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Looking Ahead
Apr. 3rd, 2010 10:03 amAt the geometry seminar yesterday, after we discussed our new results, T asked: "Okay; now what?" I'm not sure. I don't see any very promising lines of investigation; there are a couple of things I can see to do, but I'm not very optimistic about them. L's project is somewhat different; he's trying to adapt our techniques to polyhedra with pentagonal sides, and we talked a bit about the hurdles involved there.
One good thing came from the discussion: the theoretical progress we've made in the last few weeks has implications for my geodesic-search algorithm. In particular, there's one bit of the program which, as it stands, runs in quadratic time, but I now see that I can rework it in linear time. Some of the computations the algorithm makes turn out to be unnecessary. I'm going to have to go over it again and try to clean it up - and, if possible, to get rid of the kludge I used to integrate the combinatorial and numerical data.
W wasn't at the seminar, due to an unexpected obligation. We did meet afterward, and talked about publication. His suggestion involves writing two papers - one presenting the material we developed a couple of weeks ago and applying it to the eight convex deltahedra, and another, later, presenting this week's theorem together with the search algorithm. (Those two complement each other; the one says, "In this situation, you don't need to look any further than so far", and the other affords a quick way of looking at everything out to so far.)
Meanwhile, in addition to working on the algorithm, I'm going to be pursuing one of the possible ways of extending our ideas further. As I said, I'm not too hopeful, but lightning has struck a couple of times already; who's to say it won't come again?
One good thing came from the discussion: the theoretical progress we've made in the last few weeks has implications for my geodesic-search algorithm. In particular, there's one bit of the program which, as it stands, runs in quadratic time, but I now see that I can rework it in linear time. Some of the computations the algorithm makes turn out to be unnecessary. I'm going to have to go over it again and try to clean it up - and, if possible, to get rid of the kludge I used to integrate the combinatorial and numerical data.
W wasn't at the seminar, due to an unexpected obligation. We did meet afterward, and talked about publication. His suggestion involves writing two papers - one presenting the material we developed a couple of weeks ago and applying it to the eight convex deltahedra, and another, later, presenting this week's theorem together with the search algorithm. (Those two complement each other; the one says, "In this situation, you don't need to look any further than so far", and the other affords a quick way of looking at everything out to so far.)
Meanwhile, in addition to working on the algorithm, I'm going to be pursuing one of the possible ways of extending our ideas further. As I said, I'm not too hopeful, but lightning has struck a couple of times already; who's to say it won't come again?
