Here's a question for Rachel (or anyone else of course). Is a genuinely spherical spiral possible?
A familiar spiral line in two dimensions can be viewed as a circle continually increasing in radius and so expanding to surround its former self without ever completely succeeding. Can a sphere, seen as a three dimensional circle, wrap around itself likewise? I suppose it would have an opening that would partially expose a spherical surface, and that an ant could crawl into and eventually reach the centre without having to bore through anything.
I'm definitely not thinking of nested spheres, just one sphere. Nor of lines or bands spiralling over the surface from pole to pole. Nor of a loosely rolled up rectangle, that's closer but too cylindrical. A cowrie shell? Hmm, maybe that's a homeomorphism of what I'm talking about, or is it just that aforementioned rolled up rectangle fused at the ends? A Klein bottle?
The point in the article that a Klein bottle is only possible in four, not three, that dimensions is unclear to me. But I can somehow see it applying to a spherical spiral that exists as an ongoing computer animation rather than an unchanging object.