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We Can Has Paper Plz?
Mar. 19th, 2010 03:27 pmAll things considered, today has been a good day. This, despite a long committee meeting. The Chair had had difficulty finding a time when everyone could meet, and wound up scheduling it for 8:00 this morning. 80 minutes of committee, then off to class for a lecture on absolute and conditional convergence, then back for another 50 minutes of committee... Amazingly, we finished everything we needed to do, and won't need a second meeting.
The good stuff came after, when the geometry seminar met. I showed them the theorem I'd proved back on Monday, and then demonstrated some of its consequences. W was dissatisfied - he wants, perfectly reasonably, a rather different theorem, but none of us know how to prove it, or even exactly what it would look like. He and I were batting ideas back and forth when T intervened: could we, at least, solve one chunk of our problem and get the damn thing published?
After some discussion, we decided to finish off the eight convex deltahedra. The three regular ones - tetrahedron, octahedron, icosahedron - have already been thoroughly investigated, and I had just demonstrated a solution for the triangular dipyramid. That leaves four more, and we divvied them up. L, our student assistant, will take the pentagonal dipyramid; W picked the snub disphenoid; T will handle the triaugmented triangular prism, and I've got the gyroextended square dipyramid.
Basically, it means a careful - and possibly tedious - case analysis. (I've done one other example, not from this list, and it was just a bear.) There are five broad cases for the gyro, each of which could sprout subcases, and each of those could give positive or negative results. We have reason to expect a total of four positive results. I've knocked off the three simplest cases - two positive, one negative - and I've also done a little work on case four, which looks like it's going to go Hydra on me. But I've got my sword and my burning brand handy, and I will prevail!
The good stuff came after, when the geometry seminar met. I showed them the theorem I'd proved back on Monday, and then demonstrated some of its consequences. W was dissatisfied - he wants, perfectly reasonably, a rather different theorem, but none of us know how to prove it, or even exactly what it would look like. He and I were batting ideas back and forth when T intervened: could we, at least, solve one chunk of our problem and get the damn thing published?
After some discussion, we decided to finish off the eight convex deltahedra. The three regular ones - tetrahedron, octahedron, icosahedron - have already been thoroughly investigated, and I had just demonstrated a solution for the triangular dipyramid. That leaves four more, and we divvied them up. L, our student assistant, will take the pentagonal dipyramid; W picked the snub disphenoid; T will handle the triaugmented triangular prism, and I've got the gyroextended square dipyramid.
Basically, it means a careful - and possibly tedious - case analysis. (I've done one other example, not from this list, and it was just a bear.) There are five broad cases for the gyro, each of which could sprout subcases, and each of those could give positive or negative results. We have reason to expect a total of four positive results. I've knocked off the three simplest cases - two positive, one negative - and I've also done a little work on case four, which looks like it's going to go Hydra on me. But I've got my sword and my burning brand handy, and I will prevail!
