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stoutfellowThe programming bug is still biting, but seems to be diminishing. This afternoon, I made some progress on one of the two problems I mentioned yesterday, but I'm not really satisfied. (I wrote a program that I thought would work; it gave wrong answers, so I made a more-or-less random tweak, and it started giving right answers - but I don't know why, and I don't trust it.)
More important, and more entertaining, was what I did in the morning. Let me back up. What we've been working with are honest-to-God polyhedra, that we can make paper or ball-and-stick models of. A few weeks ago, though, W raised the possibility of taking a look at "abstract polyhedra" - things we can describe, saying which vertices lie on which edges and which edges on which faces, but which we can't actually construct. (Not in three dimensions, at least. If I understand things correctly, all of them can be constructed in five dimensions - not that that does us much good.) It was a casual, not-entirely-serious proposal, and we pretty much shuffled it off. Last week, though, I decided to try mining that area for data. I didn't meet with a lot of success, though.
This morning, while waiting for the bus, I got to thinking about genus-1 (doughnut-shaped) polyhedra. W has constructed two of them, and we've gotten interesting results from them. Unfortunately, in order to get polyhedra to bend into a circle, you have to make them large and complicated. What, I wondered, if we dropped the constructibility requirement - could we get something less complicated by going to abstract polyhedra? I toyed with a few ideas, but discarded them as unworkable. Then another possibility came to me.... When I got to the office, I punched in the data, after a few false starts:
Huzzah! Now to check for geodesics.
This new polyhedron is displaying exactly the same kind of behavior as the tetrahedron - what I called "off-the-charts strange" last week. After some discussion, W and I have concluded that it's for essentially the same reason.... (Part of it: the tetrahedron is the simplest of all sphere-like polyhedra; this new critter looks like it may be the simplest possible doughnut-shaped polyhedron.)
It's actually quite pretty - well, abstractly pretty; it has a number of interesting features besides having lots and lots of simple closed geodesics. I'll have to look at it some more. (Not soon, though; I'm not going in tomorrow, and on Friday none of the others will be around for the seminar, so I'll probably come home early. Next week is break week, so....)
More important, and more entertaining, was what I did in the morning. Let me back up. What we've been working with are honest-to-God polyhedra, that we can make paper or ball-and-stick models of. A few weeks ago, though, W raised the possibility of taking a look at "abstract polyhedra" - things we can describe, saying which vertices lie on which edges and which edges on which faces, but which we can't actually construct. (Not in three dimensions, at least. If I understand things correctly, all of them can be constructed in five dimensions - not that that does us much good.) It was a casual, not-entirely-serious proposal, and we pretty much shuffled it off. Last week, though, I decided to try mining that area for data. I didn't meet with a lot of success, though.
This morning, while waiting for the bus, I got to thinking about genus-1 (doughnut-shaped) polyhedra. W has constructed two of them, and we've gotten interesting results from them. Unfortunately, in order to get polyhedra to bend into a circle, you have to make them large and complicated. What, I wondered, if we dropped the constructibility requirement - could we get something less complicated by going to abstract polyhedra? I toyed with a few ideas, but discarded them as unworkable. Then another possibility came to me.... When I got to the office, I punched in the data, after a few false starts:
makePolyhedron["torus18",:data:]
"Edge mismatch."
:mess around a little:
makePolyhedron["torus18",:new data:]
"Edge mismatch."
:mess around some more:
makePolyhedron["torus18",:third try at data:]
"torus18 is a polyhedron with 18 faces, 27 edges, and 9 vertices.
It has a rotation group of order 54 and a plane of reflection."
Huzzah! Now to check for geodesics.
checkPattern["torus18",:small generator:]
":list of data:, order 3, multiplicity 9, simple."
checkPattern["torus18",:slightly larger generator:]
":longer list of data:, order 3, multiplicity 9, simple."
checkPattern["torus18",:much larger generator:]
":long list of data:, order 3, multiplicity 9, simple."
This new polyhedron is displaying exactly the same kind of behavior as the tetrahedron - what I called "off-the-charts strange" last week. After some discussion, W and I have concluded that it's for essentially the same reason.... (Part of it: the tetrahedron is the simplest of all sphere-like polyhedra; this new critter looks like it may be the simplest possible doughnut-shaped polyhedron.)
It's actually quite pretty - well, abstractly pretty; it has a number of interesting features besides having lots and lots of simple closed geodesics. I'll have to look at it some more. (Not soon, though; I'm not going in tomorrow, and on Friday none of the others will be around for the seminar, so I'll probably come home early. Next week is break week, so....)
