contracts based on the weather, but that again, only needs a small number of motors and actuators to make it open and close. How come fireworks isn't in your action model book? Is it the time I wrote it? I'd never heard of the model. But you've heard of it now, right? Oh, yeah. It's by Yami Yamauchi. It's quite well known and it's beautiful. And it actually illustrates a very kind of an amusing principle in the field of compliant mechanisms.

One of my collaborators in mechanical engineering, Professor Larry Howlett, Brigham Young University, had noted, mentioned at one point that there were, he's a world expert on compliant mechanisms, mechanisms that bend, and he had mentioned there were no compliant mechanisms that underwent 360 degrees of continuous revolution. Ha, take that. And when he talks about this now, he says, but actually the origami people had been doing this for quite some time. It's a neat pattern. I'll make sure and put a link to it. But it's kind of an origami fidget spinner. You just keep turning it in and out on itself over and over again.

It makes like a kaleidoscope pattern. Exactly. And it's closely related to a polyh makes like a kaleidoscope pattern exactly and it's closely related to a polyhedral structure called kaleidosycles oh yeah okay yeah do you are you working on more of these i want to say action models but action models with the mathematical tessellations, or are they very separate paths? They are. I continue to be very interested and working in the field of folding mechanisms. The technical name for the field is rigidly foldable origami, the idea of just rigid foldability. It arises in engineering because when we're making things from materials other than paper,

we're using materials that are usually so stiff that the panels between the folds can't bend at all. And when you fold with paper, not only are you creating folds on the folds, of course, but you're also usually sort of bending and unbending the flat layers of paper. And that allows you to do a lot more than you could if the material were perfectly rigid. Let's say the panels were metal joined by hinges. And so there's an important need in the world of mechanical engineering of being able to design folding patterns that can be made with truly rigid materials. And that's an interest of mine from both the mathematical and artistic side. So some of my recent papers in mechanical engineering journals have been on rigid foldability, designing certain classes of pattern. And some of my most, some of my satisfying personal art projects are patterns that are folded

based on these mathematical ideas, and in particular, folding from wood. So I've got on my website some photos of some folded wood patterns that are designed according to the mathematical principles that describe mechanisms and action models and the like. I guess I should look at your website, but is it like balsa wood that's bendable, or is it truly, I guess you said rigid folding. Yeah, so some of it is actually really, really rigid, like eighth inch plywood with hinges then where the folds go. And some of it is from wood veneer that has a little bit of bendability to it because it's veneer but i don't rely on that bending for the formation the wood veneer is especially nice because i can

laser score the folds into it so load a sheet of veneer into the laser cutter score the folds in and then fold it up and you get a shape that's visually interesting from the geometry of the folding and it's got the beauty and warmth of wood. One of the things that I liked about curved folding was the rigidity that you could get out of the model. Yeah. Where does that come from? Is it because of the bending and the stresses on the paper make it more evenly distributed?

Or where does that come from? Well, so this is actually an area that is, if you'll pardon the term, at the cutting edge of mechanical engineering research. And that the way using curved folds can actually give you better mechanical performance than stiff folds. And I think part of that comes from the fact that a straight fold, once you put a fold into a sheet of paper, it becomes hugely stiffer than the unfolded sheet. You know, and you can just make a zigzag sheet of zigzag folds in a sheet of paper, stand it on end and start piling weights on it. And you'll see it'll hold way more than an unfolded sheet. But curve folds can be even stiffer. And I think part of that is that the straight folds, their failure mode is a form of buckling.

And when you put curve folds in, that frustrates the buckling phenomenon and so gives you a larger, basically, a larger elastic compression interval before it gets to the failure mode of buckling. But in order to kind of mathematically describe this, we need a very robust mathematical way of describing a wide variety of curve fold patterns. And that doesn't exist yet. And that's why Eric and I are working on this book. Can I be your grad student? Wait, no, I had another question. One more.

Foil. Why do people fold it? I don't understand. People are so excited to get the gold and silver foil in the paper packs, and I'm just like, these are useless. But I know people love it. What am I missing? Well, for certain geometric shapes, gold and silver are actually beautiful. Beautiful, but they crumple so easily. Well, like with modulars, they're usually multiple layers, and if you fold it nicely, they don't crumple. But you have to put your finger on a weakness that every little buckle and imperfection in the paper is very, very visible with foil. And so the imperfections show up a lot. And I don't like that. In fact, I use foil for practice, but I fold with it inside out so that the white is on the outside because I don't like the shiny

look for most applications. But for practice, and especially practice of a model that involves shaping, it's really nice. It's quicker and easier than wet folding, and so you can sort of do development and proofreading more quickly with foil than with wet folded paper. And then once you're satisfied with the model, grab a sheet of origami dough paper and fold the final display model. It is true that the fireworks I did with foil looks really pretty. Well, there you go. So that makes sense. And doing it on the inside would actually show where my folds are going wrong.

All right. What book should I get next? What do you want to do? What type of origami excites you? Wow. I mean, boxes has always been important to me, but I feel like I've been through all of the boxes, all of the utilitarian and practical origami. You've been through Fousey's Fuse's Origami Boxes book? I'm not good with names. Tomoko Fuse is one of the most incredibly talented origami artists in the world. We're incredible artists, has done museum exhibitions,

incredible diversity of output. She's done modulars and boxes and tessellations and spirals and the like. And whenever she gets into a field, she does amazing and beautiful work in that field. So she's got a couple of books of origami boxes, origami modulars. And if you like boxes, look for her box books and try them out. This looks super familiar. Yeah. So Fuse, the last name is F-U-S-E. Yeah. Okay, cool.

And if someone is thinking about getting their, I don't know, high school age niece or nephew or son or daughter into origami in hopes to improve their math grades sneakily. Do you have any recommendations for not kids but interested young adult origami books? So I think the sort of most widespread appeal of origami or the origami that's most likely to appeal to someone knowing nothing more about that person, is probably going to be action models. Because there's a diverse range of subjects, and they have the cleverness of moving. And so I would, of course, recommend my own book, Origami in Action, but also Jeremy Schaefer's books, Origami to Astonish and Amuse. He has really inventive, clever action models, nothing terribly hard. And they're all just fun.

And I think they appeal to a wide range of people. I got that one from the library. It was really fun. But I would caution the person buying the book against trying to get them into origami, thinking it will improve their math scores. It might. It might not. It might even be a distraction from math because they get hooked on origami and spend their time folding instead of doing homework. And I don't think people should do origami as a means to some other end. They should do origami because it's fun, if it's fun. And if it's not, then they shouldn't do it. All right. Well, I have more questions about the tail on the hummingbird and how you go from an open sink to feathers.

I think maybe I have kept you long enough. Robert, do you have any thoughts you'd like to leave us with? Just one, which is kind of a reflection back on the development of my, you know, the evolution of my origami career. In going from a hobbyist to an engineer to a physicist to a full-time origami artist, all along the way, I never could have predicted any of the opportunities that came up. And, or the, you know, the forks in the road that I chose left or chose right. But but one thing I did do along the way that that I think helped to make good choices was following follow my curiosity. And so when I was curious about math, I learned more math because it was interesting, seemed interesting. Origami, I followed my curiosity. Natural history, optics,

all along, I kind of followed things because they looked interesting. And that prepared me, I think, to make good choices when the opportunity came along. And it can be encapsulated, a quote, my favorite quote from a scientist, Louis Pasteur, he said, chance favors the prepared mind. We can't predict the opportunities that are going to be presented to us. But if we've learned a lot of diverse subjects, then we'll be prepared to take advantage of those opportunities when they come along. And that would be my final thought.

Thank you so much. That's great. Our guest has been Robert J. Lange, author of many books, including Origami Design Secrets and Twists, Tiling, and Tessellations. He is an origami artist and consultant, a once-and-future physicist. You can find his books at your bookstore or make them, order them for you. Or there's Amazon. There'll be links in the show notes. His website is langorigami.com. Lang origami is all one word.

Again, a link will be in the show notes. Thanks, Robert. This was great. It's my pleasure. Thank you. Thank you to Christopher for producing and co-hosting. Thank you to our patrons for Robert's mic. We really do appreciate you.

We appreciate all of you. Thank you for listening. You can always contact us at show at embedded.fm or hit the contact link on embedded.fm. And now a quote of my own to leave you with. This one is actually from our guest. Almost any subject is suitable for an origami model, despite the stringent limitation of using an uncut sheet. Embedded is an independently produced radio show that focuses on the many aspects of engineering. It is a production of

Logical Elegance, an embedded software consulting company in California. If there are advertisements in the show, we did not put them there and do not receive money from them. At this time, our sponsors are Logical Elegance and listeners like you.