ad1: why is cylindrical universe considered flat ? because just 1 dimension is curved while others are not?
To be honest, I'm not sure why. But if you draw a triangle on a cylinder, the angles add up to 180 degrees. This isn't true on a sphere.
ad2: how do you get from 2 flat facing mirrors to cylinder ? is cylinder inwardly mirriring ?
My analogy wasn't completely accurate. In the mirror corridor, light may bounce several times off the mirrors before it reaches your eyes. That makes it look like there are infinitely many copies of any object. Similarly, on the cylinder, light can loop around several times before reaching your eyes, while still travelling in a straight line. Again, you can see infinitely many copies of an object. If you don't see what I mean, try this experiment. Roll up some paper around a cylinder, and draw two points. Now see how many straight lines you can draw between the two points.
ad1: well, topologically, you get a klein bottle (universe?) by taking your cylindrical universe and joining its 2 circular-edges with a twist. a torus with a twist. or you deform the circle into lemniscate by twisting it around any of its diameter axis, through the centre of the circle by pi, and then join the two 8-like edges with a half twist.
That's a bit too complicated mathematically. Remember, we're trying to look at gravity in this universe.
i mean, what happens to the total amount of light absorbed ? is it still finite ?
Well, physically speaking, the amount of light absorbed can't possibly be more than the amount emitted in the first place. But when dealing with infinite series, it's a good idea to be careful.
ad2: your 2 facing mirrors can be interpreted as the complex numbers. ( and, after t smith & onar aam, Reals as 1 mirror, Quaternions as 3 mirrors, and Octonions as an inwardly mirroring tetrahedron. (whatsmore, sedenions are a funky mirrorhouse !) does it help anyhow ?
The mirrors are only an analogy to explain why there are infinitely many copies of an object. Complex numbers don't really apply here. Vectors are much more useful.
Mathematica solved the equation (numerically of course), and it turned out exactly the way I hoped
. At large distances, planets move in nD orbits. At smaller distances, they move in an (n+1)D orbit. I'm surprised at how easy this is.
I might try Schrodinger's equation next.