For example, I intuitively know what the Euclidian and Manhattan metrics feel like for a space.
I don't have a good way to visualize / feel around hyperbolic space. What do distances "feel like" as I travel farther or closer?
Are there good references for this sort of thing? I want something like Flatland for hyperbolic space!
I recommend that you play the game HyperRogue.
But for a somewhat more mathematical perspective (which is evident in HyperRogue), one big difference is that if you compare the amount of stuff visible to you at a certain distance $r$, there's a lot less stuff in the Euclidean plane ($2 \pi r$-worth of stuff) than in the hyperbolic plane ($\sinh(r) \approx e^r$ worth of stuff). Those formulas are, of course, the circumferences of a circle of radius $r$.
So don't drop a blueberry on the floor, if it rolls too far you'll probably never find it... but at least you won't randomly step on it either.
As Lee Mosher states in the other answer, you cant go wrong in trying the game/visualiser HyperRouge, it's mechanically very simple so don't be afraid to try it if you're a non-gamer.
As per the question, I'll try to give you a feel for it:
Assume unless I say otherwise that I'm referring to hyperbolic space, another name is "space with a constant negative curvature" or simply negatively curved space.
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Similarities
The space is symmetric/regular everywhere, no region is special.
Locally, on small scales the space has very little curvature, like euclidian space.
The space supports paths, angles and measures such as areas, lines and straight lines. In contrast with flat Hilbert space, the equivalent to volume (units^∞) is far less useful.
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Differences
Across larger distances the space starts to behave in a more alien way.
Parallel lines need to curve towards each other, straight lines that start off parallel will diverge away from one another.
Loosely speaking there's more space around every point than in flat space, for example a regular polygon with straight edges, and right angled corners, that's centred around your point of view wont join up. To form a close shape you'll need to decrease the internal angles, or even add more sides and right angles.
Aiming in a sport (snooker, football, etc…) is much harder.
Increasingly distant objects appear increasingly tightly packed.
Circles can have the same curvature everywhere, and still be infinitely large, with up to two endpoints on the horizon infinitely far away.
It's hard to get back to anywhere without retracing your steps.
Everything becomes very "maze-like", it's easy to end up very lost.
Orientation is not preserved when moving, if you walk in a square without turning, you'll still get turned around.
Moving in a semicircle around an obstacle adds vastly more distance than it does in euclidian space.
A straight line in the right place can well approximate almost any path, such as random walks.
It's possible to move away from infinitely large regular trees, leaving them as a distant spec on the horizon. You're actually in the space between branches but it doesn't look like that, you'll need go in a very specific direction to get back to the tree.
It's easy to flee and hard to chaise, in hyperbolic space.
Don't flee to fast though as you'll feel a constant outward pressure proportional to your speed.
Maps don't work very well, you cannot just scale things up and down like that and expect them to be very coherent afterwards.
The greater the curvature the more the cells in their tilings in your body's tissue layers will be crushed together. As with altitude it may be possible for densely packed cells to die, divide, and and allow your body to gradually acclimatise. If you push this too far though simply extending and moving limbs may yield hazardous fluctuations in internal pressure.
Many forms that don't work well in flat 2D or 3D space, but do work in flat Hilbert space will fit nicely into flat 2D or 3D Hyperbolic space.
Vastly more uniform tilings and polyhedra work, for example 5 squares around every vertex.
A sphere (the edge of it) in 3D hyperbolic space with a single point on the horizon, forms euclidian space.
You can have non-intersecting regular tilings of apeirogons, with infinitely many edges per vertex and infinitely many faces per edge.
Single triangles on their own with all three corners on the horizon are all regular, such triangles have internal angles of 0 degrees and infinite side lengths, the area is PI. This property doesn't hold for quadrilaterals.
