I think one of the great shames about how we teach kids math in school, as we think it's all about memorizing rules and figuring out how to work with numbers. Math is about the study of patterns and relationships, and it's far broader than mirror numbers are algebra. Hey, I'm Christina Wallace and I'm Kate Scott Campbell, and you're listening to the Limit does not Exist? A podcast for human ven diagrams, coming at you every single Monday and hosted by US. Today, we're talking with Dr Robert Lange, who merges mathematics with aesthetics to fold elegant modern or gami. Robert had a first career as a physicist, engineer, and R and D manager. I think that's a few first careers, during which time he authored or co authored over eighty technical publications and over fifty patents awarded and pending on semiconductor, lasers, optics, and integrated do electronics. No big deal. We discuss creating tools like computer programs to create better art and delve into the practical applications of origami folding techniques as they apply to basically anything medical, electrical, optical, nanotechnical devices, even strands of d N A. Yes, And we discussed the world's most beautiful math equation and how it has that unique combination of elegance and surprise, or how I like to describe myself at a dinner party. And we learned a new word, as Robert explains, how to mathematicize. That's the new word, the design process. That's right, fun new word added to our t L d N E vocabalist that would place it currently right after kerfol uh yeah, and right before nerd gaggle see episode twenty two. Anyway, without further ado, let's do it. Let's do it. Hey Christina, Hey Kate, Hi, Robert Audi, Robert. We are so excited to have you on the show. And we are not the only ones. It was just yesterday that a Twitter follower of ours at vm C Murray, a K. A. Tory, tweeted at US Admiral Hopper a human ven diagram for your consideration, and then she linked to you on Google and we said, hey, Tori, good call. We're actually recording with Robert tomorrow night. So the listeners have spoken. We're just we're so happy to have you here. Well, I'm glad to be here. Awesome. Well, we have so much to ask you, but before we do that, we, in show tradition, like to kick it off with an article. And Christina, you found this one this week, tell us about it. I did so. I have returned to the paper edition of the New York Times the weekend. I know it's a quite quite a luxury. And this weekend, as I was flipping through my Sunday review, came upon this fantastic op ed that there's not much to discuss except to say I agree with you, Richard Friedman, but wanted to bring it up as a kicking off point I think for our conversation today. The title is the World's most Beautiful Mathematical Equation, And you know you can already tell that I'm going to love this op ed. But he talks about how sort of math is beautiful on this purely abstract level, apart from its ability to explain and understand the world. And he talks about how, you know, we already know the art, music, and nature are beautiful, but that mathematical ideas could inspire the same feelings. And there's actually a reference in here to a study where researchers used f M R I scans of mathematicians brains while looking at a range of equations. Of course, this research experiment was devised um and it turns out the most beautiful equation, no surprise to me, is Boiler's identity e to the IPI plus one equal zero, which I mean it combines two irrational numbers, an imaginary number, and the two binary integers. Of course, it's the most beautiful equal it's got at all. I mean, it really does, so that was exciting. It also was kind of fun to understand at least point of their research, the ugliest is uh ramonogens infinite series for the reciprocal of pie. It's not the most beautiful equation, but really sort of that the elegance is the utmost school of any mathematician writing approof for computer program or writing code. So there's art and mathematics, and I thought it was a perfect segue to our conversation with Robert, because, as I think we're going to hear today, there's also mathematics in art. Absolutely, yeah, absolutely. I want to chime in and say I love this article as well. And I remember when I was in college. It was early on in college I was working on the play Proof, which people may remember got turned into a movie. David, that's right with Gwyneth Paltrow. And you know, I had, as I think I've talked about before, gone through a pet calculus, but really sort of struggled at that time in my math career, we'll call it a career as a student. And I remember my it was my acting teacher who said, Kate, let's just take a break for a second and look some stuff up. And we looked up math equations. He just found all of these and I don't remember what they were, but really complex math equations, and he was able to sort of show them. It must have been a website visually on a graph and to see what happened at that level. And I remember saying to him, Oh my gosh, if I had known that math could look like this, I would probably have had a different relation ship with it. And I think it's so cool that the article talks about that. Indeed, that same area of the brain um that lights up when people find music or art beautiful also lights up when we look at these beautiful math equations. Absolutely, I mean the highest compliment you can give to any proof is it's elegant. Um. It really doesn't matter if if two proofs can prove the same thing, but one is more streamlined and more elegant, that is the better proof, um. And I love that you had that experience to Kate from the acting side. I worked on proof as well my junior year, and I was like, Oh, the best of all the world's I get to have a mathematical conversation in the middle of a theatrical one. Well, and there is that line in it where that I had been working on that I had had trouble with, where she talks about just these beautiful proofs and how much she loved them, and I had to kind of get my hands in some you know, to really understand that. And you know on the Queen of Segways talking about getting your hands and some mathematical things. Robert, you make incredible oregami and are such a connoisseur of it. Have you experienced this reaction to a piece of orgami or a math equation that you have just gone, Wow, that is gorgeous because of the math in it. Oh? Absolutely, there are I think two qualities in an equation that made those mathematicians brains light up, made them call them beautiful. And and one is one that you already mentioned elegance, the idea of saying a lot or expressing a really complicated thought in a few words or symbols. And the second is an element of surprise that you see this this little expression and that it's a surprise how much it explains or how deep it's meaning is. And so when we see a mathematical equation that solves a problem and it's actually concise and short but incredibly powerful, we have both elegance and surprise playing a role in that makes that equation interesting and beautiful. And those two qualities are what makes an oregony object beautiful as well. I think we get a surprise when we see a folded object that we know is folded from a single uncut sheet of paper, and yet it just absolutely doesn't look like it could be that way. Um, And even though an orgony figure may be very, very complex, there's also an elegance when all of the paper is used, there's nothing left over. Every bit of paper is used in what feels like a very natural way. It's so true, it seems like it's a study in efficiency. Almost When you talk about that elegance right too, that every piece of the paper is used yea. And efficiency was one of the most powerful factors in bringing about revolution in Oregony of the last years. The idea when we started saying, what is absolutely the minimum amount of paper you would need to make this or that kind of structure? What's the minimum amount of paper you could ever need to make two flaps or even just one flap? And answering that question what is the most efficient structure lead to ways of solving for very complicated structures and more importantly this precise structures that we were after. And the way you you solve an efficiency problem, well, I guess there's multiple ways, but a really good way is to figure out how to describe the problem mathematically. So if I can write equations that describe what's going on in the paper, mathematicians no know how to solve equations that asked the question what's the most efficient, what's the maximum, what's the minimum, what's the best, what's the fewest, what's the most. If we can put it into mathematical terms, then we can use the tools of math to solve the equations that describe our gamy and ultimately get us to the artistic object that we were trying to create from our artistic vision. You're not what most people might think of when they think or a gami artist. You had a first career as an engineer, working at NASA's Jet Propulsion Laboratory and as a research scientist at Spector Diode Labs, among other places. You were educated as an engineer with a bachelor's and master's degrees in electrical engineering and a PhD in applied physics. You have like eighty technical papers and over fifty patents on lasers, and so this incredible technical career came first, and yet all all the while you were doing oregamy. So what drew you to engineering in the first place, and what did origami offer you all that time that engineering didn't well, Oregamy actually came first. The engineering didn't come until I was in college. But I've been doing origami my whole life. And there's one drive that I think powered both my origami interest and my engineering interest, and that is the desire to make things. I love making things, and as a kid, oregamy was a great way to get into making things because you didn't need anything more than a sheet of paper and you didn't even need plans that you know, you had to draw lines very carefully, or you had to cut very carefully. All you needed was a sheet of paper and folding and folding. I could get better with practice. So as a kid, oregamy was a great way to make things, to make, to make toys, for example, but just to tickle the urge to create. So so that was that was a huge motivation to do origami. That's why oregami came first. That's why I pursued it my whole life. You're listening to the limit does not exist with Christina Wallace and Kate Scott Campbell. Now, when I got into high school, particularly when I started learning some recreational math through the books of Martin Gardner. Oh yes, I love Martin again. Martin Gardner is responsible for an entire generation of mathematicians inspired about math, and I was one of them. I loved math. I was because of it. I was deeply involved in math team in high school. Yeah math team, Yeah, learning go, fighting cougars and UH and learning about math. And I was interested is for the sheer beauty of math and patterns and relationships. I think one of the great shames about how we teach kids math in school, as we think it's all about memorizing rules and figuring out how to work with numbers and then figuring out how to do mystical things with letters that we call algebra, and the kids are taught that that is what math is about, is following these rules, and math is about the study of patterns and relationships. And it's far broader than mirror numbers are algebra. And if we could teach kids more of the breadth of math, that we find more of them for whom that spark is kindled. And it was kindled in me. So I figured I would go into mathematics, but in college I kind of got sidetracked when I started taking engineering courses. It was really fun making circuits and widgets, and so that took me down the engineering path. That's very, very cool. So you left your engineering career during the dot com bust of the early two thousand's. You know, it's been mentioned that your work changed from studying lasers to managing a product design group, where one of your tasks had been because of the bus laying people off, and you left that to pursue origami full time. You've said that deciding to leave was a convergence of what you wanted to do plus what was happening at your company. And there's this great New Yorker article on you that described the move as quote analogous to perhaps quitting a job as a neurosurgeon to take a shot at becoming a professional knitter. Now, I would love to hear your take on that, if that feels accurate, And also what your plan was for making origami a full time career. Well, Susan or Lean definitely has a way with work. She does she does, but and and it's it seems a little funny to say that I really didn't have any big long term plan. The specific thing I wanted to do was to write a book about how to design origami. Over the course of my engineering career, I had written by that time six books of recipes instructional books. Here's how to fold twenty different oregamy shapes that I had designed. And I could write those books while holding down a full time job and raising a family, just by working evenings and weekends and odd moments. But I've had an idea for a book that about how to design oregami, so not how you can fold something that I've designed, but how you can design whatever you want. And I've been thinking about this book for about ten years, but never really making much progress, and kind of came to the conclusion that unless I was working on that book full time, I'd really never get it done. It wasn't something I could set aside and pick up a week later. I really had to be thinking about it solidly. So that posed a decision question for me. Do I continue down the path I was going as an engineer, probably never write that book, or do I really want to write that book, And kind of weighing everything and weighing the joy of the turmoil of the dot com bust and the like, um, I decided that, you know, whatever I might do as a laser physicist, there's plenty of other laser physicists in the world who could do it without me. But I really felt like I was the only person who could write that book. So that was what made the decision. I was gonna quit. I was going to write that book and then see what happened. You know, I had savings to run for a while, so I just start writing the book, which I did, But at the same time, I also began following up on all of the leads that I'd never been able to follow up on when I had a full time job. And by the time I finished the book and delivered the manuscript, I had a pretty busy life doing origami work for advertising and sales of art and uh, teaching and lecturing and the like. Although the mix of activities has changed a lot over the years, UM it's kept me pretty busy. I'm I made the change in two thousand and one, and so here it is sixteen years now, and I think it's probably the safe safe to say I I won't have to go back. That's incredible. So I love that you were so excited to make this book to help other people, you know, make their own designs. You've also written two computer programs to help origami artists implement designs. One called TreeMaker that translates these tree forms. It's sort of like stick figures of your designs, like people or bugs, into the creased patterns so they know where to fold. And then the other this one, I think you called it reference Finder, that converts the patterns into step by step folding instructions. So have purists ever pushed back at your use of computer programs to facilitate origami or is this just considered the new wave of of how we're pushing oregamy to its most sophisticated end. There's a lot of diversity in the world of oregamy. People fold in different styles and they adopt different customs. And some of the customs are things like whether you use more than one sheet of paper. There's modular folding that uses a lot, there's uh there was a long tradition of using a couple of sheets to make like the front and back half of the animal, or whether you use a single sheet and don't use any cuts. I get a little nervous when people just start suggesting what's the purist, because if one person is a purists, that suggests that something else is adulterated. And there's already some you know, some opinion trying to be passed off as as objective. So let's take away the word purists and just ask the question do people have a problem with using computers? And some do, and I'm fine with that because if that doesn't appeal to you, then don't use computers. And in your work, I find that tools help me create better art, and sometimes those tools are physical things like good lighting, a clean working surface, high quality paper, reading glasses so I can look at things up close. These are all tools. Um, computers that help me think are also tools. And then there are sort of abstract mental tools, the theories and concepts that I use for how to represent the shape I'm after and to figure out how to translate that into a folding plan. So I don't really distinguish myself between that some classes of tools are better than others. If I'm working within the mode of folding from an uncut sheet of paper, which is usually the case, then tools that helped me do that better, I think are a good thing. But to use it to al well, you have to understand it. If you're trying to put in screws and the only tool you have is a hammer, it's not gonna going to be of much use. Um you need to know when to use a hammer when you use a screwdriver. And so I don't fault at all people who say they don't want to use computer programs. It's just choice. What tools do you use, what types of folds do you want to do? I love that concept of using the tools that you need to create better art. That's just such an interesting and accurate to me raming Robert. That's so cool. Something else that's fascinating is that kindergarten. We're talking about the early childhood education kindergarten. I think there's only one was created in eighty seven by a German educator, Friedrich Froeball, who not only introduced this radical idea of early childhood education, but also built a curriculum that included three kinds of paper folding to teach children principles of what we're talking about, math and art. So you first experienced origami at six, so curious to know what are the mathematical principles of basic origami that four and five year olds can understand and therefore people older than that, and should we bring origami back into early childhood education? Do you think there are a lot of principles that four in five year olds can understand? Things like geometric just even name what's the difference between a square and a rectangle and a triangle um and and by handling physical objects that at least for some kids can build stronger connections than just looking at pictures in a book or listening to words by a teacher. Different people learn in different ways. Some people learn better from words, some people can follow things in a book, some people need to handle objects. And so by introducing more diverse tools and education, I think we can reach a larger number of kids. Now back to the conversation with Robert, you actually were once an artist in residence at m I T is that right? That's right? That was about ten years ago, I think, and you gave this famous lecture about origami and its relationship to mathematical notions like circle packing and tree theory, and you have this great line in your Ted talk from I think just a few years after that that says, the secret to so many fields is to let dead people do the work for you. That line, this allows you to take your problem and look for a way that someone has already solved it in another field. I just find these are both really interesting theories around education, around doing your work. How do you split your oregami problems into pieces that allow you to search for a solution from a dead guy, And how do you teach other people to do that? Well? The way I would teach other people is kind of the way I learned to do it when I went to Caltech major electrical engineering. But I took classes in physics and chemistry and mechanical engineering and lots of different fields, and there was a common approach to addressing problems and all these different fields, which relies on the universality of mathematics. You figure out how to this scribe your field in mathematical terms, and once you can transform your problem in your specific field into an impure, abstract mathematical term, then you can bring to bear all the tools of hundreds of years of mathematical development onto your problem. If you can convert an oregonmy design problem into a nonlinear constrained optimization, which is a type of math problem, then you can use the work that hundreds of people have done on how to solve a nonlinear constrainted optimization, and then you solve the math problem. And then you apply that back to your field and you have a solution to the thing that you were after, whether that thing is building an electronic circuit or making a laser work or folding an oregonmy piece of art. The math doesn't know or care what it's being used for. Wow, that is so interesting. Do you have a way of looking at a piece of oregonmy or thinking about the end result of the oregonmy and figuring how what Matthew would want to use, like how to find the math to get you where you want to go. Yeah, when I'm going to fold an oregonmy figure, I start by thinking about, in broad terms, how I'm going to deconstruct it. What kind of shapes I need to build that object from. You know, if it's a unicorn, I'd want four legs, head, and the tail years a horn on the head, and so those are the parts. And then I think about mathematically what shapes of paper could I use to make all those parts in isolation? And then imagine having a little little piece is a paper for all those parts legs, arms, heads, forth? Then how do I pack those shapes into the square that I want to fold it from? And finally, mathematically, how do I connect the folds from one part to another part. There's definite mathematical rules now that over the last twenty years that we've worked out to address each of those steps. So in the end, I can plan out pretty much the whole design before I start folding. Now, I still discover things as I'm folding. I'm folding my design and I see serendipity fortuitous alignments where this fold is close to that point. So if I shifted over, I'll need fewer folds, it'll be more elegant. I'll make those changes so I do discover things along the way, But the overall layout I can achieve by using math. There's such a point in so many artistic process sees of I just want to bail, you know, like this isn't what I thought it was going to be. I thought that I had it outlined, you know, and sometimes it's like figuring out, yeah, this is when I should just scrap it and start over, or let me try to push through this and find a way out of it. Have you ever experienced any challenges like that in your folding process? Or is the question that I was going to ask, have you ever gotten to the point where you finished something and then had to still a piece of paper left over and you had to figure out where to you know, shove it somewhere. I'll just the send question because it's quicker. When I planned out a design, I know pretty much from the beginning whether and where there's going to be paper left over, and I might intentionally design it to have some extra paper and then just say, Okay, I'll figure out something I'm going to do with that bit of extra paper at the end of it. But it's rarely a surprise that there's extra paper because I've designed it all. But as your first question, do we ever get to a point and bail all the time? In fact, I would say two thirds of the time, I've worked through a design and say, you know, I did my plan, I followed my plan, I got the results, but there's something aesthetically lacking here. And the thing is, we can mathematic size structure and form, but we can't, at least not yet, mathematics size esthetic values. So I can design something and have it come out exactly the way I designed it, but once it's folded, I look at it and say, no, this just isn't up to snuff aesthetically. And I'd say that probably happens more often than not. I like mathematic size. It's not a word I've used very often, but I think I might going forward. Yes, I have a question for you. For I don't know four hundred years or however long. Oregony has been a thing. It's been you know, beautiful but pretty simple. Call it thirty steps or so according to the research that I did. But now you can have hundreds of folds, hundreds of steps. And as oregomy became more complex, it actually has also become more practical with applications of these folding techniques to all sorts of things medical, electrical, nanotechnical devices, even strands of DNA, things that have a fixed shape but need to be packed tightly and in an orderly way. Can you tell us a little bit about some of the scientific applications of orgamy folding techniques. Yeah, the practicality didn't require the complexity to come along, but it it kind of evolved in parallel. And I think both the practicality and the advance and oregonmy techniques developed because there became a critical mass of practitioners of oregonmy. It became popular enough, well known enough that you had enough people working on it and most importantly, sharing ideas, sharing concepts that accelerated each other's development. And that was what made the difference in the explosion in design over the past couple of decades. And I think that also was what made the difference in the expansion of the applicability of oregamy engineering problems. The place that oregamy plays a role in engineering problems is when you have something that's fundamentally sheet like and it needs to transform its shape. And very often that's because well, the active state is something flat and sheet like, but it needs to get smaller for storage or deployment. So examples of that some of the really big spectacular examples or things like telescopes, solar or a raise lenses, antennas. These are big flat things, but because they're going up in an airplane or a rocket or something that doesn't have a lot of room, they need to be made smaller for the journey, and oregamy, or more broadly, folding, is a way of doing that in a very well understood, controlled way, making it smaller and then making it larger. We also find oregony playing a role when you want to fabricate an object from sheet stock, and there it's probably being driven by economic reasons. You're using sheets or rolls of material, and so cutting things and folding them and then assembling them give you packages and boxes, decorative objects, but also smaller scale deployable things. You know, implements for backpacking, medical devices that you want to go into the body through as small a hole as possible. So these are all examples where oregonmi plays a role. That's interesting. I'm curious, Robert, when you're creating for yourself, meaning not for a commission or something like that, where do you draw inspiration from? What are your sort of favorite things to to fold and and to create? And uh, I guess in addition to that question, what are you curious about folding and creating right now? Well, my interests fall into two pretty different categories. I love them both. One is the natural world. I've always loved nature as a kid. Some of my happiest times we're tromping through the woods and looking at plants and animals and the like, and uh, getting excited when I saw an animal in the woods. And a lot of my folding, because animals have been one of the traditional subjects, have a lot of my folding has been trying to capture in an oregonmy figure the emotional impact that I see when I see the real creature out in out in nature somewhere. So I do a lot of animals, and they're usually things that I either have actually seen or I want to see because I want to when I see the oregonomy figure have the same emotional feeling that I did or would seeing the animal. The second category is more abstract and geometric, and there I'm driven as much by mathematical considerations as anything. That is because I'm trying to understand the mathematical laws that define what one can fold and mathematical algorithms that lead one to specific folded objects. Um. A lot of the geometric things that I fold are based on solving a particular math problem. There's a class of shapes that I want to fold. I work out the math that would describe that class, and then exercise the math and also tested out by trying to fold shapes in that class. That's so interesting. I loved when you were talking earlier about you know, can we mathematicize aesthetics and and currently we we can't, and that you can figure out how you respond to the aesthetics through the doing. I would imagine it's similar in terms of getting that emotional experience. How do you know when you found it in what you've created? Is it because you look at it and and it resonates in the way that it has with the natural object. Yeah, there is a resonance. It's not clear to me. For any given subject, what is the feature or arrangement of features or arrangement of lines that resonates. And that's maybe one of the biggest frustrations or unsolved problems is is but that I can't put my finger on it. Yeah, because if I could ahead of time, then I just bake that into the design and the whare we go and success. But but so often it's something in kuwait and I only know it once I've seen it. So I have a slightly off topic question, um, but still relevant. In our research for interview with you today, we read that you met your wife Diane at cal Tech when you both had roles in a campus production of The Music Man. So we have to ask who did you play and how well can you sing and dance? Well? I was the newspaper publisher, who who was the base in the barbershop quartet? So you do sound like you've got a voice for a barbershop quartet. So I can sing somewhat, usually bass or baritone. As far as dancing, Um, every time I tried to dance, the insurance rates went up for everyone. There's a math equation there. Yes, bonus question, what did Diane play. Diane was one of the pick a little ladies. Little talk. Yeah, she's saying that an over too higher. That is fantastic. I was just thinking that there is some math equation in terms of the direct proportion to you dancing and the insurance rate's going up. But last almost certainly. Yeah. So, Robert, what's one time where you felt limited in your education or career? And this can be anything and in any way, and how did you overcome it? An example of when I felt limited actually would be my entire approach to origami design. Some of my car I'm pretty good at origami design, but some of my colleagues and friends are just unbelievable. My friend John Montroll has the ability to completely visualize how to fold an object in his mind. He doesn't draw out a plan like I do. He just sees what the solution is. And if you say, how did you come up with that? He said, he'll say, I just saw the solution and he knows how to fold it. So he can visualize all the relationships between every part of the paper and how those different parts come together into the finished figure. And and there's others like that, and I can't visualize in that way. So the way I overcame it was to figure out how to use math to fill in that gap. Even today, I can't visualize everything going on in one of my designs, but I can draw out a plan. I can design it a little bit at a time, and as I keep working through the different parts, eventually I have a complete design. And because I followed the math, all the folds play nicely together and give me the finished shape. So I actually think that limitation was what drove me to develop mathematical tools for design, and had the additional advantage that I could teach my design techniques to lots of other people. It's pretty hard to teach someone to just see the solution, just do that. But so I couldn't teach them to design the way John can. But I could teach them to design the way I can by breaking it down into these little steps and using math as the glue to get you from step to step to step to the finished piece of art you were after. It's so inspiring that you didn't say I don't have that, so I'm going to stop. Instead, you said, Okay, what do I have? What will work for me? And then took it a step further and said, how can I teach that to others? That's just incredible, It's amazing. Okay, we have one last question for you, and then it's time to move on to the lightning round big picture. What is something really big and crazy that you want to take on in the next call it five to ten years curved folding, developing the mathematics to design curved folding. The math is a lot more complicated. Some of that I had in my educations. A lot of it I didn't. I've had to sit down with math books and I've got a stack of differential geometry texts and on my sideboard that I look through to look for the tidbits that I need to describe folding. So I'm trying to assemble all the bits and pieces of math that are out there. And the math is definitely out there. Some of the relevant math is a hundred years old, others is more recent, but it's out there. But to assemble it all into kind of a coherent hole that both I can use and that I can teach to other people. My worst grade in my entire mathematical undergraduate degree was differential geometry. So I will not be helping you not that project, but I commend you for taking it on. Nor will I since I don't recall differential geometry in my educational upbringing. But Robert, I cannot wait until you crack this, particularly to hear how you're going to teach it to others. We will wait in excitement for that. Very exciting. Well, indeed, it is time for the lightning round, Robert. The way this works is we're going to ask you a handful of fun questions, and all we ask is that you just give us your first response, whatever pops into your brain, don't think about it too much. And and then we have to remind ourselves to not ask follow up questions. We need it good. Thank you? Okay, So are you ready? Yeah? Awesome bringing, Christina, do you want to kick it off? Sure? Question one? This one's very hard, I'm sure. What are you reading right now? What am I reading right now? Is the listing for a plumber that I need leaky toilet. That has been our most honest response of the show in over forty episodes. I think I think so, I think so? Uh, Robert, what was the last thing that made you go wow? The last thing that made me go wow? I had figured out an analytic model of a pattern called the water bomb tessellation. So how to describe the motion of this pattern in in purely analytic terms? Um? So basically a mathematical expression that described how every single vertex moved as you unfolded and full to the wow that makes me go wow awesome? Okay. Question three? What is the best book or maybe just one of the best books for someone interested in getting started in origami? I would say any book by Gay Merrill Gross or Nick Robinson. They both write beginner books that are very clear, very beautiful, and very fun awesome okay. Question four? Other than oregamy, mathematics, lasers, and engineering, what's something else that's in your personal ven diagram? So maybe something that's not super obvious to people who know you professionally that you're also interested in um plants. When I was growing up, my brother and I built with my dad's help. We all built a greenhouse and we raised plants together. I was part clearly interested in ferns and vermillions, and even though I don't raise them anymore, I'm still quite interested in both. Very I know what ferns are, I don't know what the other one is, but I'm not really to ask that would be a follow up. We're going to have to save it for research later. And then you chied me for telling you a pineapple is one type of vermilion. Okay, my interests peaked. Continue Christina, Okay, last question and then we're all done. Give a shout out to a woman who is doing awesome things in the world of ore gamy. Oh there's several, but the one who I'm really impressed by is Beth Johnson. Very cool. I if I could ask a follow up question, I would ask what she's up to. I know sheet that he may or may not choose to answer exactly like Style the Wire. Well, very cool. You know what, we will do our own research on Beth Johnson and find out what she is up to. I'm sure it's a lot of really neat things. It is. It is really neat. Now say, even though this isn't a follow up, the thing that I love about her work is it's not super complex. She uses some bits and pieces of complexity when needed, but everything is elegant and perfectly balanced and esthetically beautiful. It's this thing that I said, I wish I could put my finger on and I see it in almost every one of her works. Oh I, I love that and that there is that like you said that in Kait thing right, that really just shows in the art of it. Awesome. Well, thank you for not answering my follow up question. That was not a follow up question, just chiming in some extra information about Bath. Much appreciated. Well, Robert, thank you so much for joining us today. This has been so much fun. You're welcome. You've been my pleasure to have a great night. You too,

I think one of the great shames about how we teach kids math in school, as we think it's all about memorizing rules and figuring out how to work with numbers. Math is about the study of patterns and relationships, and it's far broader than mirror numbers are algebra. Hey, I'm Christina Wallace and I'm Kate Scott Campbell, and you're listening to the Limit does not Exist? A podcast for human ven diagrams, coming at you every single Monday and hosted by US. Today, we're talking with Dr Robert Lange, who merges mathematics with aesthetics to fold elegant modern or gami. Robert had a first career as a physicist, engineer, and R and D manager. I think that's a few first careers, during which time he authored or co authored over eighty technical publications and over fifty patents awarded and pending on semiconductor, lasers, optics, and integrated do electronics. No big deal. We discuss creating tools like computer programs to create better art and delve into the practical applications of origami folding techniques as they apply to basically anything medical, electrical, optical, nanotechnical devices, even strands of d N A. Yes, And we discussed the world's most beautiful math equation and how it has that unique combination of elegance and surprise, or how I like to describe myself at a dinner party. And we learned a new word, as Robert explains, how to mathematicize. That's the new word, the design process. That's right, fun new word added to our t L d N E vocabalist that would place it currently right after kerfol uh yeah, and right before nerd gaggle see episode twenty two. Anyway, without further ado, let's do it. Let's do it. Hey Christina, Hey Kate, Hi, Robert Audi, Robert. We are so excited to have you on the show. And we are not the only ones. It was just yesterday that a Twitter follower of ours at vm C Murray, a K. A. Tory, tweeted at US Admiral Hopper a human ven diagram for your consideration, and then she linked to you on Google and we said, hey, Tori, good call. We're actually recording with Robert tomorrow night. So the listeners have spoken. We're just we're so happy to have you here. Well, I'm glad to be here. Awesome. Well, we have so much to ask you, but before we do that, we, in show tradition, like to kick it off with an article. And Christina, you found this one this week, tell us about it. I did so. I have returned to the paper edition of the New York Times the weekend. I know it's a quite quite a luxury. And this weekend, as I was flipping through my Sunday review, came upon this fantastic op ed that there's not much to discuss except to say I agree with you, Richard Friedman, but wanted to bring it up as a kicking off point I think for our conversation today. The title is the World's most Beautiful Mathematical Equation, And you know you can already tell that I'm going to love this op ed. But he talks about how sort of math is beautiful on this purely abstract level, apart from its ability to explain and understand the world. And he talks about how, you know, we already know the art, music, and nature are beautiful, but that mathematical ideas could inspire the same feelings. And there's actually a reference in here to a study where researchers used f M R I scans of mathematicians brains while looking at a range of equations. Of course, this research experiment was devised um and it turns out the most beautiful equation, no surprise to me, is Boiler's identity e to the IPI plus one equal zero, which I mean it combines two irrational numbers, an imaginary number, and the two binary integers. Of course, it's the most beautiful equal it's got at all. I mean, it really does, so that was exciting. It also was kind of fun to understand at least point of their research, the ugliest is uh ramonogens infinite series for the reciprocal of pie. It's not the most beautiful equation, but really sort of that the elegance is the utmost school of any mathematician writing approof for computer program or writing code. So there's art and mathematics, and I thought it was a perfect segue to our conversation with Robert, because, as I think we're going to hear today, there's also mathematics in art. Absolutely, yeah, absolutely. I want to chime in and say I love this article as well. And I remember when I was in college. It was early on in college I was working on the play Proof, which people may remember got turned into a movie. David, that's right with Gwyneth Paltrow. And you know, I had, as I think I've talked about before, gone through a pet calculus, but really sort of struggled at that time in my math career, we'll call it a career as a student. And I remember my it was my acting teacher who said, Kate, let's just take a break for a second and look some stuff up. And we looked up math equations. He just found all of these and I don't remember what they were, but really complex math equations, and he was able to sort of show them. It must have been a website visually on a graph and to see what happened at that level. And I remember saying to him, Oh my gosh, if I had known that math could look like this, I would probably have had a different relation ship with it. And I think it's so cool that the article talks about that. Indeed, that same area of the brain um that lights up when people find music or art beautiful also lights up when we look at these beautiful math equations. Absolutely, I mean the highest compliment you can give to any proof is it's elegant. Um. It really doesn't matter if if two proofs can prove the same thing, but one is more streamlined and more elegant, that is the better proof, um. And I love that you had that experience to Kate from the acting side. I worked on proof as well my junior year, and I was like, Oh, the best of all the world's I get to have a mathematical conversation in the middle of a theatrical one. Well, and there is that line in it where that I had been working on that I had had trouble with, where she talks about just these beautiful proofs and how much she loved them, and I had to kind of get my hands in some you know, to really understand that. And you know on the Queen of Segways talking about getting your hands and some mathematical things. Robert, you make incredible oregami and are such a connoisseur of it. Have you experienced this reaction to a piece of orgami or a math equation that you have just gone, Wow, that is gorgeous because of the math in it. Oh? Absolutely, there are I think two qualities in an equation that made those mathematicians brains light up, made them call them beautiful. And and one is one that you already mentioned elegance, the idea of saying a lot or expressing a really complicated thought in a few words or symbols. And the second is an element of surprise that you see this this little expression and that it's a surprise how much it explains or how deep it's meaning is. And so when we see a mathematical equation that solves a problem and it's actually concise and short but incredibly powerful, we have both elegance and surprise playing a role in that makes that equation interesting and beautiful. And those two qualities are what makes an oregony object beautiful as well. I think we get a surprise when we see a folded object that we know is folded from a single uncut sheet of paper, and yet it just absolutely doesn't look like it could be that way. Um, And even though an orgony figure may be very, very complex, there's also an elegance when all of the paper is used, there's nothing left over. Every bit of paper is used in what feels like a very natural way. It's so true, it seems like it's a study in efficiency. Almost When you talk about that elegance right too, that every piece of the paper is used yea. And efficiency was one of the most powerful factors in bringing about revolution in Oregony of the last years. The idea when we started saying, what is absolutely the minimum amount of paper you would need to make this or that kind of structure? What's the minimum amount of paper you could ever need to make two flaps or even just one flap? And answering that question what is the most efficient structure lead to ways of solving for very complicated structures and more importantly this precise structures that we were after. And the way you you solve an efficiency problem, well, I guess there's multiple ways, but a really good way is to figure out how to describe the problem mathematically. So if I can write equations that describe what's going on in the paper, mathematicians no know how to solve equations that asked the question what's the most efficient, what's the maximum, what's the minimum, what's the best, what's the fewest, what's the most. If we can put it into mathematical terms, then we can use the tools of math to solve the equations that describe our gamy and ultimately get us to the artistic object that we were trying to create from our artistic vision. You're not what most people might think of when they think or a gami artist. You had a first career as an engineer, working at NASA's Jet Propulsion Laboratory and as a research scientist at Spector Diode Labs, among other places. You were educated as an engineer with a bachelor's and master's degrees in electrical engineering and a PhD in applied physics. You have like eighty technical papers and over fifty patents on lasers, and so this incredible technical career came first, and yet all all the while you were doing oregamy. So what drew you to engineering in the first place, and what did origami offer you all that time that engineering didn't well, Oregamy actually came first. The engineering didn't come until I was in college. But I've been doing origami my whole life. And there's one drive that I think powered both my origami interest and my engineering interest, and that is the desire to make things. I love making things, and as a kid, oregamy was a great way to get into making things because you didn't need anything more than a sheet of paper and you didn't even need plans that you know, you had to draw lines very carefully, or you had to cut very carefully. All you needed was a sheet of paper and folding and folding. I could get better with practice. So as a kid, oregamy was a great way to make things, to make, to make toys, for example, but just to tickle the urge to create. So so that was that was a huge motivation to do origami. That's why oregami came first. That's why I pursued it my whole life. You're listening to the limit does not exist with Christina Wallace and Kate Scott Campbell. Now, when I got into high school, particularly when I started learning some recreational math through the books of Martin Gardner. Oh yes, I love Martin again. Martin Gardner is responsible for an entire generation of mathematicians inspired about math, and I was one of them. I loved math. I was because of it. I was deeply involved in math team in high school. Yeah math team, Yeah, learning go, fighting cougars and UH and learning about math. And I was interested is for the sheer beauty of math and patterns and relationships. I think one of the great shames about how we teach kids math in school, as we think it's all about memorizing rules and figuring out how to work with numbers and then figuring out how to do mystical things with letters that we call algebra, and the kids are taught that that is what math is about, is following these rules, and math is about the study of patterns and relationships. And it's far broader than mirror numbers are algebra. And if we could teach kids more of the breadth of math, that we find more of them for whom that spark is kindled. And it was kindled in me. So I figured I would go into mathematics, but in college I kind of got sidetracked when I started taking engineering courses. It was really fun making circuits and widgets, and so that took me down the engineering path. That's very, very cool. So you left your engineering career during the dot com bust of the early two thousand's. You know, it's been mentioned that your work changed from studying lasers to managing a product design group, where one of your tasks had been because of the bus laying people off, and you left that to pursue origami full time. You've said that deciding to leave was a convergence of what you wanted to do plus what was happening at your company. And there's this great New Yorker article on you that described the move as quote analogous to perhaps quitting a job as a neurosurgeon to take a shot at becoming a professional knitter. Now, I would love to hear your take on that, if that feels accurate, And also what your plan was for making origami a full time career. Well, Susan or Lean definitely has a way with work. She does she does, but and and it's it seems a little funny to say that I really didn't have any big long term plan. The specific thing I wanted to do was to write a book about how to design origami. Over the course of my engineering career, I had written by that time six books of recipes instructional books. Here's how to fold twenty different oregamy shapes that I had designed. And I could write those books while holding down a full time job and raising a family, just by working evenings and weekends and odd moments. But I've had an idea for a book that about how to design oregami, so not how you can fold something that I've designed, but how you can design whatever you want. And I've been thinking about this book for about ten years, but never really making much progress, and kind of came to the conclusion that unless I was working on that book full time, I'd really never get it done. It wasn't something I could set aside and pick up a week later. I really had to be thinking about it solidly. So that posed a decision question for me. Do I continue down the path I was going as an engineer, probably never write that book, or do I really want to write that book, And kind of weighing everything and weighing the joy of the turmoil of the dot com bust and the like, um, I decided that, you know, whatever I might do as a laser physicist, there's plenty of other laser physicists in the world who could do it without me. But I really felt like I was the only person who could write that book. So that was what made the decision. I was gonna quit. I was going to write that book and then see what happened. You know, I had savings to run for a while, so I just start writing the book, which I did, But at the same time, I also began following up on all of the leads that I'd never been able to follow up on when I had a full time job. And by the time I finished the book and delivered the manuscript, I had a pretty busy life doing origami work for advertising and sales of art and uh, teaching and lecturing and the like. Although the mix of activities has changed a lot over the years, UM it's kept me pretty busy. I'm I made the change in two thousand and one, and so here it is sixteen years now, and I think it's probably the safe safe to say I I won't have to go back. That's incredible. So I love that you were so excited to make this book to help other people, you know, make their own designs. You've also written two computer programs to help origami artists implement designs. One called TreeMaker that translates these tree forms. It's sort of like stick figures of your designs, like people or bugs, into the creased patterns so they know where to fold. And then the other this one, I think you called it reference Finder, that converts the patterns into step by step folding instructions. So have purists ever pushed back at your use of computer programs to facilitate origami or is this just considered the new wave of of how we're pushing oregamy to its most sophisticated end. There's a lot of diversity in the world of oregamy. People fold in different styles and they adopt different customs. And some of the customs are things like whether you use more than one sheet of paper. There's modular folding that uses a lot, there's uh there was a long tradition of using a couple of sheets to make like the front and back half of the animal, or whether you use a single sheet and don't use any cuts. I get a little nervous when people just start suggesting what's the purist, because if one person is a purists, that suggests that something else is adulterated. And there's already some you know, some opinion trying to be passed off as as objective. So let's take away the word purists and just ask the question do people have a problem with using computers? And some do, and I'm fine with that because if that doesn't appeal to you, then don't use computers. And in your work, I find that tools help me create better art, and sometimes those tools are physical things like good lighting, a clean working surface, high quality paper, reading glasses so I can look at things up close. These are all tools. Um, computers that help me think are also tools. And then there are sort of abstract mental tools, the theories and concepts that I use for how to represent the shape I'm after and to figure out how to translate that into a folding plan. So I don't really distinguish myself between that some classes of tools are better than others. If I'm working within the mode of folding from an uncut sheet of paper, which is usually the case, then tools that helped me do that better, I think are a good thing. But to use it to al well, you have to understand it. If you're trying to put in screws and the only tool you have is a hammer, it's not gonna going to be of much use. Um you need to know when to use a hammer when you use a screwdriver. And so I don't fault at all people who say they don't want to use computer programs. It's just choice. What tools do you use, what types of folds do you want to do? I love that concept of using the tools that you need to create better art. That's just such an interesting and accurate to me raming Robert. That's so cool. Something else that's fascinating is that kindergarten. We're talking about the early childhood education kindergarten. I think there's only one was created in eighty seven by a German educator, Friedrich Froeball, who not only introduced this radical idea of early childhood education, but also built a curriculum that included three kinds of paper folding to teach children principles of what we're talking about, math and art. So you first experienced origami at six, so curious to know what are the mathematical principles of basic origami that four and five year olds can understand and therefore people older than that, and should we bring origami back into early childhood education? Do you think there are a lot of principles that four in five year olds can understand? Things like geometric just even name what's the difference between a square and a rectangle and a triangle um and and by handling physical objects that at least for some kids can build stronger connections than just looking at pictures in a book or listening to words by a teacher. Different people learn in different ways. Some people learn better from words, some people can follow things in a book, some people need to handle objects. And so by introducing more diverse tools and education, I think we can reach a larger number of kids. Now back to the conversation with Robert, you actually were once an artist in residence at m I T is that right? That's right? That was about ten years ago, I think, and you gave this famous lecture about origami and its relationship to mathematical notions like circle packing and tree theory, and you have this great line in your Ted talk from I think just a few years after that that says, the secret to so many fields is to let dead people do the work for you. That line, this allows you to take your problem and look for a way that someone has already solved it in another field. I just find these are both really interesting theories around education, around doing your work. How do you split your oregami problems into pieces that allow you to search for a solution from a dead guy, And how do you teach other people to do that? Well? The way I would teach other people is kind of the way I learned to do it when I went to Caltech major electrical engineering. But I took classes in physics and chemistry and mechanical engineering and lots of different fields, and there was a common approach to addressing problems and all these different fields, which relies on the universality of mathematics. You figure out how to this scribe your field in mathematical terms, and once you can transform your problem in your specific field into an impure, abstract mathematical term, then you can bring to bear all the tools of hundreds of years of mathematical development onto your problem. If you can convert an oregonmy design problem into a nonlinear constrained optimization, which is a type of math problem, then you can use the work that hundreds of people have done on how to solve a nonlinear constrainted optimization, and then you solve the math problem. And then you apply that back to your field and you have a solution to the thing that you were after, whether that thing is building an electronic circuit or making a laser work or folding an oregonmy piece of art. The math doesn't know or care what it's being used for. Wow, that is so interesting. Do you have a way of looking at a piece of oregonmy or thinking about the end result of the oregonmy and figuring how what Matthew would want to use, like how to find the math to get you where you want to go. Yeah, when I'm going to fold an oregonmy figure, I start by thinking about, in broad terms, how I'm going to deconstruct it. What kind of shapes I need to build that object from. You know, if it's a unicorn, I'd want four legs, head, and the tail years a horn on the head, and so those are the parts. And then I think about mathematically what shapes of paper could I use to make all those parts in isolation? And then imagine having a little little piece is a paper for all those parts legs, arms, heads, forth? Then how do I pack those shapes into the square that I want to fold it from? And finally, mathematically, how do I connect the folds from one part to another part. There's definite mathematical rules now that over the last twenty years that we've worked out to address each of those steps. So in the end, I can plan out pretty much the whole design before I start folding. Now, I still discover things as I'm folding. I'm folding my design and I see serendipity fortuitous alignments where this fold is close to that point. So if I shifted over, I'll need fewer folds, it'll be more elegant. I'll make those changes so I do discover things along the way, But the overall layout I can achieve by using math. There's such a point in so many artistic process sees of I just want to bail, you know, like this isn't what I thought it was going to be. I thought that I had it outlined, you know, and sometimes it's like figuring out, yeah, this is when I should just scrap it and start over, or let me try to push through this and find a way out of it. Have you ever experienced any challenges like that in your folding process? Or is the question that I was going to ask, have you ever gotten to the point where you finished something and then had to still a piece of paper left over and you had to figure out where to you know, shove it somewhere. I'll just the send question because it's quicker. When I planned out a design, I know pretty much from the beginning whether and where there's going to be paper left over, and I might intentionally design it to have some extra paper and then just say, Okay, I'll figure out something I'm going to do with that bit of extra paper at the end of it. But it's rarely a surprise that there's extra paper because I've designed it all. But as your first question, do we ever get to a point and bail all the time? In fact, I would say two thirds of the time, I've worked through a design and say, you know, I did my plan, I followed my plan, I got the results, but there's something aesthetically lacking here. And the thing is, we can mathematic size structure and form, but we can't, at least not yet, mathematics size esthetic values. So I can design something and have it come out exactly the way I designed it, but once it's folded, I look at it and say, no, this just isn't up to snuff aesthetically. And I'd say that probably happens more often than not. I like mathematic size. It's not a word I've used very often, but I think I might going forward. Yes, I have a question for you. For I don't know four hundred years or however long. Oregony has been a thing. It's been you know, beautiful but pretty simple. Call it thirty steps or so according to the research that I did. But now you can have hundreds of folds, hundreds of steps. And as oregomy became more complex, it actually has also become more practical with applications of these folding techniques to all sorts of things medical, electrical, nanotechnical devices, even strands of DNA, things that have a fixed shape but need to be packed tightly and in an orderly way. Can you tell us a little bit about some of the scientific applications of orgamy folding techniques. Yeah, the practicality didn't require the complexity to come along, but it it kind of evolved in parallel. And I think both the practicality and the advance and oregonmy techniques developed because there became a critical mass of practitioners of oregonmy. It became popular enough, well known enough that you had enough people working on it and most importantly, sharing ideas, sharing concepts that accelerated each other's development. And that was what made the difference in the explosion in design over the past couple of decades. And I think that also was what made the difference in the expansion of the applicability of oregamy engineering problems. The place that oregamy plays a role in engineering problems is when you have something that's fundamentally sheet like and it needs to transform its shape. And very often that's because well, the active state is something flat and sheet like, but it needs to get smaller for storage or deployment. So examples of that some of the really big spectacular examples or things like telescopes, solar or a raise lenses, antennas. These are big flat things, but because they're going up in an airplane or a rocket or something that doesn't have a lot of room, they need to be made smaller for the journey, and oregamy, or more broadly, folding, is a way of doing that in a very well understood, controlled way, making it smaller and then making it larger. We also find oregony playing a role when you want to fabricate an object from sheet stock, and there it's probably being driven by economic reasons. You're using sheets or rolls of material, and so cutting things and folding them and then assembling them give you packages and boxes, decorative objects, but also smaller scale deployable things. You know, implements for backpacking, medical devices that you want to go into the body through as small a hole as possible. So these are all examples where oregonmi plays a role. That's interesting. I'm curious, Robert, when you're creating for yourself, meaning not for a commission or something like that, where do you draw inspiration from? What are your sort of favorite things to to fold and and to create? And uh, I guess in addition to that question, what are you curious about folding and creating right now? Well, my interests fall into two pretty different categories. I love them both. One is the natural world. I've always loved nature as a kid. Some of my happiest times we're tromping through the woods and looking at plants and animals and the like, and uh, getting excited when I saw an animal in the woods. And a lot of my folding, because animals have been one of the traditional subjects, have a lot of my folding has been trying to capture in an oregonmy figure the emotional impact that I see when I see the real creature out in out in nature somewhere. So I do a lot of animals, and they're usually things that I either have actually seen or I want to see because I want to when I see the oregonomy figure have the same emotional feeling that I did or would seeing the animal. The second category is more abstract and geometric, and there I'm driven as much by mathematical considerations as anything. That is because I'm trying to understand the mathematical laws that define what one can fold and mathematical algorithms that lead one to specific folded objects. Um. A lot of the geometric things that I fold are based on solving a particular math problem. There's a class of shapes that I want to fold. I work out the math that would describe that class, and then exercise the math and also tested out by trying to fold shapes in that class. That's so interesting. I loved when you were talking earlier about you know, can we mathematicize aesthetics and and currently we we can't, and that you can figure out how you respond to the aesthetics through the doing. I would imagine it's similar in terms of getting that emotional experience. How do you know when you found it in what you've created? Is it because you look at it and and it resonates in the way that it has with the natural object. Yeah, there is a resonance. It's not clear to me. For any given subject, what is the feature or arrangement of features or arrangement of lines that resonates. And that's maybe one of the biggest frustrations or unsolved problems is is but that I can't put my finger on it. Yeah, because if I could ahead of time, then I just bake that into the design and the whare we go and success. But but so often it's something in kuwait and I only know it once I've seen it. So I have a slightly off topic question, um, but still relevant. In our research for interview with you today, we read that you met your wife Diane at cal Tech when you both had roles in a campus production of The Music Man. So we have to ask who did you play and how well can you sing and dance? Well? I was the newspaper publisher, who who was the base in the barbershop quartet? So you do sound like you've got a voice for a barbershop quartet. So I can sing somewhat, usually bass or baritone. As far as dancing, Um, every time I tried to dance, the insurance rates went up for everyone. There's a math equation there. Yes, bonus question, what did Diane play. Diane was one of the pick a little ladies. Little talk. Yeah, she's saying that an over too higher. That is fantastic. I was just thinking that there is some math equation in terms of the direct proportion to you dancing and the insurance rate's going up. But last almost certainly. Yeah. So, Robert, what's one time where you felt limited in your education or career? And this can be anything and in any way, and how did you overcome it? An example of when I felt limited actually would be my entire approach to origami design. Some of my car I'm pretty good at origami design, but some of my colleagues and friends are just unbelievable. My friend John Montroll has the ability to completely visualize how to fold an object in his mind. He doesn't draw out a plan like I do. He just sees what the solution is. And if you say, how did you come up with that? He said, he'll say, I just saw the solution and he knows how to fold it. So he can visualize all the relationships between every part of the paper and how those different parts come together into the finished figure. And and there's others like that, and I can't visualize in that way. So the way I overcame it was to figure out how to use math to fill in that gap. Even today, I can't visualize everything going on in one of my designs, but I can draw out a plan. I can design it a little bit at a time, and as I keep working through the different parts, eventually I have a complete design. And because I followed the math, all the folds play nicely together and give me the finished shape. So I actually think that limitation was what drove me to develop mathematical tools for design, and had the additional advantage that I could teach my design techniques to lots of other people. It's pretty hard to teach someone to just see the solution, just do that. But so I couldn't teach them to design the way John can. But I could teach them to design the way I can by breaking it down into these little steps and using math as the glue to get you from step to step to step to the finished piece of art you were after. It's so inspiring that you didn't say I don't have that, so I'm going to stop. Instead, you said, Okay, what do I have? What will work for me? And then took it a step further and said, how can I teach that to others? That's just incredible, It's amazing. Okay, we have one last question for you, and then it's time to move on to the lightning round big picture. What is something really big and crazy that you want to take on in the next call it five to ten years curved folding, developing the mathematics to design curved folding. The math is a lot more complicated. Some of that I had in my educations. A lot of it I didn't. I've had to sit down with math books and I've got a stack of differential geometry texts and on my sideboard that I look through to look for the tidbits that I need to describe folding. So I'm trying to assemble all the bits and pieces of math that are out there. And the math is definitely out there. Some of the relevant math is a hundred years old, others is more recent, but it's out there. But to assemble it all into kind of a coherent hole that both I can use and that I can teach to other people. My worst grade in my entire mathematical undergraduate degree was differential geometry. So I will not be helping you not that project, but I commend you for taking it on. Nor will I since I don't recall differential geometry in my educational upbringing. But Robert, I cannot wait until you crack this, particularly to hear how you're going to teach it to others. We will wait in excitement for that. Very exciting. Well, indeed, it is time for the lightning round, Robert. The way this works is we're going to ask you a handful of fun questions, and all we ask is that you just give us your first response, whatever pops into your brain, don't think about it too much. And and then we have to remind ourselves to not ask follow up questions. We need it good. Thank you? Okay, So are you ready? Yeah? Awesome bringing, Christina, do you want to kick it off? Sure? Question one? This one's very hard, I'm sure. What are you reading right now? What am I reading right now? Is the listing for a plumber that I need leaky toilet. That has been our most honest response of the show in over forty episodes. I think I think so, I think so? Uh, Robert, what was the last thing that made you go wow? The last thing that made me go wow? I had figured out an analytic model of a pattern called the water bomb tessellation. So how to describe the motion of this pattern in in purely analytic terms? Um? So basically a mathematical expression that described how every single vertex moved as you unfolded and full to the wow that makes me go wow awesome? Okay. Question three? What is the best book or maybe just one of the best books for someone interested in getting started in origami? I would say any book by Gay Merrill Gross or Nick Robinson. They both write beginner books that are very clear, very beautiful, and very fun awesome okay. Question four? Other than oregamy, mathematics, lasers, and engineering, what's something else that's in your personal ven diagram? So maybe something that's not super obvious to people who know you professionally that you're also interested in um plants. When I was growing up, my brother and I built with my dad's help. We all built a greenhouse and we raised plants together. I was part clearly interested in ferns and vermillions, and even though I don't raise them anymore, I'm still quite interested in both. Very I know what ferns are, I don't know what the other one is, but I'm not really to ask that would be a follow up. We're going to have to save it for research later. And then you chied me for telling you a pineapple is one type of vermilion. Okay, my interests peaked. Continue Christina, Okay, last question and then we're all done. Give a shout out to a woman who is doing awesome things in the world of ore gamy. Oh there's several, but the one who I'm really impressed by is Beth Johnson. Very cool. I if I could ask a follow up question, I would ask what she's up to. I know sheet that he may or may not choose to answer exactly like Style the Wire. Well, very cool. You know what, we will do our own research on Beth Johnson and find out what she is up to. I'm sure it's a lot of really neat things. It is. It is really neat. Now say, even though this isn't a follow up, the thing that I love about her work is it's not super complex. She uses some bits and pieces of complexity when needed, but everything is elegant and perfectly balanced and esthetically beautiful. It's this thing that I said, I wish I could put my finger on and I see it in almost every one of her works. Oh I, I love that and that there is that like you said that in Kait thing right, that really just shows in the art of it. Awesome. Well, thank you for not answering my follow up question. That was not a follow up question, just chiming in some extra information about Bath. Much appreciated. Well, Robert, thank you so much for joining us today. This has been so much fun. You're welcome. You've been my pleasure to have a great night. You too,