I've enrolled in an undergraduate seminar on the subject of non-euclidean geometry. I wanted to ground myself a little before-hand, because popular media has lead me to believe that non-euclidean geometries are closely related to portals, witch-craft, and Cthulu.
In particular, the term is frequently invoked to describe phenomena that occur in the game Antichamber. Wikipedia has this to say about Antichamber:
Many of the puzzles are based on phenomena that occur within the Non-Euclidean geometry created by the game engine, such as passages that lead the player to different locations depending on which way they face, and structures that seem otherwise impossible within normal three-dimensional space.
Puzzle elements in various chambers involve maneuvering themselves around the non-Euclidean spaces, where level elements can change after passing certain points, or even based on which direction the player is facing when traversing the level.
How much of what we see in the media is an exaggeration of the implications of non-euclidean geometry? Are there only two kinds of non-euclidean geometries (elliptic and hyperbolic), or can an arbitrary space with extra rules governing movement be considered a non-euclidean geometry. It seems to me that a person's facing while moving in a space shouldn't have any bearing on there position, regardless of whether parallel lines meet at infinity... is there some tricky surface on which this is possible?