Now I want to see a game played on a Klein bottle.
To do the concept justice, you would need to make orientation meaningful. Perhaps something like a puzzle-platformer where your character has asymmetric abilities, so that you have to return to the same location in the opposite orientation to make progress.
I love the fact that it clearly demonstrates the problem with using the projective plane as a map: you can see a second copy of the snake if you approach a particular pair of points!
So wait, what goes wrong if you try to do a projective plane instead by twisting both axes? Apparently something must, but I can't offhand think of what!
Yeah, the “pouring water into Klein bottle” thing always felt completely nonsensical to me.
Around a year ago, a friend and I were toying around with the idea of making a Pokémon rom hack where some part of it was on a Klein bottle and you get sort of twisted around, but never made it off paper. It’s cool that you made an actual 2D map of it!
Why is the Hairy ball Thrm and other consequences of nonzero Euler characteristic imply something bad/unsuable for video games? What vector field does a video game need to create that cannot vanish?
I go through this in the original post, but to sum up: if you are using 2D graphics, you ideally want it to be true that whenever you visit a particular point, you only ever see it from one angle---2D graphics are, in general, not built to be looked at from different points of view, and you actually need a 3D graphics engine if that is not possible. But the choice of viewing angle is equivalent to a choice of unit vector parallel to the surface.
Therefore, to have a unique choice of viewing angle at each point, you need a vector field on your surface comprised entirely of unit vectors. This is possible on a torus, but impossible on a sphere!
The other consequence of a nonzero Euler characteristic is that you must actually have some curvature somewhere, which again isn't great if you are trying to draw things on a flat screen.
Point one is a good point, but it's enough to draw them twice, which isn't THAT much extra work (if you really wanted to do it). And you can get around the curvature requirement by stretching out the fixed points into lines. So a tiling like
ppp
bbb
ppp
is effectively a sphere with stretched poles. The "curvature" manifests as the stretching at the poles, so straight line distance on the flat map doesn't correspond to spherical distance, but that wouldn't necessarily be strange to the player. They're both seeing and experiencing flat straight-line distances so the fact that it's not accurately modeling the geometry of a sphere wouldn't be that important.
What would be annoying to, though, is that this set-up would maximize straight-line distance (and travel time) between the poles and increase the sense of needed to backtrack to get from place to place!
Now I want to see a game played on a Klein bottle.
To do the concept justice, you would need to make orientation meaningful. Perhaps something like a puzzle-platformer where your character has asymmetric abilities, so that you have to return to the same location in the opposite orientation to make progress.
There's a version of snake by a grad student in Belgrade that is on the projective plane. It's not a Klein bottle but it's a non-orientable surface.
https://ubavic.rs/snake/snake.html
Well are you in luck! How do you feel about snakes?
https://thinkstrangethoughts.substack.com/p/snakes-on-a-projective-plane
I love the fact that it clearly demonstrates the problem with using the projective plane as a map: you can see a second copy of the snake if you approach a particular pair of points!
So wait, what goes wrong if you try to do a projective plane instead by twisting both axes? Apparently something must, but I can't offhand think of what!
Taken from elsewhere in the comments: https://thinkstrangethoughts.substack.com/p/snakes-on-a-projective-plane
If you play with the head-centric mode engaged, you can clearly see the issue!
What happens at the corners? E.g. if you go in a small circle around one of the corners, will it take you a full 360 degrees?
Yeah, the “pouring water into Klein bottle” thing always felt completely nonsensical to me.
Around a year ago, a friend and I were toying around with the idea of making a Pokémon rom hack where some part of it was on a Klein bottle and you get sort of twisted around, but never made it off paper. It’s cool that you made an actual 2D map of it!
Why is the Hairy ball Thrm and other consequences of nonzero Euler characteristic imply something bad/unsuable for video games? What vector field does a video game need to create that cannot vanish?
I go through this in the original post, but to sum up: if you are using 2D graphics, you ideally want it to be true that whenever you visit a particular point, you only ever see it from one angle---2D graphics are, in general, not built to be looked at from different points of view, and you actually need a 3D graphics engine if that is not possible. But the choice of viewing angle is equivalent to a choice of unit vector parallel to the surface.
Therefore, to have a unique choice of viewing angle at each point, you need a vector field on your surface comprised entirely of unit vectors. This is possible on a torus, but impossible on a sphere!
The other consequence of a nonzero Euler characteristic is that you must actually have some curvature somewhere, which again isn't great if you are trying to draw things on a flat screen.
Point one is a good point, but it's enough to draw them twice, which isn't THAT much extra work (if you really wanted to do it). And you can get around the curvature requirement by stretching out the fixed points into lines. So a tiling like
ppp
bbb
ppp
is effectively a sphere with stretched poles. The "curvature" manifests as the stretching at the poles, so straight line distance on the flat map doesn't correspond to spherical distance, but that wouldn't necessarily be strange to the player. They're both seeing and experiencing flat straight-line distances so the fact that it's not accurately modeling the geometry of a sphere wouldn't be that important.
What would be annoying to, though, is that this set-up would maximize straight-line distance (and travel time) between the poles and increase the sense of needed to backtrack to get from place to place!
Ah that makes sense. Thanks!