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So, I want to write a story set mostly in a world using hyperbolic geometry (except brief beginning and end bits with the main characters coming from/returning to our world), but I'm a little confused on how that influences some parts of the worldbuilding. In particular: Rivers.

The descriptions I've come across suggest that it would look like everything is sloping down away from the viewpoint, does that mean all directions actually act like being downhill (unless they're actually sloped up enough to overcome that)? If so, does that mean a river can flow in any direction? And if that's true, does a river flowing in a loop actually work under hyperbolic geometry?

I feel like I've missed something that would keep that from working properly, but I can't see what. Then again, I kinda hope it actually does work that way so I can have a proper M.C. Escher waterfall.

EDIT: Some clarification and additional information:

By "A world using hyperbolic geometry", I mean that the rules space/geometry of this universe are defined by the principles of hyperbolic geometry: https://en.wikipedia.org/wiki/Hyperbolic_geometry

My rough idea of the shape of the "planet" was an endless roughly flat expanse, probably with gravity being in a fixed direction perpendicular to the plane of the "planet". But both of these points are flexible, since I was trying to start working on things that would immediately affect the opening scene so I could actually get something written. I would also accept answers for spherical planets with the same gravitational rules as our planet and only the geometry being changed.

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    $\begingroup$ There are multiple kinds of hyperbolic surfaces, none of which could exist in Real Life. Let's ignore that. Can you pick and/or provide one image of what your world specifically looks like? $\endgroup$
    – JBH
    Oct 23, 2023 at 17:48
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    $\begingroup$ @JBH I wasn't talking about the shape of the planet (which is probably an endless roughly flat expanse, though I'm undecided on that point), I was talking about the fabric of space having a negative curvature: en.wikipedia.org/wiki/Hyperbolic_geometry $\endgroup$ Oct 23, 2023 at 17:58
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    $\begingroup$ "All directions actually act like being downhill" only if they actually are downhill. Downhill is by definition the direction in which water flows. Water flows from high gravitational potential to low gravitational potential. Since you are asking about rivers, I understand that you have already figured out basic stuff, for example how gravitation works in a world with such a metric. $\endgroup$
    – AlexP
    Oct 23, 2023 at 18:15
  • $\begingroup$ @DavidMonroe Please edit your post to make that clear. Your post says, "world using hyperbolic geometry." Your post mentions nothing about space and negative curvature. At the moment, considering you haven't yet considered your planet's shape (much less its gravitational profile), you appear to be putting the cart ahead of the horse worrying about rivers. $\endgroup$
    – JBH
    Oct 23, 2023 at 18:16
  • $\begingroup$ @JBH In the questions I have previously read about hyperbolic geometry, everyone seemed to understand what the term meant. Still, I have edited the post with clarifications. $\endgroup$ Oct 23, 2023 at 19:01

3 Answers 3

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The "everything is sloping down" is an illusion caused by the geometry. When the observer lowers their eyes to terrain level, the illusion vanishes. The gravity works same as in normal geometry, you can have hills and valleys where water flows exactly as you would expect.

However, in hyperbolic geometry you have both more surface area and more volume in your vicinity. How much more that depends on the curvature. Wikipedia has the formula for area of circle which is bigger than in normal geometry. For 3d shapes it's even more. On world with high curvature both the river catchment area and atmosphere volume above it are much bigger for the same diameter. Means more water rains over bigger area thus the rivers are much more prone to catastrophical flooding.

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    $\begingroup$ Illusions are tricks of perception. Do you mean the curvature would affect how light is refracted, which would lead the observer to believe the flow is opposite what is expected, but upon closer inspection, it's actually flowing as expected? $\endgroup$
    – JBH
    Oct 23, 2023 at 21:39
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    $\begingroup$ @JBH The idea is that your perception is based on the assumption that the world lies in flat space with zero curvature. The space actually has negative curvature and once you account for that the water flows the way you would expect. $\endgroup$
    – quarague
    Oct 24, 2023 at 7:16
  • $\begingroup$ Such illusions can actually work in flat space too but for different reasons, see en.m.wikipedia.org/wiki/Gravity_hill $\endgroup$
    – Juraj
    Oct 24, 2023 at 7:29
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Generally Speaking

In a hyperbolic geometry I would hypothesize that everything looks further away, yet when you move toward it, it appears to approach more quickly than it really does.

I could be wrong in my interpretation, because my assumptions would seem to resemble how it looks to move at near the speed of light. If you're struggling to see this, consider the MIT experimental game A Slower Speed of Light ( http://gamelab.mit.edu/games/a-slower-speed-of-light/ ).

Gravity would still have to change the curvature of space from positive to negative in order to function. So there would perhaps be some strange warping of the visual field, i.e. the horizon would likely look like an infinite, flat disc, while the sky would perhaps seem to curve away, likely as though space was poking down into the sky from above while the atmosphere clings to the horizon. And the sky above would look like the beginning of time, while we would see the horizon in a way like a mirror placed at infinity ( though I submit this may be misleading, as at infinity the mirror is strangely both near and far, i.e. zero distance or everywhere, but rather would have infinitesimal perspective height, so the two may cancel ). Indeed a very strange perspective.

As for how water behaves, perhaps in a very similar manner to the way we're used to, though it would appear that a river would flow more slowly in front of us, faster toward us up stream and again faster as it moves away from us down stream. Throwing your fishing line down stream though would put the lie to this perception, not snatch the rod and reel out of the hand as one might expect.

Rain might appear to be deadly in its descent, but would still land with the same amount of momentum on the face, though the splatting of the droplets would be ever so more brief in appearance, perhaps imperceptible to the naked eye as the distance the splat travels is relatively small.

How Rivers Work

As far as how rivers behave, that's less a matter of their route along the "plane" than it is the topology of the land. Water simply flows downhill. It needn't be more complicated than that.

Conclusion

The more interesting features of this geometry are most likely going to have less to do with actual physics than of our perception of space and time in a largely distorted version of hyperbolic space.

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How big is your hyperbolic curvature?

We are on the surface of a sphere. If we draw circles with increasing radius, the larger circles will have a circumference slightly smaller than $2\pi{r}$. Eventually at the equator, the size will peak and then start to decrease.

If you were in an elliptic negative curvature space, then the surface of a sphere would be slightly less than $4\pi{r}^2$. If you were in a hyperbolic positive curvature space then the surface of a sphere would be slightly less than $4\pi{r}^2$. Our space is, as far as we can measure, on the line between the two. Light and gravity fall off as inverse squares, and no-one has reliably measured any deviation.

There is nothing wrong with having spherical planets in a hyperbolic universe. They would be much the same if the curvature was small. There would be more distant galaxies, and they would be a bit fainter.

I suspect what you want is a hyperbolic planet. This is not the familiar waisted hyperbola of geometry, but something where every point is the same. Have a look at the Poincare mapping: this may be what you are after. There are more distant points than in a flat world. You might expect ships to rise up rather than sink under the horizon as they do on a sphere, but this is a spatial mapping on a flat disc so they don't rise or sink. And rivers would still flow downhill.

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