![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
Holey Polyhedra, Batman!
Feb. 19th, 2010 06:03 pmThere was a nice surprise waiting for me when I got to work today: a genus-1 deltahedron.
Let me back up. The polyhedra that our group is studying are what are called "deltahedra"; that is, we require that all of the faces be equilateral triangles. (The standard definition is a little bit tighter than that, but what we're doing works in this generality.) Now, there are infinitely many different deltahedra, but actually constructing them takes a certain amount of work. Putting them into the right form to feed them into my algorithm is also a bit tedious (and little errors will slip in, which make my algorithm cry). Still, there's a standard list of eight - the so-called "convex deltahedra" - including the tetrahedron, the octahedron, the icosahedron, and five other, more exotic objects. I've fed all of those into my machine, and gathered quite a bit of information about them. I've also constructed half a dozen more, and run them through as well.
Last Friday, W challenged me: would my algorithm be able to handle a deltahedron "with some topology" - i.e., one that isn't topologically the same as a sphere? I shrugged: "Give me the data..." He promised to try to provide me with one. This morning, in my e-mail, was the data describing it. It's sort of a doughnut-shaped thingy, which he built by gluing together octahedra. When we gathered for our seminar, I hadn't had time to check it out, so we all trooped back to my office. I pulled up the e-mail, Control-C'd the data, and loaded my algorithm. After a couple of false starts (little errors, remember?) we got it running.
Wonderful, surprising results. Just beautiful, and unexpected - at least to W, who lost a bet with L as a result; behavior that we hadn't seen in any of the earlier examples (with the exception of the tetrahedron, which is off-the-charts strange).
W thinks we should try to assemble this stuff into a paper for one of the journals of experimental mathematics. We'll see.
Let me back up. The polyhedra that our group is studying are what are called "deltahedra"; that is, we require that all of the faces be equilateral triangles. (The standard definition is a little bit tighter than that, but what we're doing works in this generality.) Now, there are infinitely many different deltahedra, but actually constructing them takes a certain amount of work. Putting them into the right form to feed them into my algorithm is also a bit tedious (and little errors will slip in, which make my algorithm cry). Still, there's a standard list of eight - the so-called "convex deltahedra" - including the tetrahedron, the octahedron, the icosahedron, and five other, more exotic objects. I've fed all of those into my machine, and gathered quite a bit of information about them. I've also constructed half a dozen more, and run them through as well.
Last Friday, W challenged me: would my algorithm be able to handle a deltahedron "with some topology" - i.e., one that isn't topologically the same as a sphere? I shrugged: "Give me the data..." He promised to try to provide me with one. This morning, in my e-mail, was the data describing it. It's sort of a doughnut-shaped thingy, which he built by gluing together octahedra. When we gathered for our seminar, I hadn't had time to check it out, so we all trooped back to my office. I pulled up the e-mail, Control-C'd the data, and loaded my algorithm. After a couple of false starts (little errors, remember?) we got it running.
Wonderful, surprising results. Just beautiful, and unexpected - at least to W, who lost a bet with L as a result; behavior that we hadn't seen in any of the earlier examples (with the exception of the tetrahedron, which is off-the-charts strange).
W thinks we should try to assemble this stuff into a paper for one of the journals of experimental mathematics. We'll see.
