I found a few examples which explore the use of non Euclidean spaces in virtual environments - a refreshing dose of innovation in the way we have come to think about virtual spaces.
"Euclidean space - a space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional"
So then, non Euclidean space is a space which is non linear, operating beyond the bounds of Cartesian geometry and polar coordinates. This space is not topographical but a manifold of space within space, illogical to the laws of the physicality of our own world.
Feign is a game created by Ian Snyder which explores non euclidean space in a fairly simple way of navigating a maze. What makes this interesting is that there are 'impossible', overlapping spaces in this world - endless corridors, rooms that change depending on the direction you are facing, entire areas compressed within what is seemingly a single tile. You can walk around in a circle and find yourself not back where you began, but an completely new space. It could be argued that these spaces aren't non Euclidean in the sense that they are experienced through the representation of a three dimensional world on the two dimensional plane of the screen but being that what we consider '3D' games don't actually have any physical depth, I think it is reasonable to accept these representations of non Euclidean geometry as being perfectly valid.
Thirteen Gates takes the ideas expressed in Feign to an entirely new level because while the author's previous game was mind bending, the configurations of the virtual world were still rooted in an aesthetic resembling our own (walls, floors, characters). Thirteen Gates on the other hand is far more removed from the indexicallity of reality, it presents a space so foreign that it feels completely unnatural to navigate not in terms of the user interaction but in the sense of how the player understands the space and is meant to interpret it. In my view is an incredible achievement, after all, interactive space is virtual, constructed and illusionary so why must it always be representational? It is important to point out that it its not the aesthetics of this game which make it interesting or different, the surreality of the graphics undoubtedly enhance the immensity of it all but it is the mechanics of this world the very nature of this space which is so alien. The player basically navigates a maze of patterns with a distorted perspective and a non existent depth of field.
Some further examples of spacial manipulation:
Echo Chrome plays with expectations of space through perspective shifting:
Fez twists the conception of two and three dimensional space by having multiple two dimensional planes exisitng within the same world:
Thinking back to the prototype I made using multiple frames to fragment the on screen space makes me consider some of the possibilities of spacial manipulation within even a two dimensional realm. Although not non Euclidean in itself, separating the screen into multiple sections undoubtedly plays with some of the same ideas which this concept expresses. What if frames changed as new ones were entered? If 2D worlds can be made to overlap or intersect in some sense then a myriad of new potentials could unfold.