## Sunday, November 30, 2008

### 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.

## Sunday, November 16, 2008

### 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.

## Saturday, November 8, 2008

### Loxodromes!

This is a sweet word used to describe a sweet mathematical curve. And by sweet I mean awesome and one of my favorites. Speaking of favorites, I learned about these curves in my favorite mathematical book, Visual Complex Analysis, by Tristan Needham. If this post piques your interest, there is tons more about them there.

Loxodromes are curves of motion you get from certain kinds of Mobius transformations, whose general algebraic definition is the formula (az+b)/(cz+d).  z is a complex number here, that is a point in a plane. Hence, these transformations can be viewed visually by their effects on the points of a plane. However, Mobius transforms have an elegant interpretation when viewed as a corresponding transform obtained by unprojecting the plane onto a sphere (doing the reverse of stereographic projection that I described in a previous post), as the resulting motions on the sphere are much simpler! The sphere is called the Riemann Sphere in honor of Bernhard Riemann.

In this animation, I put a little white fuzzball near the north pole of the sphere to show the light generating the shadows. The shadows are the 3D->2D stereographic projection of the curve on the sphere. So this is two simultaneous views of a loxodrome, both on the sphere and on the plane (fine print: not exactly because the curve I've drawn has a little bit of thickness coming off the sphere surface, but the shadows are still close). If you remember the previous discussion in the soap bubble post about projecting from 4D->3D, hopefully that process is a little more clear by analogy now. We may not be able to look at 4D objects in our world, but we can look at their shadows! Anyway, I hope this shed a little more light (pun intended) on what stereographic projection is.

Loxodromes correspond to one of the more general types of Mobius transform motions, and the animation can help a little in explaining what I mean by that. Watch it for a bit and answer the following: Do you see the curve stretching and shrinking over the sphere like it is moving from one end of the spiral towards the other? Or do you see the curve as unchanging its shape and just rotating as a whole about an axis through the 2 spiral ends?

My perception is biased to the former, but there is no right answer because it could be viewed either way! The first kind of motion is called hyperbolic (unfortunately, the meaning here is not the same as of the last post), and the second is called elliptic. Both are special types of Mobius transformations, and loxodromes are what you get when you do both in combination. Incidentally, it is interesting that I have trouble seeing this as a rotating curve because that is how the POV-Ray script actually generated the sequence of images :)

Recognize the projected shape in this picture? A special case of the projection of a loxodrome onto the plane is a logarithmic, or equiangular spiral! Yes, the spiral of sea shells and galaxies and so much more. Are you feeling the awesomeness yet? By the way, I took advantage of this to simplify making the movie. There are no complex number calculations explicitly going on, just a function that can generate points of a logarithmic spiral on a plane, and a function that can unproject those points from the plane to a sphere. So if you were thinking it was terribly involved to generate the movie and picture, it actually was not so bad (though it did take a long time to render out the frames). The entire definition file is only about 100 lines, with half of it standard required stuff (camera position, etc.). Here it is if you care to check it out.  POV-Ray is great!