Flatland Science: Dimensions

What and where is Flatland? A Square gives us several interesting answers, many of th contradictory. We know that it’s flat, big (but how big?), and very thin, the most important question of all is “how thin?” A lot depends on the answer…

A Square himself eliminates the version that’s easiest for three-dimensional readers to understand; a world that’s thin – maybe only a few atoms thick - but nevertheless has some physical height. It would have some sort of solid or semi-solid base, with everything described in Flatland on its surface. Maybe the Flatlanders themselves would be as thin as the surface layer of chalk on a blackboard, but they would have some physical thickness. They just wouldn’t know it!

This version of Flatland would have 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 can only work 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 threedimensional 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.

Unfortunately A Square explicitly tells us that things aren’t like that. There is no ground for Flatlanders to walk on, nothing for them to attach their foundations to, no third dimension2. Flatland is genuinely 100.000% two-dimensional. While there are reasons to think that he may be an unreliable narrator in some respects, he generally lies by omission, not outright.

[...]

Could Flatland be a true 2D universe in its own 2D space, a plane exactly one atom thick? Even the atoms and periodic table would be nothing like our own3 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 seems to imply that things are a little more complicated. There’s also the odd omnipresent ligh hat bathes Flatlan – it can easily b 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 the only source of gravity is the matter inside Flatland, the place can’t be stable; all of the matter in Flatland would slowly be attracted by gravity, forming coinworlds and coin-suns, moons, etc.