Through a Glass, Darkly
Feb. 21st, 2011 10:32 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI think I'm closing in on a solution to my current mathematical problem. I'm pretty sure of the shape of the answer, but the details will require some work. The particular beasts I'm hunting are "simple closed geodesics on rectangular prisms". Here's one way to think about them.
Take a rectangular box - maybe a perfect cube, maybe some other shape. I'm looking at ways of wrapping a ribbon around the box, with the following conditions. First, when the ribbon goes around an edge, it has to do so smoothly - no kinks. Second, I don't want the ribbon to cross itself anywhere. Finally, I want the end of the ribbon to connect to, and line up smoothly, with the beginning. The question is, given the dimensions of the box, how many different ways are there to do this?
If the box is a cube, there are basically only three ways. First way: begin the ribbon so it's perpendicular to one edge. Then, just follow it around four sides until it comes back where it started. The ribbon is in the shape of a square. Second way: begin the ribbon at a forty-five degree angle to a side. It'll wrap around all six faces, cutting off a corner on each one. If you started at the middle of the side, the ribbon forms a regular hexagon. The third way is harder to describe. Look at the face in front. Mark the bottom edge two-thirds of the way from the lower right corner, and the right edge one-third of the way up from that corner. (It doesn't have to be exactly those distances; what's important is that one distance is twice the other.) Run the ribbon between those two points, and follow it around. Again, it will cross all six faces; it'll cut off a corner on four of them, and run all the way across the other two. This one's different from the other two; the ribbon doesn't lie in a plane.
Now suppose your box isn't a cube; in particular, suppose all three dimensions are different. The square ribbon always works, but now there are three different ways to do it, depending on which edge the ribbon runs parallel to. The hexagonal ribbon works provided that the longest side of the box is shorter than the sum of the other two; in that case, there's essentially only one way to do it. The third one... well, the conditions are complicated, but there are boxes where it won't work, boxes where it'll work one way - i.e., starting on one particular face, boxes where it'll work two ways, and boxes where it'll work three ways.
Each of these is special, in that the ribbon crosses each face at most once. There are other ways; there's a standard "gift-box" wrap that crosses the top and bottom twice each and the other four sides once each. That one, though, doesn't work on the cube.
What I think I'm seeing is this: the other ways, the ones that don't work on the cube, fall into half a dozen infinite families; each family falls into (at most) three subfamilies - and the subfamilies seem, for some reason, to be related to the three ways of wrapping the cube!
The subfamilies may themselves break into subsubfamilies; that's my next task, to figure that out, and then to figure out which boxes allow which wrapping methods. I'll make a prediction: I will have this wrapped up (heh) by the end of March. The game's afoot!
Take a rectangular box - maybe a perfect cube, maybe some other shape. I'm looking at ways of wrapping a ribbon around the box, with the following conditions. First, when the ribbon goes around an edge, it has to do so smoothly - no kinks. Second, I don't want the ribbon to cross itself anywhere. Finally, I want the end of the ribbon to connect to, and line up smoothly, with the beginning. The question is, given the dimensions of the box, how many different ways are there to do this?
If the box is a cube, there are basically only three ways. First way: begin the ribbon so it's perpendicular to one edge. Then, just follow it around four sides until it comes back where it started. The ribbon is in the shape of a square. Second way: begin the ribbon at a forty-five degree angle to a side. It'll wrap around all six faces, cutting off a corner on each one. If you started at the middle of the side, the ribbon forms a regular hexagon. The third way is harder to describe. Look at the face in front. Mark the bottom edge two-thirds of the way from the lower right corner, and the right edge one-third of the way up from that corner. (It doesn't have to be exactly those distances; what's important is that one distance is twice the other.) Run the ribbon between those two points, and follow it around. Again, it will cross all six faces; it'll cut off a corner on four of them, and run all the way across the other two. This one's different from the other two; the ribbon doesn't lie in a plane.
Now suppose your box isn't a cube; in particular, suppose all three dimensions are different. The square ribbon always works, but now there are three different ways to do it, depending on which edge the ribbon runs parallel to. The hexagonal ribbon works provided that the longest side of the box is shorter than the sum of the other two; in that case, there's essentially only one way to do it. The third one... well, the conditions are complicated, but there are boxes where it won't work, boxes where it'll work one way - i.e., starting on one particular face, boxes where it'll work two ways, and boxes where it'll work three ways.
Each of these is special, in that the ribbon crosses each face at most once. There are other ways; there's a standard "gift-box" wrap that crosses the top and bottom twice each and the other four sides once each. That one, though, doesn't work on the cube.
What I think I'm seeing is this: the other ways, the ones that don't work on the cube, fall into half a dozen infinite families; each family falls into (at most) three subfamilies - and the subfamilies seem, for some reason, to be related to the three ways of wrapping the cube!
The subfamilies may themselves break into subsubfamilies; that's my next task, to figure that out, and then to figure out which boxes allow which wrapping methods. I'll make a prediction: I will have this wrapped up (heh) by the end of March. The game's afoot!
