Friday, September 5, 2008
Hey! It's been pointed out to me that I haven't written anything here for ages, and I realised there's a couple of projects I haven't found the time to mention yet.
So, first off, Alien surfaces!
This project actually belongs to local mathematician Madeleine Shepherd, but I helped out with some of the maths, so I'm sure it's worth a post :o)
As part of a festival exhibition on 'alien surfaces', artwork inspired by descriptions of alien planets in science fiction, Madeleine knit a model of , the surface you get if you take the curve y=1/x and rotate it around the x-axis(for the region x>1 - although I think we took x>1/30, or thereabouts, to make the curvature show up more).
This surface has some very cool properties mathematically - it turns out that it's surface area is infinite, while it's volume isn't, which means that you could, hypothetically, fill one with paint, but it would be impossible to paint it's entire surface. Which is a bit mind-boggling. (The trick is that in comparing a volume with a surface area in this way, you're kind of assuming that you're covering the surface with a layer of paint of uniform thickness, the volume is only finite because the trumpet tapers off quickly as x becomes large.)
Since the surface has a rotational symmetry, writing down a pattern for it is relatively simple - you just need to work out the circumference at each row, convert this into a number of stitches, and work out how many stitches you need to decrease each time.
The difficulty is that the rows in this case do not correspond to the coordinate x, but the arclength - the distance you've travelled along the curve from the first row. Now, in differential geometry, this isn't really a problem, you can just change coordinates without any difficulty, but actually doing this in practice takes a bit more work because while it's easy enough to find the arclength from the position, inverting this formula is quite hard, and needs to be done numerically.
Happily though, Maple actually cooperated with this, so I now have a bit of code which is capable of doing this more or less automatically. If anyone's interested, or is keen to knit there own surfaces of revolution, I'd be happy to go into more detail on this. I was considering tidying up the code a little and convincing it to print out real honest-to-goodness knitting patterns, but I'm not sure how many people would be interested in this and have access to Maple?
And I love how the surface turned out, there's something amazing about writing down a bunch of maths and getting to see it suddenly turned into a piece of knitting! I'm quite keen to read the book it comes from too, I'd be interested to see how far the author was able to take this idea, and how this unusual geometry affected the people living there :o)
(Oh, and the tapir is because Madeleine seems to be quite keen on them)