Or, How I spent my Friday evening.
Somehow (okay, I'm working on designing hats, so it's not exactly out of the blue), I got a bee in my bonnet the other day about hats.
Let me back up for a second and explain a bit about hats so that the non-knitters are with me on this. A hat is round. A hat is knit in what amounts to a very, very shallow spiral, so in some ways you can treat each round as a circle of stitches stacked on top of another creating a cylinder with a cone on top, but you can also treat the stitches as a single long line.
For this particular thought experiment, we will begin with a hat that has 100 stitches. It measures 20 inches around. So, there are 5 stitches per inch. Measuring vertically, there are 7 rows per inch. We have already knit a cylinder that is the appropriate height (6 inches or so) and are ready to shape the crown of the hat.
To shape the top of a hat you have to get rid of all the stitches over some number of rows that is at least as big as the radius of the hat. Since the hat measures 20 inches around, it's radius is 3.2 (2 pi r = C). So we need to decrease our 100 stitches over some number of rows greater than 22. 25 rows would give you a flat top like a pillbox hat. 200 rows would give us an absurdly long (maybe in a good way) floppy, pointed hat. Usually, decreases are done on some sort of regular schedule so you decrease (a decrease turns 2 stitches into 1 stitch)at 2 or 8 or 6 points in a single round every other round. Changing the number of decreases in the decrease row or changing the rate at which you repeat the decrease row (every row, every 12 rows, etc) changes the shape of the hat. That's pretty much all you need to know to design hats.
So my "bee" was this: What if, instead of decreasing in the usual way, I decreased on the 100th stitch, then the 99th, then the 98th and so on? What would it look like? I didn't have access to paper at the time so it took me longer to figure out (plus I was driving -- both good and bad for thinking). Later I asked Leo about it and also how could you make the line of decreases into a spiral? I figured I'd mark each decrease with a purl stitch or bead to mark it. What would happen if you start with 100 stitches, but do your first decrease at 110? 150? 60? What shapes would these hats have? What if you decreased every 100th stitch all the way up?
So we spent some of Friday evening discussing this problem. Leo fell to writing code while I pursued the empirical test. I have a pretty solid picture in my head now of what happens in these various cases, but I think it would be great to actually make some or at least do some computer modeling. Now you can all just sit there for a moment and either a) shake your heads at the fact that people exist that discuss such things at length, or b) smile quietly because my mate and I are so shockingly well suited to one another.
And now, the mathematicians among you can fall to work on hat design.
Yoav is to thank for "topology."