Topology / complex analysis / what

[jack]

For a plot bunny (yes, really :)):

You have a multivalued function from a sphere onto "some surface", continuous everywhere except two points. (Or, equivalently, a function from "some surface" to the sphere, I guess?)

If you look at points on the surface which map onto the same point on the sphere, and connections between them of "paths" on the sphere (up to continuous deformation), I feel like they end up acting like the integers, where "+1" and "-1" correspond to a clockwise of anticlockwise circumnavigation. Or possibly some subset, a cyclic group of some finite order, if there are repeats. Is that right?

If you have *three* points, what can the relationship between the points look like? What about more?

I remember doing something like that but not what it's called.

I'm trying to put something like the shadows of amber onto a more concrete mathematical footing :)

Comments

[simont]

Eg. that there's a discontinuity at the N pole, where you could run round and round it and go quite a distance into the parallel worlds.

I don't quite know whether you think that's a bug or a feature – of course it would depend on what kind of story you're setting in the result of this worldbuilding. (A game-breaking exploit like 'run round exact point location of north pole 1000 times in an hour' might make some kinds of story too easy because it would provide a trivial solution to problems that the plot needs to be a big deal, but on the other hand in a different kind of story the process of hypothesising, confirming, finding and exploiting that loophole might be the whole point of the plot.)

But if you think it's a bug, then here's a possible fix you might like.

In my other comment I mention that the fundamental group of basically anything in the class of spaces you're talking about is the free group on some finite set of generators, i.e. any word formed out of those generators and their inverses gives a group element, and the only distinct words that represent the same group element are due to cases where a generator appears right next to its inverse – after you cancel those obvious redundancies to reduce a word to 'lowest terms', the remaining set of words are all distinct elements of the group.

But any other finitely generated group arises as a quotient of a free group, by imposing additional relations between the group elements. And the quotient map is a homomorphism, i.e. behaves sensibly and consistently and respects the composition law. So you could easily rule that the actual group of all parallel worlds was not actually the free group, but merely its quotient by some set of relations of your choice. And a nice choice might be to assign each individual group generator a finite order (i.e. for each generator g, add some rule that says g^2=e or g^3=e or g^99=e or some such), but without adding any relations between the generators.

Result: you still have an infinite branching tree of parallel worlds available if you're prepared to do a lot of arduous travelling back and forth between 'poles' of the world, but no matter where you currently are in the group, running round and round the same 'pole' of the world – whichever pole it might be – only cycles you through some small finite set of worlds.

[simont]

Really horrible afterthought:

There is another well-known kind of group I can easily bring to mind in which each individual generator has a small finite order but the group as a whole is ginormous and you need to combine those simple generators into very long words to get between an arbitrary pair of points.

Omit seven points from your sphere, and arrange that the six resulting generators of your worldwalking group correspond to quarter-turns of the six faces of a Rubik's cube :-)

[jack]

:)

[simont]

(I vaguely recall someone telling me once, on that subject, that there's a text adventure game set on the interior of a Rubik's cube – the rooms are organised in a 3×3 grid and there are some puzzle actions you can take to perform face turns...)

[jack]

Oh, cool.

I wasn't sure between "bug" and "feature" :)

My first thought was, if you have something like this, there must be points of discontinuity, and what are they physically like, why hasn't anyone noticed? But then I thought, if they're N and S pole, well, maybe just no-one's noticed.

My second thought was, a good follow-up story would be an expedition to one of those points, and what you do see if you experiment with it?

I assumed the characters would not initially have the resources to go to one of those remote places, but there might be far-flung parallel worlds where scientists have expanded understand of them and governments trade across boundaries etc.

But I like your answer too, that it's plenty infinite, but only so much on any particular loop. Or maybe, they're all infinite, but some of them dead-end in uninteresting worlds without life.

I'm not sure I want everywhere to have a *fixed* number of points. If it's more than 2, I'm not sure I want to fix it. I'm not sure if that works mathematically, I guess, there's a point where if you keep travelling, the S pole discontinuity splits into two, initially too close together to traverse, then a tight figure-8, eventually it migrates to the bermuda triangle?