As we all know, one cannot pull oneself up by the bootstraps, except for Baron Münchhausen, who is told to have lifted himself out of a swamp by the simple device of pulling himself up by his own hair. As for myself, i’ve pulled my hair lots of times to no avail and, as a result, i have some empirical reasons to question the Baron’s deeds!. You probably know the that the reason for our boostrapping inability is, basically, Newton’s Third Law, from which it is easily derived that the center of mass of a set of bodies can only accelerate as the result of external forces. Thus, no matter how much wriggling, you won’t be able to swim in empty space. Or will you?

Center of mass in a sphereWell, actually there’s a twist: you won’t be able to swim in empty, flat space. As it happens, curved space gives you rope enough to boostrap yourself! This well known phenomenon is explored in detail in the delicious Swimming in curved space or the Baron and the cat, recently published by the New Journal of Physics (which, by the way, is an open-access publication, although it maybe falls a bit short of what every scientific journal should be). The authors provide a detailed account of how Riemannian curvature can be used as a sort of handle to pull you around. Even if you don’t feel like following all the mathematical details, you can see movies of the swimming in action over a sphere or a hyperboloid, showing how the periodic motion of three particles on constant curvature spaces gives rise to a net displacement of their center of mass (which carries the particles around in the process). As the article explains, the gist of this effect hinges on the fact that the center of mass is only well-defined in Euclidean space: if you take for instance three equidistant (yellow) points on a sphere, their center of mass would be located at the red point, but if you move the two bodies on the right to the red dot, the new ‘center of mass’ is displaced to the blue dot.

Swimming in a circleThere are other simple examples of boostrapping. For instance, in spaces with non-trivial topologies like a circle. In there, a particle (red) can split in two parts (blue) that recombine in the opposite side (hollow red) without violating Newton’s law. A similar effect is possible in spaces with intersecting geodesics, as i’m sure you can convince yourself with a little thought (or just cheating and looking at the article’s appendices). Note also that this fun magic works on the basis of geometry alone: there’s no need of relativity or anything, just plain Newtonian mechanics.

Over at MathTrek, Ivars Peterson is talking about Edwin Abbot’s classic Flatland. Taking into account how fun this little book is as it is, just imagine how much more interesting could the history have been had flatlanders lived in a curved two-dimensional space. Any takers? :)


One Response to “Bootstrapping”

  1. Michael Says:

    The edition of Flatland I bought came with a sequel called Sphereland, which is exactly that: the surveyor-general discovers that his triangles don’t contain 180 degrees, and slowly it emerges that flatland is in fact curved. (If I remember right there were also some attempts to give women a post-Victorian role.)

    I think there have been many sequels written, although I haven’t read any others. Sphereland didn’t live up to the original, and then I stopped looking.

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