Saturday, February 20, 2010

Universal cover scarf


Hi folks!

Continuing the mathematical theme, my next plan is to knit the universal cover of a punctured disc, arranged to make a ruffly scarf.

So, what is a universal cover? The idea is to take a space, and form a new space which is locally the same, but in which any loop can be shrunk down to a point. This is useful because when we try to extend local properties of spaces to global ones, it's usually these kind of loops which cause problems.
As an example, consider a disc with it's centre missing, the `punctured disc'. A loop in this space can be shrunk down to nothing provided it doesn't wrap around the hole. To remove these loops, imagine we take an infinite number of these discs, and declare that a path which crosses the x-axis in a clockwise direction moves one disc up the chain, and if it crosses anti-clockwise, one to the right. Now if we take a loop in this space, if it wraps round the centre point we've moved up or down the chain, so the loop is no longer closed. Then any loop in this new space corresponds to a loop on the punctured disc which doesn't wrap around the hole, and can be shrunk to a point. This space is called the universal cover of the punctured disc.
Another way to see the same space is this -- take an infinite sequence of discs, each slit along the x-axis. Glue the top edge of each slit to the bottom edge of the slit on the preceding disc. You can then stretch this out to make a sort of spirally chain, which I think would make a neat scarf.

So, how to knit this? If you imagine following a circle around the origin in this spirally chain, it will form a helix (assuming you've `stretched' the same way I have, this isn't entirely fixed) in space. My plan is for these helices to form the rows, working from the outside in. Of course, since I don't want my rows to be infinitely long I'll only have a finite number of discs in my chain.
One of the cool properties of helices is that it's quite easy to calculate their lengths (which is really quite rare among curves). Since the space is symmetric under shifting along the axis of the helices (after a suitable rotation), it's then just a matter of working out how many stitches to decrease in each row and spacing these evenly along the length. This is the same problem as knitting surfaces of revolution (see also), and it will be easy to adapt the solutions from there.
The actual maths I'll carry out using Ruby, this seems like the kind of problem it's very good at.

I should say too that I'm fairly sure this kind of scarf already exists, though without the mathematical intent. Must make sure to look around and see if I can find some links to compare with. I'm hoping that not being flat will make it nice and warm, but will have to see how it stands up to Edinburgh's winds.

Happy knitting, and/or calculating!
Hugh.

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