Biomechanics: March 2009

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Indian star tortoise

Illustration by Joe Sharkey

There is something about a turtle on its back that twists your heart. With neck craning toward the ground and legs waving to no effect, it is the image of helplessness. But, malicious kids aside, turtles almost never end up upended. And it turns out that the apparent risk factor for that predicament—the turtle’s rigid carapace—is less a liability than an asset, surprisingly well-suited to the turtle’s goal of righting itself. The secret is in the mathematics of its shape.

Many a truly difficult mathematical problem can be stated in an approachable and deceptively simple way. Can you, for example, cut a convex shape from a uniform sheet of plywood that will only sit one way when balanced on edge? By “convex,” I mean that there are no dents in the outline: a straight line drawn between any two points along the edge will stay inside the shape. A plywood square, for example, has four edges on which it will happily balance (if you ignore the third dimension and forget that it could flop over sideways). In addition, it has an unstable balance point on each corner—a point on which it can balance, but so precariously that even the smallest perturbation will send it tumbling. A triangle comes one step closer in the quest for a shape with only one balance point: it has three stable positions and three unstable ones. An ellipse has two of each. It turns out that this is the very best you can do. No amount of trimming or trickery will get you a single stable equilibrium point in this essentially two-dimensional case.

Resembling the Gömböc, the shape of its shell gives an Indian star tortoise only one stable configuration: on its feet!

Illustration by Joe Sharkey

You might come to that conclusion based on trial and error; proving it mathematically, however, was not a trivial exercise. Gábor Domokos, of the Budapest University of Technology and Economics, successfully tackled that challenge. He then sought to extend the proof into three dimensions. It is tempting to believe that, as in the 2-D case, the best you can do is two stable equilibria and two unstable ones, but Domokos came up with a counterexample. Consider a rod with both ends cut off obliquely [See Figure 1]. It rolls around on the table and stops with the long side down every time. It has just one stable configuration. On the other hand, it has three unstable equilibria: balanced on either pointy end or with the shorter side down. Is that the best we can do? What about a shape that has just one stable equilibrium and a single unstable one—a mono-monostable object, in mathematicians’ parlance? In the 3-D world we are all familiar with Weebles™, those cute plastic toys that wobble but don’t fall down. They fit the bill, but they violate a key condition of the plywood problem: they have a little metal weight in the bottom. Is it possible to construct a Weeble that isn’t weighted, one that rights itself just by virtue of its rotund little shape?

The trial-and-error answer is "no," but that seldom impresses a mathematician. Domokos, joined by his graduate student Péter Várkonyi (now also a professor at the Budapest University of Technology and Economics), began to work seriously on the problem. Using a spherical coordinate system, they constructed formulas that slightly squashed a sphere, added a pair of extra flattened planes on the surface (which they call "raceways"), and pressed one side into a sharp edge. For reasons that are clear only to native speakers of Hungarian, they called this sensuous shape a Gömböc (pronounced something like "gumboots"). Though it existed as a mathematical construct, Várkonyi and Domokos did not truly appreciate how interesting the shape was until they had a model made with a 3-D printer. With the Gömböc in hand, they were struck by the similarity of its shape to the highly domed shell of the Indian star tortoise (Geochelone elegans).

Illustrating the difference between an unstable and a stable equilibrium, a plywood triangle (top), regarded as a two-dimensional object, may balance precariously on one of its three points or come to rest on one of its three long sides. Similarly, a rod with both ends sliced off at oblique angles (middle) teeters on its short side—as it would if stood on either pointed end—but finds stability on its long side. Designed by mathematicians, the Gömböc (bottom) never rests for long on any point except its one and only stable surface.

Illustration by Joe Sharkey

Domokos and Várkonyi turned their attention to the possibility that the Gömböc shape was a naturally occurring example of a mono-monostatic object. They looked at seventeen species of turtles, including high-domed tortoises and flatter pond turtles. By determining the curves that captured the 3-D shape of each turtle shell, they translated the animals into the same mathematical language as the Gömböc. It quickly became clear that tortoises are indeed quite close to being monostable: when placed on its back, a tortoise will roll right side up with just a little perturbation. But pond turtles are quite stable when placed on their backs and must use their necks and ninja-turtle footwork to lever themselves over. (In reality, that rarely becomes necessary, since they stay in or near water, where sudden inversion is not common.)

Of course, tortoises have an advantage over the Gömböc in that they have four legs to flail around, which moves their center of gravity enough so that they will wobble right side up without an outside influence. They are also not really of uniform density, and the placement of the lungs high in the dome certainly makes them more like a Weeble than a Gömböc. The similarities are striking enough, however, both for tortoises and for certain species of beetles, that it seems natural selection has been hard at work smoothing their form into a Hungarian tongue twister.