I have an advanced facility with many floors containing experimental technology. One of these floors goes on forever. What shape is it?
More specifically: within the room, you can move as far as you like in any given direction. It is possible to travel an infinite distance in a straight line, in the sense that you'll need to travel that same distance in order to get back to your starting point; travelling costs time and energy; there is (eventually) a speed of light delay in communications, etc etc. There are no observable edges to the room, so upon entering you appear to be have emerged the centre of a vast featureless space. Objects do not visibly appear distorted when moving through the space, but everything in the room is small enough to have been brought in through the door (let's say vehicle-garage-size).
This room is however not particularly useful, because despite its apparent size, you can't put very much in it – the storage capacity of the room is still limited by the containing building:
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the volume within the room is no greater than the volume of any other floor of the building with an equivalent external footprint. Trying to put more objects into the room than would fit into a conventional space will result in Bad Things™ happening (certainly not anything a protagonist would want to experience first-hand).
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the mass that can be contained within the room is limited by the supporting structure of the building, the same as on any other floor. Overload it and the floor will collapse. This is likely to result in problems connected to the first point, too, unless whatever process is maintaining the spatial distortion is turned off quickly by the destruction.
Trying to flood the level with water or cement would certainly be a very bad idea.
The cause of this effect is obviously waaay beyond the scope of a SE question, and anyway the protagonist isn't interested. However, for us creators with weaker math-fu, what would be a rigorous mathematical description of the effect on space that's being observed within the room?
I'm leaning towards something like a four-dimensional version of a shape along the lines of a Menger sponge, which demonstrates a similar effect to the room, in that it has infinite surface area but zero volume (or perhaps its inverse, which – I assume – also has infinite surface area, but unit volume). Moving on the surface of a Menger sponge would be slightly similar to a 2D version of the above, in that you could move infinitely on certain paths, but remain within an enclosed space.
Is there a fractal shape that fits the results described above and can be used to describe the space within the room? i.e. infinite lengths possible between points in the space it encloses, finite 3D volume for the whole shape.
The space doesn't have to appear empty, if unpacking a 4D shape into 3D would necessarily result in singularities or mathematically correct terms creating areas that can't be moved through cleanly / spaces that don't line up (it would actually be a bonus to have "free" floor and support struts throughout, that aren't real objects with mass), similarly to how you can't travel in an endless straight line in any direction on the infinite surface of the Menger sponge (I think).
(solutions that allow an infinite volume to be contained within the space are acceptable, if this is an otherwise impossible request – I can always use the weight limit to prevent the space from being useful, but limiting volume seems more interesting for the story)
Best Answer
One possibility is a flat $3$-torus. This is the shape you get by taking a cube (or more generally, a parallelopiped) and declaring that the opposite faces of the cube are actually equal to each other. This means that when you walk far enough in one direction, you "wrap around" back to the other side. In particular, this means you can keep walking infinitely long in any direction, and room has no "edges". The total volume of the torus is the same as the volume of the cube you started with, so it is finite. And locally, the geometry is the same as the cube, so it is just ordinary Euclidean geometry, so it looks just like normal space as long as you don't look far enough to see things wrapping around.
However, this might not be very satisfying for the property of being able to "walk infinitely far", because while you can walk for an infinitely long time, after you walk a little bit, you'll find yourself right back where you started (or, technically, arbitrarily close to where you started, unless the direction you're walking is a rational linear combination of the edges of the cube). And there is a uniform bound on the length of the shortest path between any two points.
Alternatively, if you're willing to drop the assumption that you can move infinitely far in every direction and just want there to exist directions you can go infinitely far in, it is easy to get such a shape without having any sort of weird topology. Just take any nice ordinary subset of $\mathbb{R}^3$ like a ball, and attach some "tendrils" to it going out to infinity. If you make the tendrils get thinner and thinner fast enough as they go out, the volume will still be finite. I'm not sure this really captures your intuition though, as the directions in which you can go out to infinity are very restricted, and if you try to actually travel along them for very long, they will quickly become too thin for you to fit down them (assuming you are an actual physical object rather than a point with no volume). The latter problem seems unavoidable though: if you can move an object with positive volume down an infinitely long path that doesn't loop back around to itself (as happens in the torus), then by tracing its path, you can see that your space must have infinite volume.
By the way, I'm assuming here that when you say there are "no edges", you're including that there is no floor or ceiling to the room (so up and down are among the directions you can move in). If you want to have a floor and ceiling, you should probably just take the two-dimensional versions of these shapes (a flat $2$-torus, or a planar shape with tendrils), and let the vertical directions have ordinary geometry (in more mathematical terms, this means you take a product of the 2-dimensional shape with a closed interval, with the interval being the vertical direction).