Non-Euclidean Architecture is how you build places using non-Euclidean geometry (Wikipedia's got a great article about it.) Basically, the fun begins when you begin looking at a system where Euclid’s fifth postulate isn’t true. When that happens, you are talking about a system where parallel lines don’t remain the same distance from each other.
Two basic ways of describing Non-Euclidean spaces: are elliptic and hyperbolic.
In elliptic geometry, two parallel lines will eventually curve towards each other (think of the outline of a football). Space iscurved, and the degree of that curvature affects how long it takes the parallel lines to intersect, and what angle they make when they do.
In hyperbolic geometry, the opposite is true. Space is curved the other way. Parallel lines move further away and will never intersect, only grow farther apart.
Non-Euclidean geometry is weird because it looks like normal space as we know it on the local level, but on the global level it is much different.
Here's an example of "locally normal, globally weird": The globe can be a non-Euclidean space if we assume that the surface of it is actually flat. A man standing at the equator travels to the north pole. He turns 90 degrees to the right and travels back to the equator. He turns 90 degrees to the right again and travels back to where he began. If you map it out, he has made a three-sided figure with three 90 degree angles. He has made a three-sided square! If the surface of the Earth were actually flat, the man would be in a non-Euclidean geometry, probably running from eldritch abominations that he discovered at the north pole.
Actually, most physicists believe that we already live in a non-Euclidean space. Like how the surface of the Earth is 2-D locally (and squares are squares) but exists in a 3-D space (and three right angles make a triangle), the universe is probably 3-D locally (where cubes are cubes) but 4-D globally (and cubes are not cubes).
The Pillar Room
How to apply this to a tabletop game? I like to introduce it with the Pillar Room.
Imagine you go into a normal room with a square pillar in the middle. You walk 360 degrees around the pillar, noting that it has four sides with 90 degree angles for the corners, and you are back at where you started. Sound good? That's a normal room.
But what if it took more than 360 degrees to get back to where you started? What if you had to go around it twice, and it took 720 degrees to get back to the door? Picture this: the party enters the pillar room from the only door (on the S wall). The rogue decides to walk around the pillar and look around, but when the rogue gets back to the S side of the room, the party is gone. The rogue can still hear the party asking him why he's hiding behind the pillar (the sound is bouncing off of both of the N walls) but he can’t see them. In fact, the door is gone too, even though he is on the S side of the room. Of course, he has only to walk 360 degrees in either way around the pillar in order to get back to them.
With non-Euclidean architecture, a 10’x10’ room can hold 200 sq. ft.
You might notice that this looks a lot like hyperspace, having many things occupy the same space. In fact, the room I just described could be duplicated by putting a discrete, two-way portal from the pillar to the middle of the north wall. This portal would lead to an identical room (that doesn’t have a door or any party members in it). By walking around the pillar, the rogue walked through the portal into the identical room and didn’t even notice it. But another 360 degrees around the pillar and he’ll be home.
But that’s still simple stuff.
What if it was 270 degrees to go around the pillar to get back to the starting point? The rogue would go ¾ of the way around the pillar before getting back to the party, even though the pillar has square corners. In fact, the rogue could stand in the NW corner of the room (after leaving the party on the S wall) and see the party in two places. And the party could see the rogue in two places. Note that they aren’t seeing copies, they’re actually seeing the rogue from two directions because space is curved and parallel lines meet here. This is an elliptic geometry, and the apparently square room has three corners. This 10’x10’ room has an area of 75 sq. ft.
If it was 180 degrees around the pillar, the pillar would be a two-sided square, and the rogue could do weird things like shoot himself in the back as he peers around a corner. Highly elliptical spaces get weird fast, and I'll cover them in the next sub.
What if space was highly hyperbolic? What if you had to walk around the pillar 10 times before you got back to where you started? A 10’x10’ room on your dungeon map suddenly has 1000 sq. ft. in it (and the square pillar has 40 sides).
What if you put two of these pillars in the same room and called it a maze? Depending on how the party twisted and turned around the two pillars, they could get very lost, and end up very far away from the door that they entered. A 10’x20’ with two pillars could be . . . hell, as big as you want it, with as many branches as you feel like mapping. If you put a monster in a smallish 2-pillar maze, the party will probably be less than 20’ away from the monster at any given time. It'll be roaring like a giant garbage disposal and the party will be screaming like cheerleaders, but neither the party nor the monster will know how to reach each other (since the noise is coming from all the different paths to the other party). Spooky, huh?
Fun Tip: When trying to map simple hyperdimensional mazes, just think of each center of the room as a single location. Then just figure out where each of the four directions takes you (each direction around each of the pillars) and which location it leads you to. Just because it confuses the hell out of your players doesn’t mean it has to confuse you.
Time to think big.
Don’t be afraid to extrapolate the pillar room to the whole dungeon. Maybe a spin around the pillar takes them to a very similar dungeon—the party might not realize that they’re in a different one for a while, nor will they realize that the pillar can take them back.
Or picture a main room between two pillars, as in the two-pillar maze. Depending on where you are in the maze, the central room can have different themes or purposes. With the price of real estate the way it is, you can fit a 20-room dungeon in a 50'x50' area.
The pillar doesn’t have to be a pillar, either. It can be a square hallway, where the party must travel around it three times to get back to where they started. (This hallway has 3 north halls, 3 east halls, 3 south halls, and 3 west halls.) It can be a hole that party jumps down into a pool of water. It can be an arch or a mousehole. It can be a building where the windows lead somewhere the front door doesn't. It can be a gazebo.
Lastly, you finally have some justification for making some truly nonsensical maps. If five (90 degree) left turns equal a right turn, you are allowed to put two rooms in the same space and confound logical attempts at map-making.
Interfacing Non-Euclidean Spaces with Euclidean Ones
You can’t. As soon as you start trying to put three-sided squares onto your battlemap, you’re going to run into problems. Technically, you should be mapping those sorts of spaces with weird tessellations and not graph paper.
But Non-Euclidean spaces can work well in confined starships and dungeons, where there are a limited number of ways into and out of a room. You can have a lot of fun mapping out a room with Non-Euclidean geometry. The trick is to remember is that they are Euclidean locally (squares still look like squares), but not on a bigger scale (a big enough squares doesn't have 4 sides anymore).
Start simple. Maybe one lap around the pillar room leads to a hallway that curves a different way than the hallway you you came from, and leads to a different area. Maybe clockwise turns lead you into older and older iterations of the ship, until after four turns, it dead-ends, and you are left in a decrepit corpse of a starship (and maybe the turns took you back in time, if you want to get stupid).
And if your party starts hacking at the walls between non-Euclidean space and Euclidean space. . . well, breaking the things that keep an impossible object in our universe can’t be a good thing. Options for the discriminating DM include (but are not limited to): Explosions (hyperbolic spaces), Implosions (elliptical spaces), Sucking Vortexes, Sentient Itches, Cthulhu, etc.