Topology / complex analysis / what
Dec. 6th, 2016 10:50 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackFor 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 :)
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 :)
no subject
Date: 2016-12-07 01:35 pm (UTC)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?