World maps and their topologyAn idea that has been wandering around in my head for a long time is the creation of world maps living on top of an interesting 2-dimensional topological varieties. Most games have a world map that is a rectangle (topologically a disc), and there might be some world maps that carry our own earth's topology of a 2-sphere. These are rather boring varieties, and it would be interesting to allow for worlds with more exotic topologies: For instance the topology of a torus (the surface of a donut) or even of objects with a higher genus (the number of holes, so the torus has genus 1). The classical pacman game in which the top & bottom and left & right are identified is the torus.
One can triangulate such a variety in order to obtain a decomposition in tiles. Locally, on the player's screen that is, part of this world would then just look like a square as usual. But when walking/driving/flying around in one direction, you can end up at the same place (leaving the player puzzled ;) ). Part of the gameplay would then be understanding the world map and in particular its topology, allowing for more complicated mazes. It would be interesting to carry over some of the phenomena of topology to such a computer game (identifying, cutting), thereby surprising the user now and then.
One advantage of keeping a strict structure in the topology rather than inserting some portals here and there is the superior ability of NPC's to deal with it. Another advantage is the possibility to visualize the topological phenomena in a spectacular way (for example identifying two worlds with each other in a point and then smoothly stretching this identification to a bigger area).The concept could also be generalized to 3-dimensional varieties for 3-dimensional "shoot 'm ups" (like for instance Descent). For three dimensions the phenomena get even more exotic. Why always make the same game, if one can do spectacular things without much extra effort?