Before getting too specific about the minutiae of Flatland and the Flatlanders, there’s an important question that needs to be answered; where is Flatland, and what exactly is it?
Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows - only hard with luminous edges - and you will then have a pretty correct notion of my country and countrymen.
Flatland (Ch. 1)
Space, my Lord, is height and breadth indefinitely prolonged.
ibid (Ch. 16)
You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, or in, the top of which you and your countrymen move about, without rising above or falling below it.
ibid (Ch. 16)
WHAT and where is Flatland? Abbot gives us several interesting answers, many of them contradictory. We know that it’s flat, big (but how big?), and very very thin, the most important question of all is “how thin?” A lot depends on the answer…
The easiest version for three-dimensional readers to understand is a world that’s thin – maybe a few atoms thick – but nevertheless has some physical height. There’s some sort of solid or semi-solid base, with everything described in Flatland on its surface. Maybe the Flatlanders themselves are as thin as the surface layer of chalk on a blackboard, but they have some physical thickness. They just don’t know it!
This version of Flatland has advantages. Things stay put if they’re attached to a solid base, and if the base slopes downwards to the South that explains why rain always falls from the North. But there are problems – it only works if Flatland is physically part of a larger three-dimensional world, and it implies the possibility that one object can move on top of another. Many of the things Abbot describes, such as a three-dimensional being’s ability to see inside the body of a Flatlander, wouldn’t be true in such a world – as one of many problems, the contents of internal organs would leak out if they didn’t have membranes, or some equivalent, to hold them in.
I am in no jesting humour. I tell you that I come from Space, or, since you will not understand what Space means, from the Land of Three Dimensions whence I but lately looked down upon your Plane which you call Space forsooth. From that position of advantage I discerned all that you speak of as solid (by which you mean "enclosed on four sides"), your houses, your churches, your very chests and safes, yes even your insides and stomachs, all lying open and exposed to my view.
Flatland (ch. 16)
Another possibility is that Flatland is a true 2D universe in its own 2D space. There is no up or down (except as alternative names for North and South), and it’s a plane exactly one atom thick. Even the atoms and periodic table would be nothing like our own and there could be no ground – just a thin film of air, fluids, and solid matter, all more or less free to drift. But there are problems with this idea too. It almost works, but the intrusion from 3D space makes it seem a little unlikely. There’s also the odd omnipresent light that bathes Flatland – it can easily be explained as coming from an external source if Flatland is in a universe of three or more dimensions, but not if there is nothing outside the plane of the universe. A final important snag is that if there’s no external gravitational force Abbot’s version of Flatland can’t be stable; all gravity would come from objects in Flatland, and over billions of years they would be attracted to each other, slowly forming coinworlds and coin-suns, moons, etc.
This game assumes that Flatland is a two-dimensional section through a three-dimensional universe. It’s large if not infinite in extent, and gets its gravity, or something that behaves like it, from an external source. Gravity or an equivalent force is needed to explain why rain falls from North to South; it doesn’t explain why houses, Flatlanders, and everything else don’t fall with them.
Another problem is the variable attraction felt by Flatlanders, a pull to the South that is stronger the further South they go. If this pull is gravity then this behaviour seems very strange; one unlikely explanation is that Flatland gets its gravity from a source to the South that is so close that there is a noticeable gravitational gradient; gravity is actually stronger the further South you get.
An interesting model for this is a flat plane with a rod-like gravitational source, such as a superstring, at one end. Gravity obeys the inverse square law, so would be stronger to the South than to the North. Since the forces described seem relatively gentle we’re probably not dealing with a black hole or neutron star here; in human terms it might be equivalent to a small percentage of a gravity in the temperate zones, rising to (for example) 0.25 or 0.5g in the tropics.
Another possibility is that Flatland is actually cylindrical, or some other form of curve, with the “temperate zones” where gravity is negligible representing a region where the slope is relatively gentle. If there’s a uniform gravitational source underneath (such as a planet) the pull from the South Flatlanders experienced would be weak in the temperate zones, strong in the tropics, simply because the slope is steeper. The illustration below shows both of these models.
A semi-plausible explanation for the stability of the Flatland landscape is that certain solid objects – the mountains etc. of Flatland – are actually 3D objects protruding through the plane of the 2D universe. They’re solid, and remain in fixed positions, because they’re attached to the 3D universe. It sounds good – but it doesn’t explain why Flatland houses stay in place, and mining would be a little risky. Imagine a Flatland miner excavating a tunnel one atom thick through a 3D boulder – even as he was working the molecules above and below him would be vibrating and re-forming bonds to close off the tunnel. Soon any opening would be gone, and there would be no evidence that the mine had ever existed.
Scenario Idea: What if the whole of Flatland is drifting South, like the rain, only more slowly, heading towards some ultimate disaster? If all of the reference points were moving at the same speed there would be no way to know. At the moment things seem calm, but what if there are rapids or a waterfall ahead…? Is the impending catastrophe survivable? If not, is there any way to prevent or transcend it? Maybe that mad old polygon who believes in a third dimension has some ideas…
These are usable explanations, but there are many others. None of them really explain where the rain comes from, and where it goes, or all of the odd properties of Flatland. Finding out which theory is correct could involve exploration and some interesting adventures.
I think that some of the ideas in here, especially the curved space bit, could explain why I had my scenario idea...