HyperRogue app for iPhone and iPad

You are a lone adventurer in a strange, non-Euclidean world. Gather as much treasure as you can before the nasty monsters get you. Explore about 50 different lands, each with its own unique treasures, enemies, terrain obstacles, and mechanics which challenge your tactical and navigation skills. Your quest is to find the legendary treasure, the Orbs of Yendor. Collect one of them to win! Or just ignore your quest and collect smaller treasures.

The twist is the unique, unusual geometry of the world: it is one of just few games which takes place on the hyperbolic plane. Witness a grid composed of hexagons and heptagons, straight lines which seem to be parallel, but then they diverge and never cross, triangles whose angles add up to less than 180 degrees, how extremely unlikely is it to reach the same place twice, and how the world seems to be rotated when you do return. All this matters for the gameplay. The game is inspired by the roguelike genre (although in a very minimalist way), works of M. C. Escher, and by puzzle games such as Deadly Rooms of Death.