Game Space

[SwirlGlome]

Hello.
I wish to create a 3D game with a certain spatial layout.
This isn't in the programming section because I don't have the skills to code it yet.

For now, I only wish to find the proper words to describe my space,
so anyone familiar with standard dimensional terminology will understand me easily.
So please proofread my description and edit anything that needs changing,
to make it more concise, precise, and obvious.

The space itself is relatively easy.
it is a torus(?) formed by extruding a half-sphere instance of the real projective plane into a ring.
Half-sphere because it's smooth unlike the disk version, and round unlike the wraparound square version.

One thing about this space that interests me is that it has positive curvature in the planar directions, and zero curvature "vertically", the direction that progresses around the ring. I'd like to know what such a hybrid curvature is called.

The hard part to describe is the 2D "land" that inhabits this space.
Ok, clockwise around the ring is "down", and counterclockwise is "up".
The land has 3 levels, distributed evenly around the ring.
Each level is shaped like the real projective plane, but because it's embedded in 3D,
it has a topside, the "floor", and a bottomside, the "ceiling" of the next level down.
Each floor is connected by a pillar to the ceiling above, and by a hole to the underside of the next level down.
The holes are the insides of the pillars.
The pillars curve smoothly up from the floor, and smoothly back to the ceiling.
So you can walk from floor to ceiling in a smooth motion.
"gravity" draws you toward whichever surface you're closest to,
in case you wonder why you can walk on the ceiling, lol.
What is the best word for a smoothly curving pillar of this sort?

Anyway, because the 3 levels of the land are an odd number,
you can eventually navigate from the topside of a level to the bottomside of that same level.
So not only is the land non-orientable in the sense that
you can walk away from your friend who is right handed and return to meet the same friend who is now left handed,
it is also non orientable in the sense that
the land has only one surface, the top side connects smoothly (tho circuitously) to the bottom side.
I believe these two types of non-orientability are often thought of as the same thing,
but without the pillars and holes, each level would have 2 distinct sides which are both non-orientable.

Is that all? I think so. Did I make sense to anyone? Is there a way to improve my description?

ZV

[SwirlGlome]

Ok, no responses yet. Um, that's unpleasant (my mom disapproves of swearing).

Well, another question I had was how many euclidean dimensions do I need to embed my game-space in so that it doesn't intersect itself? Am I correct in thinking the real projective plane (half-sphere version) can exist in 4-euclidean without distortion or self-intersection? If that's the case, then I only need two more to make a torus of it, so 6D total, right?

Well, luckily for my game, I won't technically need to embed it, it can just exist in and of itself. But the embedding is the only excuse I have for posting in the "higher dimensions" category, and I really don't see any other place for it to go.

ZV