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Let's say our world is set in a hyperbolic space with a curvature $K$ of $-1$ per $9$ square meters. The surface of our world is a hyperbolic plane (approximately, there are still hills and valleys and stuff). The sun is surface 100 million miles away that turns on and off every in a 24-hour cycle. The amount of energy it gives the world per unit area is similar to that of our sun. Earth-like life lives on the hyperbolic surface, including a human-like species.

My question is, how would humans navigate this world?

Before you say "the same way as ours", let me note some problems:

  • Whereas spheres in our world have a surface area that grows quadratically with radius, in hyperbolic space, the surface area of a sphere grows exponentially! That means, assuming energy is conserved, long distant wireless communication does not work (since the signal given off is distributed over a larger area).
  • Maps also are much harder to use. This is because you can't make images of things that are smaller than the original object without some distortion (the bigger the difference in size, the bigger the distortion). The curvature in this world is so severe that maps of even smallish regions distort quite a bit. Here's an example (the curvature of the world that map is portraying is also about $-1$ per $9$ square meters).
  • You might say that navigating by the stars might work (if they exist). But, you can't see the stars (due to the first point) nor map them (because of the second point).

So, given all these challenges, how would someone navigate in this world? The solution should work both within cities, and allow travel from one city to another.

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    $\begingroup$ As I understand it, light rays on a hyperbolic surface would follow the shortest path on that surface. It's not like you would look to a region, start walking in that direction, and end up farther from the region than prior to starting, because you would follow the path of the light you see, which also happens to be the shortest path regardless of what it may look like from an extended, outside Euclidean observer. To a human on that surface, I would imagine hyperbolic space would not look any different from Euclidean space. $\endgroup$ – B.fox Jan 28 '18 at 4:16
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    $\begingroup$ @B.fox Navigating to places you can see generally isn't challenging. $\endgroup$ – PyRulez Jan 28 '18 at 4:18
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    $\begingroup$ @B.fox Uhm, how would you do that without a map? People in our world don't navigate to places by just figuring out which direction they are and then blindly walking in a straight line. $\endgroup$ – PyRulez Jan 28 '18 at 4:23
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    $\begingroup$ @B.fox The problem isn't that geometry stops working (it doesn't). The problem is that maps and GPS don't work. Without that, we would have trouble navigating in real life. $\endgroup$ – PyRulez Jan 28 '18 at 4:33
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    $\begingroup$ @PyRulez Why do you think GPS won't work? True, there are (probably) no satellites, but there could be still radio towers - and on a planar world, a radio tower has an excellent coverage. And your world has much less distance per area, so even with exponential decay the emitters would be receivable. You would not be able to measure time delay with sufficient precision without high tech, but triangulation based on signal strength is easier. $\endgroup$ – Radovan Garabík Jan 28 '18 at 21:58
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Roads and Road Signs

As long as you are on a road, the curvature of the world doesn't matter. The fact that you could get lost really fast if you diverged from the road by even a tiny angle just doesn't matter, because you wouldn't do that--you'd stay on the road! The only thing you have to worry about is making the correct turn when you get to an intersection of two or more roads--exactly like navigating a road system in our world.

That works perfectly well within cities, and perfectly well for going between any two cities that are already connected by some road network. So if those are the only cases you need navigation to work for, you're all set!

The problems only really start when you need to travel off-road, explore new place or blaze new trails--and expect to be able to get back home again. And to handle those cases, you'd probably just literally blaze a trail, or unravel a proverbial ball of string behind you in the labyrinth. Once you've gone a certain distance, you set down a marker. What kind of marker it is will likely depend on the specifics of the terrain you are traversing. Then, you just make sure to stay close enough to that marker that you know you can find your way back unassisted, until you set down another marker, and so on. Long-distance surveyors and road-building crews might use laser guides, or simply theodolites, just like we do in our world, with the difference that their angular tolerances have to be much tighter, and the maximum distance they can sight accurately would thus be considerably smaller than it is in our world. I.e., you wouldn't try to lay out a 50-km road in one go; you'd do it in small chunks, expecting to run into someplace you recognize at the end of the new road. If you don't run into anything in the expected distance, you back up along the path, adjust the angle of the last few segments, and try again. If you run into something you know, but not the right thing, then you can make better adjustments.

If you need to navigate over water... well, don't. This was a historical problem in the real world, as well. You just don't sail out of sight of land. Not until you can set down well-anchored buoys, spaced closely enough together that you can always find the next one before you go out of detection range of the last one (whether that's by sight, short-range radio, or whatever). Effectively, building roads in the sea.

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  • $\begingroup$ "If you need to navigate over water... well, don't." They likely wouldn't need too. See alchemist lab in this article as for why. $\endgroup$ – PyRulez Jan 28 '18 at 3:00
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    $\begingroup$ @PyRulez I don't think your conclusion follows. If the only reason for travelling over water was to expand your territory, then yeah, you wouldn't need to do that, since you can probably (but not necessarily!) go as far as you'd ever want on land. But it's not. There can still be disconnected regions of land, and people may want to get between them, e.g. for purposes of trade, or to exploit resources that don't exist on their home continent. Or, they may want to find faster, more direct routes between lands that are connected, but only circuitously. $\endgroup$ – Logan R. Kearsley Jan 28 '18 at 3:15
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    $\begingroup$ Also, as long as you don't have too many cities, you can make a topological map of the connections between the cities. $\endgroup$ – PyRulez Jan 28 '18 at 3:29
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    $\begingroup$ @NofP the one City probably started as a settlement of the other. When making the settlement, they blazed a trail. $\endgroup$ – PyRulez Jan 28 '18 at 22:22
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    $\begingroup$ Would you even have roads in our sense? You can pack our Earth to the disk with 60m radius - everything is within walking distance, all the natural resources they ever need, road building never develops. And the tribe that walks away buds off their own civilization, never to be seen again. $\endgroup$ – Radovan Garabík Jan 28 '18 at 22:52
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Riverboat pilot style; during the days of the paddle steamers on the Mississippi and Missouri Rivers in the US boat pilots apparently traveled up and down very small, sometimes less than ten miles, sections of the river steering for different captains that came their way. Each pilot specialised in their home patch and didn't know what was around the next bend beyond their section of the river but they knew every sand bar, sunken log and tying off tree that was on their bit of the river.

My suggestion is that long distance travelers don't navigate for themselves over long distances but rather they rely on locals who know small areas of the world very intimately either as professional guides through a particularly difficult piece of territory or just asking the locals "which road to Travistoc?" Now an established road, or track, network will help a lot when it comes to getting from A to B with a minimum of fuss on well traveled routes (you can in fact map such a network without much fuss using a Schematic Map that is accurate in routing and labelled for distance, as surveyed by chain) but off the beaten track you need local knowledge since you can't have accurate long range mapping. Surface shipping will have to use a similar approach; hopping along the coast between established ports guided by local specialists where necessary. Coming back alive after getting out of sight of land will be a cause for massive celebration and epic boasting.

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Dogs

Dogs' sense of smell could help humans find their way. Humans have a long and beneficial relationship with canines, and in your world that relationship would be even more precious since a dog's sense of smell could help provide direction and get their human home.

Dogs sense of smell is easily 1,000 times more powerful than a human's. I found places that rate it even higher than that. Humans in your hyperbolic world would likely create their own breeds of dogs that are gifted with tracking and guiding and always take them with them when they venture out of town. Roads develop their own scents from the various travelers that use them. As such if a dog hears the command road they know that their human wants to be guided back to the road. If the dog is told home, the dog will try to pick up the scent of home if it is within range. If it is, then they are set.

Distance of smells

This is much harder to determine. There has been a documented case where a dog could smell whale feces from 1.2 miles away. If your dogs are breed with a high focus on smelling things that are far away, then your dogs should be able to exceed that mark reliably.

Culture based around incense and aroma

Since people know that their dogs are likely the only thing that can get them home if they get lost, humans likely will try and make their towns easier to find by their dogs. As such towns may burn large amounts of incense, or use wood like cedar in buildings since it has strong natural aroma to it. That way a dog should have an easier time picking up the smell of home. Much like how coastal locations build light houses to guide ships, towns could have scent houses to guide dogs. Note the smell does not have to be pleasant, just strong. So places that do not have access to anything nice, they could use rotting carcasses, could convert an outhouse into their scent house, or use a large pile of whale feces.

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    $\begingroup$ Keep in mind that scents also drop off exponentially. Dogs could probably follow a trail, but finding one would be difficult. $\endgroup$ – PyRulez Jan 28 '18 at 19:05
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Is this about a life on a 2D hyperplane or a sphere in a hyperbolic 3D space (the question mentions both)? The life will be drastically different in these two cases. Note that curvature $\frac{1}{9}m^{-2}$ is kind of extreme. I assume the spacetime is a "hybrid" of hyperbolic 2D (or 3D) space and absolute Newtonian Euclidean time (if the time is hyperbolic as well, we get anti-de Sitter space which means geodesics (worldlines of two points in rest) will diverge and you get quite a good model of a rapidly expanding universe - your planet will have rather short life!)

First, the Sun - 100 million miles, let's say $160\cdot 10^{9} m$ means the light (and gravity) is reduced by $sinh^2 \frac{160\cdot 10^{9}m}{3m} \approx \frac{1}{2}e^{2\cdot 5\cdot 10^{10}} \approx10^{2\cdot 10^{10}}$... just forget about it, there is no way to even compare it with anything (and literally anything, like shining a match across observable universe is immensely brighter) from our universe.

The vision will also work with an exponential decay - say you see object up to 10km in Euclidean space. Then in the hyperbolic space you'll see objects up to 20m.... (note however that's a lot of space in there)

2D hyperplane

Let's assume the area you are interested in is that of a very small country, say inside a circle of 100km radius (approx. $30000 km^2$. In the hyperplane, the country is inside a circle with radius a bit over 60m... So your map has to depict a circle with 60m radius and the area of $30000 km^2$. Or, in 1:1000 ratio, your map will have to display 6cm circle with an area of $30000 m^2$. I guess any "map" will be purely topological, displaying (or even only describing) major terrain features, more in terms of very precise angles (in hi tech society) from other terrain features - like "10 m from the tower in the azimuth 12°10'12.57664'' there is a well". Note that all topographical features will be necessarily rather small in their diameter.

For sufficiently technological civilization, use radio beacons. Since the world has much less distance per surface area compared to Euclidean one, beacons should be more or less receivable even if the signal decays exponentially, especially if the transmitters use directional beams. Then it's just a variant of fox hunting. Even a form of GPS is possible with triangulation and sufficient computational resources.

3D hyperbolic space, spherical planet

A sphere in hyperbolic space is a sphere, with a positive (2D) surface curvature.... so nothing different from mapmaking of Earth surface.

The problem is that with the space curvature radius 3m, standing on the surface of the planet will look like standing on a very steep hill, rapidly disappearing into the infinite abyss below (remember vision works very differently). So map will be inherently more navigable than the reality.

(also, the planet with Earth surface will have a radius of about 40m)

EDIT: it's a hyperplane & add radio beacons

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  • $\begingroup$ The planet isn't spherical, it's planar. The world is a hyperbolic plane embedded in hyperbolic space. Also, time is Euclidean. $\endgroup$ – PyRulez Jan 28 '18 at 20:54
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    $\begingroup$ @PyRulez Then I'd expect the maps to follow major landscape features - in Hyperrogue, I guess the crossroads would be numbered by explorers and the map would consist mainly of tables saying which road intersects which at what milestone and where it is closest to which parallel road. And since for "not getting lost", angles are more important than distances, I'd assume maps in the form of conformal Poincaré disk or half-plane model, very necessarily with a lot of details around the "centre" and just major objects at the border. Assuming you know how to follow a straight line in the real plane. $\endgroup$ – Radovan Garabík Jan 28 '18 at 21:24
  • $\begingroup$ you could also glue a bunch of band models together, which preserves a line. $\endgroup$ – PyRulez Jan 28 '18 at 21:25
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    $\begingroup$ @PyRulez Yeah... if there are rivers (if your world is big enough to have rivers) then the natural "coordinate system" will probably develop along rivers. $\endgroup$ – Radovan Garabík Jan 28 '18 at 22:24
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    $\begingroup$ Hyperplane is not a correct abbreviation for "hyperbolic plane", because this word already exists and has a different meaning: a (d-1)-dimensional subspace of a d-dimensional space ( en.wikipedia.org/wiki/Hyperplane - so technically it is what PyRulez meant, but still). $\endgroup$ – Zeno Rogue Mar 26 at 19:33
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When you look at a distant object in hyperbolic geometry, the main problem is that it will be tiny. You're using HyperRogue as a base, which, in its standard tiling, can be considered to have absolute unit of 3 meters. A circle with diameter of 15 meters will then have circumference around 1400 m. In Euclidean space, a circle with this circumference would have radius 222 m, so in other words: you would see an object 15 m away just as small as you would see an object 222 m away in our world. Since the circumference of the circle is the same, the angular size of the object (which is basically how big part of the circumference it takes) is also the same.

Let's say you can see an object (perhaps a mountain) that is 50 km away on Earth. A circle with radius of 50 km has circumference 314.16 km. In HyperRogue world, this circumference would correspond to radius of only around 31.3 m; in other words, if you could only notice something as big as a mountain at THIS distance, there's not much chance of seeing anything that is farther.

And this assumes that you could see the whole mountain, which you actually can't -- and this brings me to the next point.

In Euclidean world, we take it for granted that when we look at a distant 3D object, we see a half of it. You look up to the Moon and you see half of the whole Moon. If we ignore the pesky fact that we have two eyes for a moment, then a single eye can actually never see a full half of the Moon, but the difference is negligible.

In hyperbolic world, it's not negligible. If you have a sphere in this world, it turns out that you will always see just a small part of it -- the bulk of it will be so diminished by the insane hyperbolic perspective that it will actually hide behind the nearest bit.

When you calculate it, you will find that there is an upper limit on an AREA of a sphere that is visible. And that area is pi -- that's what you'd get with infinite distance and infinitely large sphere (horosphere). In HyperRogue, an area of pi would mean around 28 square metres. Let that sink in. If there was, say, Jupiter near you (no matter whether we're talking Jupiter's radius, circumference, mass or whatever -- in this case there's no real difference), you couldn't actually see more than 28 square metres of it! This also means that you couldn't find a meaningful difference between a small sphere and a large one -- as the radius grows, the curvature approaches horosphere, and it would be very hard to distinguish how big the sphere is, exactly -- or whether it's even finite.

That means you couldn't actually see a mountain at the distance of over 30 m as previously mentioned -- the bulk of that mountain would be hidden and you could only see a very small front of it, which could, of course, be seen at smaller distance.

Not only would be distant objects small, but they would also move insanely fast when you move. Hyperbolic parallax doesn't work the same way the Euclidean one does; distant objects would not seem to move more slowly. If you walked under a sky full of stars (and you could see them for some reasons), the stars would seem to move -- regardless of their true distance.

So, given all this, how would you navigate in such a world? The best bet probably is to put together a set of local coordinate systems, but... not even that wouldn't work in HyperRogue. The curvature of that world is simply so big that I have to question whether things like cities or civilization are even possible in there.

One idea I came up with was to have square map plates, each corresponding to a square from a hyperbolic tiling like {4,5} or {4,6}. You could then put the plates together to get a limited view. But even a {4,6} square -- square with inner angle of 60 degrees -- would have an edge of only around 5 meters in HyperRogue. What's worse, no square at all could cover an area bigger than 56 square meters (2 pi square units) because that's the maximum area of a square in hyperbolic geometry. If we used hexagonal map plates, we could get to double of that, but that's all.

Tricky.

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