The purpose of this article is to explain what's the deal with hyperbolic geometry in HyperRogue. It is intended for mathematicians interested in hyperbolic geometry who would like a summary of HyperRogue, and for roguelike players who would like to know what hyperbolic geometry is -- some of them think that HyperRogue takes place on a 3D sphere, or that you will return to the same location if you move in a particular direction for a long time, which is not accurate at all, so I explain it here in more detail.
HyperRogue is a roguelike computer game. Roguelikes are a genre of computer games that have evolved separately from the mainstream -- even though they are commonly viewed as a subgenre of RPGs, they are in many senses closer to board games, like Chess where you have just one piece and the "chessboard" is randomly generated by the computer, than to other computer games. (Recently the elements of roguelikes, such as procedural generation and the challenge of permadeath, are added to more and more computer games which do not really have much in common with true roguelikes, causing even more confusion.) In case of HyperRogue, the "chessboard" is an infinite hyperbolic plane. You can read more about it on Wikipedia. This is not a sphere, but rather a hyperboloid (but also not in a normal space). The 3D object you see when you activate the "eye distance" option is this hyperboloid.
Hyperbolic geometry seems to be underused in art. M.C. Escher has been using it in some of his work (like Circle Limit). I have seen Maze, Pool, and Sudoku on a hyperbolic plane, but here the map is looped, and thus IMO less interesting than the real infinite world of HyperRogue. The original concept of Hyperbolic Rogue got quite popular despite being an extremely simple game, so I think this is a great area to explore.
Normally the game presents the world in the so called Poincaré disc model (also used by the art mentioned above). The closer you are to the edge of the disc, the bigger distances are (five pixels close to the edge is a much bigger distance than five pixels in the center). On the Euclidean plane, the boundary of a circle of radius r is linear in r. On the hyperbolic plane it is exponential. This means that it is very hard to return to a place where you have already been in HyperRogue. You would have to return almost the same way.
Triangles in hyperbolic geometry have angles which sum to less than 180 degrees. For example, if you take two hexagons and one heptagons next to each other on the HyperRogue map, and construct a triangle with vertices in centers of these cells, the sum is (360/6) + (360/6) + (360/7) = 171.4 degrees, so there is a "defect" of 60/7 = 8.6 degrees. The defect is proportional to the area of the triangle. (Similar thing happens on a sphere: a triangle with a vertex on a pole and two on the equator has 90+90+90 = 180+90 degrees, which is π/2 more than 180 degrees, and the total area of the sphere is 4π, which corresponds to eight of our triangles.) The visible in-game effect of this is that if you go from A to B, from B to C and from C back to A, the screen will (probably) appear rotated, making it easier to get lost.
This is the map structure I have found most suitable for a roguelike (or other similar grid-based game). Three hexagons would sum up to 180 degrees and we would get a plane. Three heptagons in each corner would be possible, but they would be too big (each of them would take the space of one heptagon in HyperRogue plus 1/3 of each hexagon surrounding it).
Now how does this relate to the actual gameplay? As already mentioned, an important result of using the hyperbolic plane is that you almost never get into a place where you have been before. Some of the areas make other aspects of the hyperbolic geometry important, and some are just for fun.