Monday, February 26, 2007

Alexander horned sphere

More maths geeking for this project, and a return(sort of) of the tori!

In two dimensions, the Jordan-Schonflies curve theorem says that if you have a shape in the plane which is topologically equivalent to a circle, then there is a homeomorphism(topological equivalence) of the place making this into the standard circle in the plane. In higher dimensions this isn't true, and to my mind this is one of the big differences between 2D and higher dimensions.
The Alexander horned sphere is the standard counterexample to this. To construct it, you take a sphere, then you pick two discs on it, and stretch them into a kind of claw. Then you pick two discs on the end of each 'horn' and form another claw, with the two claws interlocking but never touching. You can do this infinitely many times, and you need to add in a few points(actually these points form a kind of Cantor set. I'm not quite clear on how this step works) to get a complete sphere. This picture probably makes the construction clearer.
Then if you imagine passing a loop through the middle of this thing and trying to take it out, you find it's impossible - you need to bend it through infinitely many horns or ever smaller size, and topology won't let you. This can never happen with the standard sphere - any loop on the outside can be pulled down to a point, so the 'outside' of this sphere is very different from the outside of the standard one.
Because it has this hole in the middle, I like to think of this as a kind of honourary torus.

In theory, the knitting of this is made easier by self-similarity - each of the smaller horns should be a scaled down version of the big of the big one, so you should be able to write a pattern for the big one, then the smaller ones are all multiples of this. Unfortunately I can't figure out quite how the horns need to be arranged to make this work - the crucial question is what angle they need to make with each other for this to happen. So to get around this, I'm planning to just knit it one layer at a time - I'll make the big piece first, then I'll be able to measure and find out how big the next level of horns should be. It's a little less satisfying than working it out directly, but maybe once I've made this I'll be able to get my head around how it's arranged enough to work it out explicitly.
I'm planning to knit the first three levels of horns, each in different colours to highlight some of the hierarchy, then represent the fourth level with sewn stitches which will hold the thing together.

I'm also toying with the idea that at some point in the future this could be used as part of an Alexander horned deer, because the sheer horror on people's faces when I explain that pun would be worth the effort alone.


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