Thoughts on configurations of at most n points

A research talk at the meeting on Topological Invariants and Their Applications, at Gdańsk University of Technology, January 29-31, 2024. This is preliminary work together with Daciberg Goncalves.
Probably hard to follow for nonexperts, but some nice pictures at the end.

[@ChrisStaecker]

Some nice pictures at 22:30.

[@BravoCharleses]

Young man I appreciate your commitment to putting hardcore math content on your channel once in a while. Monkey calculating machines are pretty great candy, but once in a while I like to eat my vegetables.

[@MattMcIrvin]

I liked the surprise cameo by the Möbius strip

[@jdmr4815]

I clicked the like button because I enjoy your analog calculating device videos, but I have no idea what what Nielsen theories and compact polyhedrons are.

[@ChrisStaecker]

Yeah this one makes my obscure calculator videos look mainstream.

[@Gunbudder]

You have entered the... DONUT SPACE i don't fully understand what is going on other than i can see you are drawing on an unrolled torus's surface. i love some torus surface fun. i always like to imagine the space ship from asteroids jumping through the third dimension to teleport by moving to the inside of the donut (the donut hole) and then flying up off the surface of the torus, into the third dimension, crossing the hole in the torus, then landing on the other side of the donut hole back on the surface. what would that translation look like from inside the ship? from outside the ship (still in the 2d perspective)? presumably you would see the ship vanish as it lifted into the third dimension, then reappear somewhere else on the map. then it makes me wonder, where exactly would the ship come back? how would you calculate your course to reappear where you want to? also you would not be able to fly "up" into the third dimension from the outside of the donut. or rather, it would make more sense to move to the inner hemisphere of the donut first before going into the third dimension. at any rate, this is how i imagine traveling through a 4th dimension to "teleport" in 3 dimensions would work.

[@ChrisStaecker]

Yes you have noticed without even noticing that the product S¹×S¹ is homeomorphic to the 2-dimensional torus! Here is my opinion about the most sensible way to think about "jumping across the hole" of a torus represented in this way: the ship will disappear and reappear somewhere else- where exactly you reappear would depend on the details of exactly how you had wrapped up the rectangle to make it into the torus. To me it's most natural to wrap it up so that the horizontal line across the middle of the rectangle ends up as the innermost "ring" on the inside of the torus that defines the hole in the middle. So to jump across, the ship first needs to fly to some spot on that line. Then the actual jump looks like disappearing and reappearing at the "opposite" point on that same line, where "opposite" means traveling along the line by a distance of half the width of the screen (in either direction, since we wrap around you'll end up at the same point). This distance of the jump is very predictable (it's always the same), so you could easily calculate your course to reappear where you want to. What would this look like from the ship's point of view? I guess it would just look like they were suddenly in a different spot from where they were before- kinda depends on exactly how their technology is accomplishing this task.

[@dimitar4y]

argh, my head.