Picture me and Sarah sitting side by side in bed around 11:30 pm, her knitting, me crocheting, and you'll have a pretty good idea of a number of evenings of ours over the last few weeks. Maybe not the most exciting image, but we've been having fun. I've finished my first crochet project now, a portion of a hyperbolic plane with radius of curvature of about 5 cm, and I think it turned out pretty good. The crocheted rows are quite visible in this picture I snapped. Are they geodesics of the surface? *
I'm sure I'll learn more by playing with it, but I've learned some already, not the least of which is that I can't count to 5. All I had to do was 5 normal stitches for every doubled-up stitch, and I'd say 60% of the time I lost my place! And boy are programmers spoiled with undo. Too bad that functionality isn't available in the physical universe.
Something noteworthy about this particular construction (but not a property of hyperbolic geometry itself) has to do with the fact that the number of stitches in successive rows forms a geometric sequence, that is the length of each row is a constant multiple of the previous row. That has some unintuitive side effects. I did 23 rows total, the first had 20 stitches and took maybe a minute, but the last had over 1000 stitches and took almost 3 hours! If I were to do another 23 rows, the final 46th row would take me over 28 days (no sleep, no breaks) and who knows how many skeins of yarn. Add yet another 23, and the final row would take over 5 years. This reminds me of "the magic of compounding interest", and what I've been told the value stocks are supposed to do in theory.
Speaking of economic unraveling, this leads to something else intriguing about my hyperbolic plane. Instead of tying off the end when I was done, I could have undone the entire uber-knot in one fell swoop just by pulling out my crochet hook and gently pulling on the yarn. It's like the whole thing is a house of cards, a deceivingly stable form that is actually no more substantial than the first slip knot that started the whole thing. This reminds me of the axiomatic foundations of mathematics.
While working on this, I couldn't help but focus on a possible useful application. I haven't figured it out yet, but my mind can't let go of the idea that this could solve the widespread problem of competition for blankets when couples sleep. The extra material seems like a perfect candidate to provide some benefit here :)
* Nope. If they were, I would be able to fold the surface so that they appeared flat and straight.