How objects move, how materials change shape over time, and the physics of how all of this happens can be most naturally described in terms of geometry. That's the natural language we're talking about in the motion of objects. When you shake your head and your hair goes all over the place, or when you pour water from a cup into another cup, all of that motion can be written down in terms of a language called differential geometry.
If I had to distill what it is I look for when I try to understand the laws of motion, it's a couple of things. One, I think it's really interesting to look at geometry that can be explained in an integral, holistic sense rather than a differential or specialized sense. For example, if you built up a tensegrity structure, you don't know if it will be a stable structure by looking at any one connection. Looking in that way just tells you about that specific connection. But the only reason the overall structure can stay up and be stable is because everything is collaborating together. If you go back to Kepler's Second Law of Planetary Motion which describes the planetary orbit around the sun, it's not looking at the instantaneous velocity of the planet and the forces, it's looking at what happens over finite pieces of time and furthermore, how it applies to any finite piece of time anywhere. And so, in that sense, it's also a sort of global description of what's going on.
Some of the earliest work I've done in graphic animation was about refining and adding more detail to a simulation where it's needed, in other words, converting computer power to the most important aspect. I have a philosophical struggle with some of that work because it can serve as a band-aid for getting the geometry wrong. If you don't get the geometry right in the design of a physical simulation things will basically look ok and the more and more computer power you throw at it, it'll look better and better, so you can just decide where things don't look ok, throw more computing power at it and cover up the fact that you didn't get the geometry right to begin with. So after doing some of that early work and thinking about the geometry, I started thinking how much can you do without adapting and without diverting computer power, because that exposes you to how much of the geometry you got right or wrong. And now we're coming back full circle, and I've been very interested in the deformation of surfaces.
A piece of paper bends, but doesn't stretch or compress. And if I take the paper and I confine it to be in a small space it starts to take on very characteristic shapes. Not that we've seen this particular shape before, but if I showed you a photograph of it and asked if it was paper, you'd say, yes it's paper. Your eye can tell that this shape has come from something that was flat and has developed into something not flat without stretching or compressing anywhere. That's what I mean by a developable surface -- something that deforms without compressing or stretching anywhere and starts flat. It also has the characteristics at any given point that there's a straight line that's not bending. So if you pick any given point, you can always draw a straight line. It could be curved everywhere, but it always has a straight direction everywhere. It could have some singularities -- some special points -- where the curve seems to go into infinity. So, a piece of paper is something that would traditionally have been an ideal candidate for adaptive computation, because it is really hard to model, especially the singularities.
With adaptive computation we would refine our meshes to tackle the more complex bends and use big meshes to model the flatter areas. But another way to approach this would be to realize that the paper has a bunch of straight directions everywhere and that maybe you can come up with a discrete representation of this phenomenon that's adapted not by amping up the computer power or refining certain regions, but it's adaptive just in the sense of being smart about how it forms the mesh. You could imagine if I told you to take some plywood -- certainly something that doesn't bend -- and approximate a shape of bent paper, then as architects you could do it. But if I told you that the winner is going to be the person who approximates this shape with the fewest cuts to the plywood, then someone would get clever about it and get just a few pieces of plywood to make this shape. So, now it's not about more computing power, it's about intelligence reallocating the computing power by choosing the mesh that actively tracks the geometry of the surface. And that's challenging because as the surface changes, where you want to place the edges on the mesh is going to change. So then it doesn't become about refining the mesh, but about sliding the edges and singularities -- and the singularities can move too from one place to another. So then it becomes about how can you update the mesh as singularities and the edges of the mesh slide around. But it all boils down to the same thing, how can you help the computer to best allocate its resources, but also respect the geometry of the model.
A lot of the projects we're doing now are in collaboration with experimental physicists. They'll say, here's the data we're getting and we'll say, here's the data we're getting, does it match up? If it does, great. If not what are the differences? What are they overlooking? What are we overlooking? We're doing a beautiful project right now with Pedro Reis at MIT on coiling spaghetti. When you cook spaghetti and you take it and drop it on the table it's going to start coiling. But it doesn't only coil in one direction; after a while it switches and starts coiling in the other direction. And it seems to switch in periodic patterns. Why can't it just pick a side and stick to it? It turns out that as it's falling and turning one way, it's accumulating twist at the top end like a capacitor. And once it accumulates enough twist at the top, it overcomes the initial direction of movement and it starts going the other way to undo that twist like a torsional pendulum. The interesting thing is our simulations were not able to repeat this. So we were trying to match up with MIT's experiments and when we ran the simulations the spaghetti always coiled in one direction and stayed in that direction. This made us ask: is the thickness right? Is the height we're dropping it from right? Are we using the right type of material for spaghetti? Is the friction correct? Everything seemed to match up.
The problem was that we were assuming at the beginning of the simulation that the spaghetti is perfectly straight, but usually when you've had a cable for a while and you store it, it's never perfectly straight. It always has some amount of curvature. One of our graduate students decided to try and give some natural curvature to the spaghetti, as if it was a cable that had been stored on a spool for a while. And sure enough, as soon as he put the natural curvature on it, we started to see these inversions happening. So he just took the natural curvature and adjusted it until it was matching the rate of inversion in the video from MIT. He was in New Zealand visiting a studio, so he was across the world. And he emailed MIT and asked, by any chance are you storing the cables on a spool of such-and-such diameter? And they went back and measured and responded that he had predicted the spool they was storing the cable on to less than 2 percent.
And that would not be easy to discover in the lab because you never have the option of comparing to a perfectly straight cable. Whereas on the computer you can represent perfect things, you may not want to all the time, but for purpose of understanding, if you want to factor out something like the effect of natural curvature, you can represent the perfectly straight thing. On the other hand in the lab setting you have both the aches and pains, but also the beauty of not being able to neglect any of the physics. You can't turn off friction, which means any effects you're seeing are real effects. Whereas on the computer, if you forget to turn on friction, you might get an effect that is unrealistic. Or if it doesn't occur to you that the air might play a role, then you might get the wrong physics. So the lab and the computer play hand-in-hand because the lab experiments work by starting with all of the physics and trying to control as much of it as possible. And the numerics or computers start with a blank canvas, with no physics, and add more and more physics until it matches the experiments; and they need to meet. That's where the understanding starts to happen.
The only reason we have Internet access between here and Europe is because somebody put a cable on the ocean floor. And the cables on the ocean floor are miles and miles away from the ship moving along and dropping the cable. This is the problem of coiling spaghetti. If it hits the floor and it's all stretched out, then, when an earthquake hits, it snaps. So, it needs to hit the floor with some give, but how much give? If the ship moves too slowly and drops too much cable, then it's a giant waste of cable and there's lots of coiling in place and it can get knotted and that's bad for reasons of cost and communication speed. So, the ideal is dropping the cable with a nice, meandering pattern that has some give but not too much give. So, you need models on how fast the ship moves. But now we're realizing that it's not just how fast the ship moves, but it also depends on how they store the cable on the spool on the ship and how much the cable has been altered in the storage process. That's a new discovery based on this observation in this very simple spaghetti experiment. And now we can predict how much a cable has been altered by being on a particular spool.
To another extreme, if you want to make stretchable materials with electronics embedded in them, then you want the wiring to have give just like the telecommunications cable. So now when you drop the molten wiring onto a substrate and it falls into a line and you strengthen the material later the wire will snap. If it has loops, that's bad because then it short circuits. So again, what you want is this meandering pattern with some give that doesn't short circuit. You could do this with a tiny little print head that moves back and forth in a meandering way, or you could have a model that says if I drop the wire from a certain height at a certain rate while moving laterally, the necessary pattern will form. That's a little extra work up front for a much less expensive engineering later. And if you want to keep working at these smaller and smaller scales, you can't design print heads that move at those small scales. So you have to rely on this passive formation of this weaving pattern. So, any of these very basic experiments we're doing with motion will have applications to stretchable electronics or telecommunications or packing of shampoo bottles or transportation or visual effects or medical surgical simulation or hair products -- it doesn't matter because they're such very basic questions.