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okay so i'd like to implement coordinates that basically correspond to this tiling of the hyperbolic plane. (incidentally i always found it weird that people say 'the hyperbolic plane' when depending on the tiling it's a plane that's more or less curved, which kind of changes which plane it is since now the plane itself has attributes.)
the issue is... i have no clue how to do that? like okay sure, have all coordinates as paths radiating out from a central origin, fine. i don't plan on having paths that are like hundreds of tiles distant from the origin so the unbound nature of that representation is probably not a practical concern. the issue is: i'd need these coordinates to be in some canonical, reduced form, so that going uh N E S W N E to get back to the origin actually does take you back to having an empty path. and for the inverse path of W S E N W S (i'm also not sure that that's even the correct inverse path b/c i have no real sense of how this space connects)(i do realize it's incorrect to call the movements 'north' or w/e tho. constant left turns etc) to also return the same exact coordinate values, so the movement math can't just cut loops when it sees them because that would still yield the 'same point' having different representations. like i guess i should be possible to represent each coordinate with the shortest path from it to the origin, so the coordinates would be like...
as you move around that circuit. but wow i have no sense whatsoever what the math for that would look like, even a little.
i guess if i stop using fake directional coordinates and accept that this can only be described by (L)eft (R)ight and (T)hrough movements (with (B)ack just popping the last movement off the stack) then the coordinates look like, uhhhh
okay so now we have a problem already b/c there are same-length paths to the same coordinate. which... makes sense, since like, even in a flat plane stuff like TR and RL gets to the same coordinate. so. the theoretical canonical reduction would also need to handle that.
basically i have no clue how possible / feasible any of this is :/
the issue is... i have no clue how to do that? like okay sure, have all coordinates as paths radiating out from a central origin, fine. i don't plan on having paths that are like hundreds of tiles distant from the origin so the unbound nature of that representation is probably not a practical concern. the issue is: i'd need these coordinates to be in some canonical, reduced form, so that going uh N E S W N E to get back to the origin actually does take you back to having an empty path. and for the inverse path of W S E N W S (i'm also not sure that that's even the correct inverse path b/c i have no real sense of how this space connects)(i do realize it's incorrect to call the movements 'north' or w/e tho. constant left turns etc) to also return the same exact coordinate values, so the movement math can't just cut loops when it sees them because that would still yield the 'same point' having different representations. like i guess i should be possible to represent each coordinate with the shortest path from it to the origin, so the coordinates would be like...
- []
- [N]
- [NE]
- [NES]
- [WSE]
- [WS]
- [W]
- []
as you move around that circuit. but wow i have no sense whatsoever what the math for that would look like, even a little.
i guess if i stop using fake directional coordinates and accept that this can only be described by (L)eft (R)ight and (T)hrough movements (with (B)ack just popping the last movement off the stack) then the coordinates look like, uhhhh
- [] / [TRRRRR]
- [R] / [TRRRR]
- [RL] / [TRRR]
- [RLL] / [TRR]
- [RLLL] / [TR]
- [RLLLL] / [T]
- [RLLLLL] / []
okay so now we have a problem already b/c there are same-length paths to the same coordinate. which... makes sense, since like, even in a flat plane stuff like TR and RL gets to the same coordinate. so. the theoretical canonical reduction would also need to handle that.
basically i have no clue how possible / feasible any of this is :/
