Monday, February 19, 2007
Surface of genus three
This is going to be a birthday present for my (maths geek) older brother. The surface of genus 3 is made of three tori stuck together. Alternatively, you can take a sphere and stick three handles onto it.
There's a theorem in geometry(the Poincare-Hopf theorem) which tells you what kind of vector fields you can put onto different shapes, and particularly about the kind of zeroes they have to have. One consequence of this is that only very specific shapes can be combed without leaving any bald spots or tufts, and the surface of genus 3 is not one of these(in fact the torus is the only compact orientable surface for which this is possible).
So the plan is to make this surface out of combable wool, so you can physically demonstrate this.
Niftily, there's a similar result which says the odd dimensional, but not the even ones, can be combed, is (sometimes) known as 'hedgehog combing'.
Knitting-wise, I'll make this in three pieces, each one consisting of a torus with a strip cut out, and a small tube sticking out to attach them together. This probably isn't the best way to do this, but I'm quite squeamish about doing anything too complicated with fluffy wool.
Sorry about the rubbish pictures, I haven't found a good drawing program for my diagrams yet.