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.
It impacts how far you have to go to reach a variety of terrain. In euclidean space there is a fixed amount of surface area with any given walking distance. In this game there much more space within that same walking distance which allows more user choice and variety in areas without having to decrease the size of areas. Thus any given biome can border many more biomes without making the distance between those biomes larger.
To clarify, "For instance, if you were project the space on to a rectangle, what would fundamentally change about the game?" didn't make much sense in the context of this game, and made it sound like the game's world could have been represented in a simpler, more obvious way. All I'm saying is, as you might understand better now, there is no simpler way, and you can't simply fit hyperbolic space onto euclidean space. I would approach this game partly as a math experiment, and partly as a game, with the goal of exploring the possibilities of this combination.
* In Euclidean open space, if you are attacked by two adjacent monsters at once, you cannot escape, because the monsters would just move in parallel lines. In hyperbolic, you can, because parallel lines do not work.
* In Euclidean open space, if you are ambushed by many (say, 12) monsters on all sides, you cannot escape. In hyperbolic, you can. There is always more directions than it seems.
There is a land Hunting Grounds which teaches the tactics above. It simply does not work in Euclidean mode. There is also this article: http://zenorogue.blogspot.com/2012/03/hyperbolic-geometry-in...
When I started working on HyperRogue, I expected it to be a "gimmick or tech demo" as you say. But after implementing the basic roguelike gameplay, it felt surprisingly good, better than what I had actually planned. So released it, and other people liked it too.
So it is not surprising that people expect it to be a gimmick (given that I did expect that hyperbolic geometry would make basic roguelike gameplay feel better myself). Some people do not get it on their first try, but love it on their second try.