Here is a challenge and some answers to previous concerns.

It is possible to have a flat earth and mathematics can support it and not support it all the way. I love mathematics and it does show that you can completely map a 3d universe unto a 2D plane. The paradox is the fact that even if you observe the earth from a 2nd plane it will still have turth values for third dimensional effects. There is no reason to neglect 3D effects it just not as obvious in a 2D plane. A great story to read is Flatland it is a famous book made by a famous mathematician that has a 2d head explore his world and comes into contact with third deminsional beings and objects which he fully doesn't see. The moral of the story is three fold and could help answer some of you questions between 3D and 2D mapping

On this map, what would happan is the side you see is obtaining sunlight while the side you don't see is not. There is a rotation of the earth on a two-dimensional disc it is just the fact you can not see it. You could represent from each point on the map as it round earth's equivilant giving the effect of rotation.

also the 3d north and south poles do exist if you don't neglect 3D rotation on a 2D plane. remember if you do compress the earth you still have not remove rotation that is a given therefore the earth rotates as it had in 3D. It just now you would not see the south pole that does not mean you could not travel there!

All the round earth theory is a point of reference, but even in a 1D universe where none of the laws by commen sense could be there. The laws are still there from the 3D universe. all you see the part of the dot you are in, and the dot would observe the same rotation as the 2d space. The thing that math brings to the table is we can map are universe in nth dimensions. I am awaiting the Hyperearth theory or 4D earth