Pretty swirly things

[simont]

I've just put up a web page of pretty swirly fractals, one of which is animated and thus doubly swirly. Anyone who's into that sort of thing, have a look at http://www.chiark.greenend.org.uk/~sgtatham/newton/. Gets a bit mathematical in places; read the maths if you care, skip down to the pretty pictures if you don't :-)

(Yes, as it happens, I have been spending a lot of today waiting for long test runs to finish. How can you tell?)

Comments

[kaet.livejournal.com]

Thanks!

My main concern was that I thought it was obvious with the extra dimensions and if there's no way to do it in three, I shouldn't waste time trying to warp my brain around it.

I'd come across this bijection definition, but I thought that the two things were equivalent (given everyone 'knows' that's what topological deformation is). I shall endeavour to abandon it, :). It makes more sense when you're dealing with immersions anyway, I think, to think of them as 1-to-1 mappings of neighbourhoods rather than the screwy one about intersections in n-2 dimensions, or whatever it is, so the deformation thing probably gives up even the advantage of being more intuitive when you get to more involved stuff, anyway.

I had also suspected that it was that my brain wasn't warped enough to untie the knot in three, :). Some of those shapes are just, just not natural, :).

[simont]

It may be the case that any two surfaces equivalent by the bijection definition are also deformable into each other given sufficiently many dimensions. Certainly I can't immediately think of a counterexample...

I was wrong about the internally knotted torus, incidentally. The book I mentioned does contain diagrams for untying a knot in the hole of a torus, but only in the case where the torus has two holes and the unknotted one goes through the middle of the knot in the other. An internally knotted one-hole torus can't be untied (in three dimensions) any more than an externally knotted one; as a simple proof, consider a line drawn along the surface such that it goes through the knotted hole, loops round the outside of the torus and joins back up with itself. This closed curve is tied in a simple overhand knot, and hence no continuous deformation of the torus can transform it into an unknotted curve. (I feel silly for not having spotted that to start with.)