While digging around, it seems there's also an online version for lighter-weight tire kicking https://www.roguetemple.com/z/hyper/online.php
Don't overlook the author's YouTube channel, which digs into the non-Euclidean parts: https://www.youtube.com/@ZenoRogue/videos
The coolest non-Euclidean game ideas I've seen involve violation of the triangle inequality of metric spaces, i.e. the shortest path between two points is not necessarily a straight line (e.g. Portal).
Just from the article, I can't tell how the projection fundamentally impacts game play.
That being said, I standby my initial comment. After playing the game, I see how hyperbolic geometric affects movement in the world. What still remains unclear, however, is how movement in a hyperbolic world impacts game play in a meaningful way. Games are not complete happenstance; they're a collection of intentional choices to create an experience through purposeful mechanics. While the choice of hyperbolic geometric was intentional, the impact of this decision on the game play feels more or less random. That's not to say hyperbolic geometry couldn't be used to make an interesting game, but one would need to design the game in a way that creates a compelling experience by intentionally exploiting the characteristics of hyperbolic geometry. Taking an existing thing and recreating it with hyperbolic geometry is not cutting it for me, at least not in this specific instance.
Another effect is that "parallel lines" don't exist in quite the same way. That is, any given line has infinitely many non-intersecting" parallel lines, but they all diverge from it rather than maintaining a constant distance. So one entity pursuing another along a "parallel" course will have to constantly turn* to keep following them, which lengthens their path and slows them down.
It doesn’t violate the triangle inequality, but it does mean that there’s way more than one parallel line through a given point off an initial line. I think that’s Euclid’s axiom.
Intuitively, the mechanics make it easier to run away from groups of enemies, since the angle it would take to follow you is more precise. It also makes it harder to find and return to earlier locations. In fact, to win the game you have to find an “orb of yendor”, then a key spawns 100 tiles away with an arrow helping you find it. The hard part is getting back to the orb once you get the key.
it's like.. suppose your space is discrete like in this game (if it's continuous it's the same argument basically), if you walk N steps in arbitrary directions in euclidian space, you can be sure you are confined in a N x N square so there are N^2 possible tiles you could be. in hyperbolic space your possibilities are much larger, space grows exponentially so you easily get lost
anyway that's why parallel lines diverge in hyperbolic space: if you have two parallel trajectories that go in the same direction they get further and further apart because as you go, there is more space around each trajectory
this means that unless you retrace your exact steps, it's very very hard to get back to your starting position after you wander for a while. navigation becomes almost impossible and it's not a matter of recognizing landmarks because you may never get to see the same landmarks again
so in this game you are always walking towards new stuff; even if you go back, you won't find the placew you were before
good thing it is procedurally generated then, and it basically doesn't matter much where you are because the game is pretty much the same everywhere (it has biomes but they just determine your enemies and stuff like that)
a game like this but with a plot would be much harder and maybe require some sort of teleport or transport network
Still, there are many very polished roguelikes on Steam (Dungeons of Dredmor, Crown Trick, Sproggiwood, Moonring, etc.).
I've been thinking about what it would like to express something like the holographic principle in this form factor ... haven't made much progress!
