Life inside a hyperbolic plane?
May 20, 2011 8:44 PM Subscribe
How does walking around on a hyperbolic plane differ from a euclidean one?
Based on this post I was drawn back to an idea that I've been tossing around my head for a while. I want to create a dungeon that drops the players onto a hyperbolic plane. They'd step off a teleporter and start walking down a hallway, reach a 90 degree bend and then...
I have the imagery from Not Knot to start me on my way, but I have a bunch of questions about visualizing such a space, like whether it'd be possible to have a room that is a regular cube (or something close to it) in a space that otherwise has five or six 90 degree angles to navigate a loop. I'd happily take answers to that particular question or a description in general, but I'm more looking for a resource or two.
Based on this post I was drawn back to an idea that I've been tossing around my head for a while. I want to create a dungeon that drops the players onto a hyperbolic plane. They'd step off a teleporter and start walking down a hallway, reach a 90 degree bend and then...
I have the imagery from Not Knot to start me on my way, but I have a bunch of questions about visualizing such a space, like whether it'd be possible to have a room that is a regular cube (or something close to it) in a space that otherwise has five or six 90 degree angles to navigate a loop. I'd happily take answers to that particular question or a description in general, but I'm more looking for a resource or two.
Response by poster: I guess I didn't explain myself enough in the question- flunkie the idea would be that the strange topology would not be instantly visible, but would, over time, show itself as they found themselves in places where they shouldn't have been. (eg, taking 4 lefts to get back to a room and finding something else)
posted by Hactar at 10:35 PM on May 20, 2011
posted by Hactar at 10:35 PM on May 20, 2011
I don't think walking around in a hyperbolic geometry would be that interesting. There's way too many "straight" paths that just diverge infinitely.
What about a toroid geometry? Move right or left and you quickly come back to your starting point. Move one room forward, then right or left you go through different rooms that also circle back. Keep moving forward and you also circle back.
posted by sbutler at 11:46 PM on May 20, 2011
What about a toroid geometry? Move right or left and you quickly come back to your starting point. Move one room forward, then right or left you go through different rooms that also circle back. Keep moving forward and you also circle back.
posted by sbutler at 11:46 PM on May 20, 2011
"Oh dude, do a Klein bottle!"
You don't want to do that - you'll reverse your parity on traversing the bottle and the next potion of healing you quaff will have molecules of the wrong chiralty and give you a hell of a stomach ache.
posted by edd at 5:29 AM on May 21, 2011 [4 favorites]
You don't want to do that - you'll reverse your parity on traversing the bottle and the next potion of healing you quaff will have molecules of the wrong chiralty and give you a hell of a stomach ache.
posted by edd at 5:29 AM on May 21, 2011 [4 favorites]
Its a little bit of a tangent, but the novel Inverted World by Christopher Priest kind of deals with this premise, albeit in a more temporal sense than a geometric one.
posted by TheOtherGuy at 6:55 AM on May 21, 2011
posted by TheOtherGuy at 6:55 AM on May 21, 2011
Best answer: There's way too many "straight" paths that just diverge infinitely.
So a group of people set out to walk in a bunch, and there is a constant tendency to spread out, have their paths diverge from each other.
As Flunkie says, it really manifests itself on larger scale phenomena, but I think it would be a neat effect that would slowly get creepier over time if each character was thinking more and more that the other guys were moving away from them. And in a world with spherical geometry other people would be constantly bumping in to you. (Though I know people who walk like that in plain old R2.)
One thing you have to take care of (or get to play around with) is that, while flat space has no characteristic length, i.e. no scale, curved spaces do have a "radius of curvature". And there's no reason it has to be constant over the whole world; you can have places where it curves a lot, and places where it is barely noticeable.
It sounds like a fun conceit. Good luck implementing it.
posted by benito.strauss at 7:16 AM on May 21, 2011
So a group of people set out to walk in a bunch, and there is a constant tendency to spread out, have their paths diverge from each other.
As Flunkie says, it really manifests itself on larger scale phenomena, but I think it would be a neat effect that would slowly get creepier over time if each character was thinking more and more that the other guys were moving away from them. And in a world with spherical geometry other people would be constantly bumping in to you. (Though I know people who walk like that in plain old R2.)
One thing you have to take care of (or get to play around with) is that, while flat space has no characteristic length, i.e. no scale, curved spaces do have a "radius of curvature". And there's no reason it has to be constant over the whole world; you can have places where it curves a lot, and places where it is barely noticeable.
It sounds like a fun conceit. Good luck implementing it.
posted by benito.strauss at 7:16 AM on May 21, 2011
Best answer: It sounds to me like you want to lay out the rooms & paths according to a order-5 square tiling of the hyperbolic plane. The "tiles" in that diagram would map to the rooms in your dungeon. All of them would appear identical, and there are four exits from each room, but it takes you five "rights" or "lefts" to get back to where you started from.
Equivalently (though perhaps easier to visualize, lay out the rooms as the nodes of a order-4 pentagonal tiling, if that helps you visualize it. You could put a square "room" that would not be perceptibly distorted at each node, and have long, straight hallways (the blue lines in that diagram) leading from room to room.
Hopefully this is enough to get started — the references listed in there (including the paper ones) might also help in your research.
posted by Johnny Assay at 7:29 AM on May 21, 2011
Equivalently (though perhaps easier to visualize, lay out the rooms as the nodes of a order-4 pentagonal tiling, if that helps you visualize it. You could put a square "room" that would not be perceptibly distorted at each node, and have long, straight hallways (the blue lines in that diagram) leading from room to room.
Hopefully this is enough to get started — the references listed in there (including the paper ones) might also help in your research.
posted by Johnny Assay at 7:29 AM on May 21, 2011
« Older Learning about algorithms and data structures from... | methods or products to query objects? Newer »
This thread is closed to new comments.
If the person walks far enough and takes studious notes, they have a chance of noticing differences, just like a person walking around on the surface of the earth would have a chance of noticing that (say) he arrived at the same spot he started from, all the while walking straight forward.
In the hyperbolic case, one such thing that a person might notice would be that if he's walking in a straight line, back and forth and going further each time, painting a line as he goes, and two other people are doing the same but on two different lines, and the two lines of the other people hit each other, his line might never hit either of the other two lines. That can't happen in a Euclidean plane - his line will definitely hit at least one of the other two lines.
But I think the important thing is that, really, it doesn't matter and wouldn't be noticeable for most purposes - again, just like people in real life don't naturally notice that they're walking on a non-Euclidean surface.
I think perhaps a better idea for a dungeon which has strange but noticeable topological properties would be to embed it in four dimensional space instead.
posted by Flunkie at 9:38 PM on May 20, 2011 [2 favorites]