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A Conversation With Erik Demaine


Erik Demaine took his bachelor's at the age of 14 and at 20 became the youngest professor in the history of MIT. Erik received a MacArthur fellowship in 2003 which cited him as a "computational geometer tackling and solving difficult problems related to folding and bending—moving readily between the theoretical and the playful, with a keen eye to revealing the former in the latter". Erik is one of this year's Katayanagi Prize winners. His website (erikdemaine.org) includes Demaine's work in Computational Origami, and highlights the great beauty found in mathematics. Erik's recent book (with collaborator Joseph O'Rourke) is Geometric Folding Algorithms: Linkages, Origami, Polyhedra (www.gfalop.org)—read the theory, proof, and algorithm and then copy 'em, fold 'em, and cut 'em!

Within two minutes of talking with Erik, the conversation had arrived by its own inscrutable logic at game theory and veered off from there towards epistemology, biology, parenting, and origami.

DDJ: You are noted for your work in folding geometry. Is there a folding geometric solution to the game of chess?

ED: There is no good solution to chess.

DDJ: How can there not be?

ED: It's a matter addressed in the study of computational complexity.

DDJ: The pieces' moves are regular, the board is regular, and there is a certain large number of possible checkmates and many symmetric reflections of them around the board. The Knight's Tour is algorithmically solved since the 18th century, why isn't the game solvable?

ED: The game is harder. You can prove you need exponential time to solve the game. You need to brute-force search through all possible moves.

Erik Demaine. (Photo courtesy of Donna Coveney/MIT)

DDJ: I've been familiar with that assessment since the 1970s. I just can't believe it! It seems to me that if you, say, had an infinite number of origami cranes you folded just right, you could describe chess that way.

ED: That's probably true, but that relationship is not size preserving. You would need a huge number of cranes, or bits, or whatever you want to think about, in order to represent the full game of chess. So even though chess positions can be very succinctly described, the whole game, the tree of possible states, is "gi-normous".

The 8×8 game is trivial. It's not interesting from a theoretician's point of view. I assume it will eventually be solved. Checkers was solved last year.

DDJ: Checkers was pretty much solved by human beings by the mid-20th century.

ED: The solution is not geometric, and I don't know how algorithmic it is, but it is exhaustive and provably correct. They do some tricks to show that certain moves need not be considered because they're guaranteed not to be useful plays. And from the initial position, they've tried everything.

DDJ: Computers can exhaustively examine every move. But is there a mathematical answer, like, "Considered in 15 dimensions a king looks like this, a queen looks like this, starting from a certain position, ergo..."

ED: Computational complexity theory tells you there is no clean answer for games like Chess, Checkers, and Go. If you believe in the perspective of complexity theory, it tells you that all you can do is sit there and do the bookkeeping and exhaust the exponential possibilities.

DDJ: But there's an algorithimic solution to Nim.

ED: Yes and computational complexity theory says that Nim is "easy." There's an easy algorithm for Nim. You can prove that there is no easy one for Chess, and by that I mean an n×n chessboard. Complexity theory only applies to certain scalable problems. But the problem of Chess as it is played is not theoretically interesting...we know the answer.

DDJ: What is "interesting"?

ED: The problems we don't know the answers to. Finding the answer, or proving that there isn't one. That's the name of the game in algorithms: Find an interesting problem that no one has solved before, nor shown that it's unsolvable.

DDJ: Epistemologically, is it really that there is an Ideal against which you can compare a problem, or is it just that, "Our scaffolding is consistent, we're building a scaffolding out to this problem, and we find no way to bridge the gap"? Is there an answer to these sort of questions as mathematically certain as "1 = 1" or is there a shape to our intellectual approach that just doesn't mesh with certain kinds of problems?

ED: I would say some of both: That mathematics in general are as close as you can come to Truth; but still, you do rely on your set of beliefs and axioms as to what makes physical Truth. Most people, at least modulo ["until you reach the domain of"] quantum computing, believe the Church-Turing Thesis that computers are just Turing machines, and that is the limit of computation. It's a bit of a philosophical statement, there's no sense in which you can establish it. You might find the yet-unknown counterexample.

DDJ: Pauwels and Bergier in Le Matin des Magiciens pointed with glee at the establishment science press of the 1890s and its assurance that all the big problems were solved and that all that remained was measurement—this a decade before Relativity.

ED: I think that's a different issue. I hope it's a different issue! There's a difference between that and showing that a certain problem as formulated cannot be solved. There's no shortage of problems that can be solved.

DDJ: : What are you solving now?

ED: We'd like to solve protein folding. That's a hard one, not purely mathematical, but biological, how nature folds proteins.

DDJ: How does nature fold proteins?

ED: How? [laughs] No one knows! Biologists will happily tell you lots of theories.

We'd like to know how it works. The practical matter is that you can already design proteins and build them. The trouble is that when you build them, they will fold, and you'd like to be able to predict how [they are] going to fold and see if [they] do what you want [them] to do. This is usually in the context of drug design. Does it kill the virus without killing humans? You could tell that if you understood how proteins work, and one piece of that is what it looks like geometrically, which dictates in part how it interacts with other proteins and other structures.

How does nature convert DNA into this three-dimensional shape? If you understood that process, then you could create custom drugs that do the things you want. Can we predict it? If I give you a protein sequence, if I give you the corresponding strand of DNA, then using a computer, can you tell me what shape it will make? Is there an algorithm, which explains what happens? Is there a fast algorithm?

DDJ: Do they discover that, given a certain protein sequence, it will fold in just this way?

ED: Again, it's not totally certain. There are definitely environmental factors. That's part of the challenge. You need chaperones to help protein fold in a certain way. But generally it's pretty consistent, the same protein usually folds in the same way.

DDJ: And the behavior of proteins is governed in part by the shape.

ED: It's their shape, and how they fold. They're usually flexible after they form their main shape. But they can still wiggle in certain ways and that controls how they interact.

But the big challenge is, how does that initial folding happen?

And the converse question is reverse engineering. Suppose I want to build a particular 3D shape because it works so well. What DNA should I manufacture that will fold into that shape?

So "How?" is a pretty broad question—we know it involves physics, etc. The more practical problems are as I have stated.

DDJ: You get paid to ponder these imponderables. Most of us have to do it sitting on our porches on weekends!

ED: It's a pretty awesome position to be able to think about these basic mathematical truths and what's solvable and not solvable...

DDJ: Aside from the Secrets of Life, what else are you working on? Are you a multitasker, or a one-project man?

ED: I work on many things at once. I could go on quite a while on different things. Recently, I've been working on decoding Khipu, a conjectured Incan writing system that used knots tied in ropes. What do they say? It seems clear they recorded some numerical data. We see numbers that add up in patterns. But there is linguistic information in there, too. They tell stories. That's the tantalizing thing.

DDJ: How does your interest in geometry, folding, and algorithms come into it?

ED: The geometry is a bit superficial. The interest is from the computational perspective, there's information in the knots, it's fairly digital, 13 or so different knot types, there's color information, other things like the handedness of the knot. How would you encode information into that thing and how do we extract information from it?

Unlike the mathematical things I do, this is not exactly a well-posed problem.

DDJ: Are you very visual?

ED: Yes. I use visualization a lot, of course in geometry, but in everything else I study. I visualize what ought to be true intuitively and then try to prove it's true. But I also learn from listening. I learn a lot from other people and what they do and what their tools are, and what they're experiencing.

I learned to read early, but it never was as interesting to me as personal experience. I didn't read textbooks as an undergrad. My father, Martin Demaine, had home-schooled me until I went to university. He was against the whole school thing, [and] wanted to be engaged in my education. Also, my father wanted to travel, so around Grade 2 we started traveling around North America, Canada, and the United States. I got to see a lot of different cultures, meet lots of different kinds of people, different backgrounds, different ages...

My father and I still work together on mathematical and visual things. His background is in the visual arts. He pulled me into the arts and I pulled him into the mathematical stuff. We have some joint sculptures at MOMA right now as part of the Design and the Elastic Mind exhibit.

DDJ: So tell us how all this beautiful art resulted from your work in geometry.

ED: It started for me with the fold-and-one-cut problem. You take a piece of paper and fold it as many times you like, then you take your scissors and make one complete, straight cut. Then you unfold the pieces and see what you get. What shapes can you make?

Before he was an escape artist, Harry Houdini would do a trick where he folded and made one cut and out came a regular five-pointed star. The general mathematical question was posed by Martin Gardner—my dad and I are great fans of Gardner, we were just at the Gathering for Gardner. In the 1960s, Gardner asked, "What is the limit?" This was the first geometric problem I worked on, with my dad and my adviser Anna Lubiw. We proved that everything is possible, any polygon, any shape made of straight lines, it's possible to make arbitrarily many pieces of arbitrary shapes with folding and one straight cut. You need some pretty complicated folds, but there's an algorithm that says, "If you want that shape, here's how you fold the paper."

It's all on my Folding and Unfolding page (erikdemaine.org/folding).

DDJ: Not only do you tackle the Secrets of Life, but you also fold paper, and children can understand this!

ED: That's what I like about this. The problems I work on are pretty accessible. The solutions are hard, and that's what keeps the customers happy and pays the bills.


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