Hyperbolic Planes & Pseudospheres

Dear LuLu, the news about your sis is certainly disheartening. I'm thinking about you, and praying for strength for you both, and wisdom on the part of her medical team.

I've made a little progress on Sleeve #1. It's quite phenominal, the way it expands from the long cuff. (That's the cuff, the skinny tube sticking up in the middle.) The pattern calls for increasing to 180 stitches around, but I've called a halt at 120. Just look at the size of that sleeve! Don't you think that'll be enough?!

One of my favorite fiber books of the past year is "Crocheting Adventures with Hyperbolic Planes." The author, Daina Taimina, is a professor of mathematics. She crocheted examples of hyperbolic planes to demonstrate to students their properties. (Be patient -- there's a connection here.) Do you remember the definition of parallel lines, that you learned in geometry? Through a point not on a line, there is one, and only one, line that is equidistant from the first line at every point on the line. Something like that. Well...that definition works on a flat plane. But what about on a sphere? If straight lines are the shortest distance between two points, on a sphere those would be great circles. And there are NO great circles that won't intersect every other great circle on the sphere.

So what about the opposite of a sphere? If a sphere is a surface that's always closing in on itself (mathematicians say it has positive curvature), then the opposite (negative curviture) would be a surface that's constantly expanding. And guess what? On such a surface (called a hyperbolic plane), there are an infinite number of lines that will not intersect the first line.
Mathematicians (men, of course) have been trying for the several centuries since they started thinking about such things, to figure out how to SHOW such a surface and its properties. It took a woman mathematician, with a crochet hook. You could do it with knitting, but eventually you'd have too many stitches on the needle, so crochet works best. Make a chain, close it, and as you go around, increase the number of stitches in every round. That's a hyperbolic plane!

And here, dear LuLu, is the point of all this: Doesn't my KF sleeve look for all the world like the beginning of a pseudosphere, which is a special kind of hyperbolic plane?

I'm off to Central America for a few days, wearing my accounting hat, on behalf of my favorite charitable organization. I won't take the fuzzy, woolly Romeo & Juliet sleeve: it'll be in the 90s, with 95% humidity!