Knitting Math

In the January issue of the Notices of the American Mathematical Society, I fortuitously saw an A K Peters publishing ad for a new book by Daina Taimina titled Crocheting Adventures with Hyperbolic Planes. A whole book by an author of the paper I had previously found!  It wasn't then available, but I preordered a copy immediately, and today I excitedly received it.  This is a beautiful book, and I'm going to love it.  After flipping through and enjoying the multitude of pictures, the forward by William Thurston started off the reading experience perfectly.

Many people have an impression, based on years of schooling, that mathematics is an austere and formal subject concerned with complicated and ultimately confusing rules for the manipulation of numbers, symbols, and equations, rather like the preparation of a complicated income tax return, where there are myriad unexplained steps, rules, exceptions, and gotchas.
        Good mathematics is quite opposite to this. Mathematics is an art of human understanding.

My first impression is that this is a gentle, real-world-examples and hands on introduction to hyperbolic geometry.  My sneak peek also showed there's more crocheting projects for me to learn in Taimina's work, and I'll surely be breaking out my hook again.  I have to say I'm liking having this bridge between Sarah's and my interests.  

By the way, "Knitting Math" was how a friend of ours described my crocheting adventures to some others when the term hyperbolic plane slipped her mind. I thought it was great :)

Hi, my name is Roice... and I crochet

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.

Sarah Goes Hyperbolic

Sarah has been knitting some pretty scarfs lately. She was showing off her latest project to me, and lo and behold it turned out to be mathematical! As she was knitting successive rows, she added incremental stitches to give it a ruffled appearance. I told her I thought this was especially cool because the extra stitches were giving the scarf a negative curvature. It was hyperbolic! That is what happens when you try to put extra material into what would otherwise be a flat 2 dimensional surface.  She lovingly rolled her eyes :)

Every time I think I have a new idea, it turns out someone has already been there, done that.  On the plus side, the article I then tracked down already contained developed information and instructions for crocheting your very own hyperbolic plane.  Following the directions will result in a hyperbolic surface of constant negative curvature (Sarah's scarfs don't adhere to the constant part).  I also found this site with some nice pictures of completed crochetings (the site mentions the work of Daina Taimina, who is one of the authors of the paper above).

It is interesting to note that if you build a constant negative curvature surface large enough, it will necessarily end up intersecting itself in our 3D world.  Models living in our physical universe are limited in their representation.  This is in contrast to models of constant positive curvature surfaces, which do fit nicely into the world.  The surface of any ball will do.

Sarah and I just returned from Hill Country Weavers, where Sarah bought me a crochet hook, so I'm now off to attempt creating my own hyperbolic plane!

update:  Sarah did not aprove the cuteness factor of my first picture, so I've uploaded an improved version.