Some Assembly Required
Aug. 17th, 2010 04:22 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowFall Semester begins next week; I'll be teaching Euclidean and Non-Euclidean Geometry, Linear Algebra II, and Calc I.
W and I are working to get our first geodesics paper written up before the semester begins. He's doing the actual write-up of the first draft; then he'll circulate it among the four of us for critique. We've been meeting sporadically, as opportunity arises, to discuss the structure of the paper.
This paper will have two parts. The first will present and prove a number of theorems about simple closed geodesics on deltahedra; the second will apply those theorems to find all of the scg's on the five non-regular convex deltahedra. (The regular ones - tetrahedron, octahedron, icosahedron - have already been dealt with by others.) There's been some disagreement over how to handle the first part; W wanted to use the most powerful theorem we've got, but I feel that it's too powerful - the proof is quite complicated, and the complication deals with situations that simply don't arise in the five examples we want to work. There's a less powerful but much simpler theorem that, I think, does everything we need, and W has agreed to look it over and see if it'll do.
The second part of the paper has proven difficult to write. For each of the five polyhedra, we have to carry out a fastidious case analysis. ("Look at this vertex. In the vicinity of this vertex, one of the following five things must happen. Case I: assume the first possibility. This has the following consequences... This edge, now, must be marked in one of two ways. Case IA: assume the first marking.... Case IB: assume the second marking....") In one case that I worked out (the "gyroextended square dipyramid"), the case-tree is binary except at the very top, and has height five - that is, I'm down to (e.g.) Case IA1ai before I'm done. We could write this up purely in words (after displaying a labeled picture of the polyhedron), but it would be long, tedious, and difficult to follow. Forget that! Instead, we decided to present the case analyses pictorially. I devised a color-coding that explains what happens, and W came up with a labeling method that indicates the order in which things happen (in other words, the direction the line of argument follows). It's still going to be long, but it should be much easier to follow.
So. We get this done. Then we start on the second paper - the one that really does need the more powerful theorem I mentioned....
W and I are working to get our first geodesics paper written up before the semester begins. He's doing the actual write-up of the first draft; then he'll circulate it among the four of us for critique. We've been meeting sporadically, as opportunity arises, to discuss the structure of the paper.
This paper will have two parts. The first will present and prove a number of theorems about simple closed geodesics on deltahedra; the second will apply those theorems to find all of the scg's on the five non-regular convex deltahedra. (The regular ones - tetrahedron, octahedron, icosahedron - have already been dealt with by others.) There's been some disagreement over how to handle the first part; W wanted to use the most powerful theorem we've got, but I feel that it's too powerful - the proof is quite complicated, and the complication deals with situations that simply don't arise in the five examples we want to work. There's a less powerful but much simpler theorem that, I think, does everything we need, and W has agreed to look it over and see if it'll do.
The second part of the paper has proven difficult to write. For each of the five polyhedra, we have to carry out a fastidious case analysis. ("Look at this vertex. In the vicinity of this vertex, one of the following five things must happen. Case I: assume the first possibility. This has the following consequences... This edge, now, must be marked in one of two ways. Case IA: assume the first marking.... Case IB: assume the second marking....") In one case that I worked out (the "gyroextended square dipyramid"), the case-tree is binary except at the very top, and has height five - that is, I'm down to (e.g.) Case IA1ai before I'm done. We could write this up purely in words (after displaying a labeled picture of the polyhedron), but it would be long, tedious, and difficult to follow. Forget that! Instead, we decided to present the case analyses pictorially. I devised a color-coding that explains what happens, and W came up with a labeling method that indicates the order in which things happen (in other words, the direction the line of argument follows). It's still going to be long, but it should be much easier to follow.
So. We get this done. Then we start on the second paper - the one that really does need the more powerful theorem I mentioned....
