You are here

Interview: Tom Hull - Beautiful Mathematics

Edited by Sara Adams

This interview was conducted by Liliana Badillo at the British Origami Society Convention Autumn 2012 for Mini Neo. It was translated into Spanish and published in Mini Neo Issue Number 13. The editors of Mini Neo have generously allowed us to reproduce this interview in English, the language the interview was also originally done in.

-- Sara, February 2013

During the BOS convention, I had the opportunity to talk to Tom Hull. He is an associate professor of mathematics at Western New England University and is well known for his expertise in the mathematics of origami. We spoke about many interesting things, such as his experience in designing modular origami, his inspiration, and the way he uses origami for teaching mathematics.

What is the meaning of origami to you?

When I think about what origami means to me, I always look at the math behind origami. The reason for that is that I find math really beautiful. When people look at origami they see something that they think is pretty because origami is a way of simplifying things; it's a minimalist art form. But a lot of it is what, I would call, the underlying mathematics in origami. It makes patterns emerge and actually lends itself to simplicity. And so to me it is all kind of the same stuff, math and origami. The things that I find pretty in math and the things I find pretty in origami are kind of very similar, but that's hard to describe without actually teaching some origami.

How did you first see the connection between origami and mathematics?

I first saw it when I was really young, I think I was maybe 12 or 10. I had folded a lot of origami by that time. At some point, I made a model and I thought, "Well let's unfold it." And I remember looking at the crease pattern and thinking there's math here! And I didn't understand any of it. I thought "I don't know what this math is, it is all way beyond me," at that time. But there was clearly some geometry going on. There was something about those crease lines that told the paper how to fold into this shape. And I thought it'd be really cool to understand that. At the time there was no way I could learn it. I was really young, but it kind of stayed with me. Then by the time I got to College and then went onto Graduate School, I started trying to look for things that people had written about the connections between math and origami. There were a few things out there, that got me started. There were a couple of origami books that hinted at the connection. They mentioned things like Kawasaki's Theorem or Maekawa's Theorem but they didn't prove them, they weren't math books. But they gave references to it, and that got me excited. Now it's gotten a lot easier, there are a lot more references out there and the internet helps a lot. Back in the late 80's early 90's it was rather difficult, but it all started around then I think.

When did you actually start creating origami?

Actually, fairly early on. The first origami book I owned was a book called “Origami” by a Japanese origamist named Toyoaki Kawai, I think, and he specialized in making origami masks. I really liked them, they were very angular Japanese masks like devils. That got me thinking that I should try to make some of my own masks. The challenge of making an origami mask is a good way to start creating, because a mask can be anything. You can start folding paper and try to see a face in it, you know? I think it was a really good exercise for me to explore origami creatively. And then, after that, I think when I was in College, I started making my own models. But I also remember learning about Tomoko Fuse in College, and really liking modular origami and thinking I would never be able to do something like that. I did not understand how she could have come up with these things. It seemed way beyond me, but there was something about modular origami, and I guess the math side of it too. The more you study it, the more you do modular origami, the more you absorb it. Before I knew it, I was designing my own modular models. So it's been going on for a long time, I guess. Starting with masks and some animal models and then getting to modular stuff, which is what I mostly do now.

One of your most famous models is the five intersecting tetrahedra. How did you come up with that idea?

That object is a classic mathematical object. You can even see it in some of M.C. Escher's pictures. I had a poster in my office in Graduate School that had that object on it. I remember thinking one day: I should try to make that out of origami. The picture was not perfect, it was tetrahedra frames. It was computer generated and so it was more artistic than practical. I looked at that and thought, "Well, first I have to make a tetrahedron out of frames." I had a book by a friend, Francis Ow, and I knew that he had a unit that would do that perfectly. So that was step number one: finding a unit that would do the work. It wasn't mine but who cares? And then I thought, "Well, now I have to figure out what width I have to use," and that took a bunch of math to figure out. But luckily I started with a one by three rectangle, which is close enough. And then it was just a matter of putting it together, and getting it to work. I got a bunch of my fellow Grad students together and we folded all the units and then tried to see if we could figure out how to put them together. It took a group of us to experiment while looking at the poster and trying to achieve this, but eventually we got it and that was it. So I had a lot of help. Once we got one done, then I was able to say, "O.k., let's do more colors, something more difficult, or make it more interesting and diagram it." And very early on, I put it on the internet. I had a webpage back in 1996. I think one of the reasons why it is so popular is that it's been on the internet for a long long time, and that seems to help a lot nowadays.

And right after that came your PHiZZ unit?

Right around then, maybe a little bit before but with the PHiZZ unit I wanted to achieve something very deliberately. I wanted to make a new unit whose lock was really strong so that I could make large things with it. That was my only motivation. I was a huge fan of the Sonobe unit, so I wanted something like that, but that would make different kinds of shapes. And I don't know how I stumbled upon the PHiZZ unit. But I was deliberately trying to find a new unit with a really strong lock, and that's what I came up with. Again, I think the reason why it's so popular is because I put it on the web right away. Those instructions have been on the internet since, I think, 1995 or 1996, which is ancient by internet standards. It's been Google-searchable for as long as Google has existed.

What is your inspiration for creating units and modular models?

Math and in particular geometry. I think, half of the time when I want to try to create something new, I see something in mathematics first that I think is really pretty, and I want to try so see if I can express it with origami. That's what the Five Intersecting Tetrahedra was. Once I designed the PHiZZ unit I realized I could make buckyball structures, and I was like, "Oh, let's see what that looks like!" On paper buckyball structures are just like molecules, but when you make them in origami it looks very different. I like that. I like trying to see what different representations of classic mathematical things would look like out of origami. One of the things I tell people about is that math really is all about studying patterns. Math is not just about algebra and equations; it is about understanding patterns. Those patterns could be in numbers or could be in geometry or they could be in nature or who knows. And it doesn't matter which part of math you're looking at, you just want to study all types of patterns. So I find that in origami I'm trying to do the same thing. I'm trying to take interesting patterns and express them with paper and there's lots of ways to do that. You can do that with tessellation folds, you can do that with modular origami, with crease patterns and other things. So that tends to be my inspiration, finding new patterns and thinking, "How can I do that with origami?"

Do you have some ideas for new designs? Are you currently working on some new design?

I have been experimenting with something new, and you can see this in the exhibit downstairs. I like very geometric folds, but something I discovered happens a lot is that I make some geometric origami, and it's cool. But then I can twist it around into even more interesting shapes. However, when I let go the model just goes back to something flat or whatever. So the way to make paper stay in an interesting shape is by wet folding. But wet folding often makes the paper stretch. I mean, that's why people do it, because they want to make their animal models to be more expressive and stuff. Whereas with geometric origami, you tend to want the lines to be crisp and pure. So I have been trying to explore ways to use wet folding in geometric origami, so that I can capture some of these twisted strange forms. And there is a way. By talking to people who are wet folding experts, like Michael LaFosse in the United States, I've been able to learn techniques that allow me to do exactly that: to capture geometric shapes with geometric origami, but without the paper getting so wet that looks like classical wet folded stuff. Many of my models downstairs were created like that. There's a hyperbolic cube model that's been wet folded, even though it doesn't look like it. But that's why it is staying in that shape. There's a bunch of models nearby that are also twisted around and I find that really fun and sort of pretty difficult. So that's taken a lot of practice, to get that far, but there are lot more things to explore, too. I think there are a lot of interesting shapes that can be made that way. I can't think of anything else right now where I think, "That's something I want to fold." Otherwise, I'm just play around, meet people, and have fun.

Is that the way you create things, by playing around?

Usually yeah. Sometimes I have lots of ideas and I think, "Well, someday I'll try this out," especially things that I know will take a long time to fold. So I have to wait around for time to do it, or an excuse to do it. But a lot of it is exploring. You never know if it's going to work. For most things I may have an idea: "Oh that would be very interesting, let's take this tiling patterns, and just put that one into it, and then try to wet fold it in this way," and half of the time it just doesn't work. You try and say, "Oh no, this is not going to work at all." Or you have an idea for modular folding, and you come up with a good lock that would work, but once you put everything together it doesn't really look the way you wanted it, and you give up. So it's a mix. Before I came here, there were a couple of ideas I wanted to explore. Some of them worked, some of them didn't. The ones that didn't, I left them at home. But it's hard to say specifically I was trying to fold this or that. It's more like I was playing around with this idea and it worked, or it didn't work. So, yeah, I do a lot of playing around.

Whose work do you admire?

Certainly when I was younger, even today I mean, Tomoko Fuse's work is fantastic. I think anyone who does modular origami is probably inspired by her work. But I also really like the work of Robert Neale, I met him when I was in College. The first origami book I wrote was just trying to diagram his models. That was a book called “Origami Plain and Simple”, which is out of print now, and which is all about Bob's models. I like his work because it is very simple. He's very good at capturing things like frogs, owls, birds, whatever, really really simply in just a few folds. And I think that's the hardest origami to design. Sometimes I try to strive for that kind of simplicity, but it's also difficult given what I like to do with geometry. Also, Robert Lang is an incredible inspiration. The things I like about his work are the math that he uses in a lot of his models or that he explores. He and I talk about stuff a lot, but certainly he is a big inspiration.

What do you think your life would be like if you hadn't met origami? What do you think you'd be doing?

I don't know. I might still be a math professor, because I like that a lot, but it's hard for me to imagine what kind of math professor I would be without origami, because I use origami in my teaching so much. I don't know. It's been going on for so long that is hard for me to imagine. I mean, if I hadn't found origami, I probably would have found something like knitting or making string figures, I don't know, something to capture that side, because I do like making things with my hands and I like mathematics. So I don't know, it's hard to know.

You mention that you involve a lot of origami in your teaching. What is the message that you want to communicate to your students?

Well, it depends. A big one is that at least in the United States, probably all over the world people, students - especially in high school - get the impression that math is just factoring quadratic polynomials and doing all of this mechanical stuff that seems very divorced from the real world. It's no wonder that students think "What's this good for?" And the thing I think is a shame is that mathematicians don't look at mathematics that way at all. Mathematicians look at mathematics as studying patterns and I think origami is an easy way to have students get a glimpse of what that's really like. Math is more than just solving equations or factoring numbers or whatever is actually being studied. It is often about studying something from the real world, looking at its patterns and then trying to, say, make theorems about it, make conjectures and prove them. Any time I can get students to see that I feel like I'm doing something very worthwhile, because that's really what math is about. And if you understand that, then it's much easier to go back and tolerate all the factoring polynomials, because you understand what all of these tools are for. So that's the perspective of what math really does. It's difficult to see that. So that's the big thing. But another one is that I often teach people who are teachers as well. Teachers are always looking for ways to get their students excited, and origami is an easy way to do that. There are lots of ways to have students fold paper where they will be doing and learning math without knowing it. So it seems like a really practical thing to do as well, I think a viable way to teach math is getting students interested in ways that other things didn't.

What do you think is the future of origami? Do you think it is going to become more and more popular, especially when teaching math?

I hope so, it does seem to be becoming more popular. It's certainly becoming more popular among teachers in the United States. I don't know about the rest of the world as much, but I do know in England there's actually a lot of effort to incorporate origami into the math curriculum. Ever since I have been teaching at College-University level I ask my students, "How many of you have folded paper or done origami before?" It used to be only a couple of people who would raise their hands, but now it's definitely more. I don't know why, but more people - maybe in school - are doing origami. Or at least everyone knows what origami is, which used to not be the case. So it's definitely gotten more popular. It's very hard to say how far it's gonna go. Because even in Japan, where origami is very popular and everyone knows what origami is, most adults in Japan view origami as something that only children do. So there can be problems with it becoming more popular, if it becomes associated with just a child's pastime. I don't encounter that in the United States much, because there has been a lot of interesting applications in science and it seems that, at least in Colleges and Universities, people understand that origami is very useful in different aspects, as well as being something that's fun and artistic. There's interesting math, interesting science in it so it has more legitimacy and I hope that grows, instead of it becoming, "Oh that's just something kids do.".

There seems to be two different trends in origami, one of them is the one that tends to do more realistic origami and the other tends to make a more simplistic and expressive origami. Do you have a preference?

I don't think I have a preference and it's not something I actually ever worried about. I understand the debate you're talking about. For a long time, especially when Robert Lang was developing his tree maker algorithm and there were more and more people making more and more complicated models. But a lot of them were kind of ugly, because they were just lots of points and crumbled paper involved. But that's changed. If you look at the really super complex models that people are making now, they are more artistic than they were ten or twenty years ago. I find that people are not satisfied with just pure increasing complexity, they want the models to also to look good. It means having an artistic component. For example, there are a bunch of people who have invented lions and have tried to capture the mane of a lion in different ways. Some designs have a mane with lots of layers, use pleated paper to represent the mane, something really complicated, but very artistic. I think Joseph Wu invented a lion that uses some kind of weird box pleating to capture the mane. It looks fantastic, and it's very complicated. But it's one way that is very inspiring in terms of how pretty it is. At the same time I know how difficult it is to create elegant, simple models and have it still work and look good and be something that no one has ever done before. And some things that look really simple are incredibly hard to fold. I'm thinking of, what's his name? There's a guy who does really simple looking things with just a few folds. I'm thinking of Giang Dinh. So I don't have a preference. I want to see both continue, to go on in their own directions. Because they both are valuable. I think people worried that origami was getting too complex and too ugly, but what I've found is that no one is satisfied with ugly origami. Even people who want to do ultra-complicated things always look for ways for it to be beautiful in the end. Whether by wet folding, or using special paper that allows you to have really lots of layers, but still thin limbs for insects. People have been finding ways to still have it be gorgeous, but really difficult. So it's all good.

Links

You can find more information about mathematics and origami at Tom’s website: http://mars.wne.edu/~thull/ and his Flickr album: http://www.flickr.com/photos/33761183@N00/. He also has some video instructions at his YouTube channel: http://www.youtube.com/user/tomhull17