The Math of Folding Maps
NOAH ADAMS, host:
Frances Hill's essay got us to thinking about creases and folds, and the properties that govern the edges. And so that is subject for this week's Science Out of the Box.
Dr. L. Mahadevan teaches applied mathematics at Harvard University. He has taught a lot about wrinkles and folds, and he's kind enough to join us to talk about them. Welcome, sir.
Dr. L. MAHADEVAN (Applied Mathematics Professor, Harvard University): Thank you very much.
ADAMS: Well, let's get to the big one here. Here's a roadmap of Pennsylvania and New Jersey. It's all pristine, but if I unfold it and just look at a little bit, it's going to be a mess when I start to fold it back up. Why is that? Why is it so hard to put it back together?
Dr. MAHADEVAN: When you unfold a map, at every fold you just have one possibility, which allows you to unfold it. But when you're tying to fold the map, every fold has two possibilities, which is why when you try unfold the map - since there are so many possibilities, unless you know exactly the sequence in which you do it and how it's arranged, relative to how the fold arranged directed to each other - you know when to get it right.
ADAMS: It's kind of diabolical, the design.
Dr. MAHADEVAN: It is. But the surprising thing is that it only works when the folds essentially independent of each other, which means, then the folds are the right angles.
ADAMS: Right. This map has two horizontal folds and nine going cross vertical folds. How many possible ways could you fold this map back up?
Dr. MAHADEVAN: So there are nine plus two, total of 11 folds, sort of 11, which is about 2,000, roughly, a few thousand possibilities. Of course, that's not how you do fold a map because there is some memory, you know, each fold really doesn't have both possibilities unless you forced it, which is what happens when you get frustrated.
ADAMS: Right. So you should be looking at the memory at the fold line, the crease, right?
Dr. MAHADEVAN: Precisely.
ADAMS: Okay. Now, here's another one. And a colleague brought this in who is a pilot, Charlie Myer(ph). And this is an aviation map. And he points out that this particular map is so nice because it only has one fold in the center. That's a big advantage.
Dr. MAHADEVAN: Absolutely. Or if you have many folds in a map, and if the folds are not at 90 degrees to each other, then it turns out that, in fact, you can arrange it so that the whole map will simultaneously fold up, and simultaneously all the folds will also unfold up when you pull, for example, on two edges, two corners.
And this idea was recognized by Japanese Aeronautical Engineer Koryo Miura, maybe about 35 years ago.
ADAMS: We tried to follow these directions and make a map here in the studio, a Muira map. It's beautiful, but it took 20 minutes to fold up this piece of paper. Is there a way to do this sort of elegant folding without computers and without a machine and without taking 20 minutes?
Dr. MAHADEVAN: Yes. And the way to do it would be by taking a thin sheet such as your map or a thin - a piece of paper just to heed down on to a soft material, and then do this, following in two steps. First, I compress that material along some - in some direction. And when I do that, I will form large number of shallow wrinkles. Now, compress it in a direction perpendicular to the wrinkles that's formed. And if you dot that, each one of those wrinkles will break up into a zigzag pattern. And that zigzag pattern that you essentially find is the per-cursor of the folded map that you hold in your hand.
ADAMS: You're talking about the very elegant Miura map.
Dr. MAHADEVAN: Yeah.
ADAMS: And would you find the same sort of process replicated in nature?
Dr. MAHADEVAN: We think so. When you look at some insect wings, they have a similar problem to the map folding and unfolding. The insect wing is typically hidden bellow a shield. And when the insect wants to deploy its wings just before it starts to fly, it has to take that folded structure because the insect wing is typically as large, sometimes, even larger than the body itself. So they just take that folded structure out from the shield and deploy it and unfold it.
ADAMS: All right.
Dr. MAHADEVAN: So how did they do that? And the answer turns out to be the that it uses essentially this simple Mountain-Valley Miura kind of fold in a repeated way, which allows it then to unfold the entire wing in one shot or two fold it.
ADAMS: So you get - deepen the nature here, it is some of a directed, really, out of more practical way to make a roadmap for us in the car?
Dr. MAHADEVAN: Well, people use GPS devices nowadays, don't they, who has roadmaps.
(Soundbite of laughter)
ADAMS: Right. L. Mahadevan is a professor of applied mathematics at Harvard University. Thank you for your time.
Dr. MAHADEVAN: Thank you very much.