Not Alone
In a few weeks, my half-year sabbatical will come to an end. Though I’ve spent much of the time in an international collaboration, I haven’t travelled beyond Nova Scotia except for a few vacation days in New Brunswick. Our work took place by email, and it seems to have produced results. In some ways, it would have been nice to travel: Budapest is beautiful (though the language totally defeated me on my previous trip.) But relocating for months would have made no sense, and the work progressed slowly, so that a couple weeks might have made little difference. So I saved a bit of carbon and worked from Halifax. Not in Budapest, but not alone: modern technology kept me in touch with my coauthors, often daily.
In earlier work, we’d shown that a properly-shaped tetrahedron can be weighted to be stable on only one face. Weirdly, if this can be done for one face of some particular tetrahedron, a different weighting can make it monostable on any other face. Early this year, my Hungarian colleagues had built a working model. It had been surprisingly difficult: the outer tetrahedron was a feather-light skeleton of carbon-fibre rods, the weight was precision-machined tungsten carbide, and it _just_ worked! The design had made use of several heuristics and hunches, as well as innumerable machine cycles of numerical optimization. Now, “the proof of the pudding is in the eating,” and the _Bille_ (from the Hungarian word for “tip over”) certainly worked. But we wanted to know why we had to push the limits of available materials to make it happen—and why two of the four falling patterns would (it appeared) have required materials known only to science fiction!
While our work was based on ideas (solid geometry, centres of mass) familiar for centuries, it rapidly took a turn into new territory. Nonetheless, we were not in an untracked wilderness: when we needed guidance, there were often blazes on the trees. We got one basic idea from a puzzle-problem in an old Martin Gardner column, though it needed a lot of work to adapt it for our purpose! At one point, it looked as it we would need a long and complicated piece of multivariate calculus to prove a technical point, but (after translating some notation to work with what we were doing) we found that result (up to a scaling constant) in a paper on integral geometry. Though we were working on a quirky new problem, the literature kept us in touch with the larger mathematical community, even across the decades. We were not alone.
And my wish for all our readers in the new year is this: may you never be alone in your mathematical endeavours.
