Imagine a world with only two dimensions—length and width, but no height. The inhabitants are geometric shapes who slide around on a flat plane, unable to conceive of "up" or "down." In this world, your shape determines your social class, and the more sides you have, the higher your status.
Abbott wasn't celebrating this hierarchy—he was mocking it. The rigid class system of Flatland, where birth determines destiny and women are oppressed, mirrors the Victorian England of his time. The book asks: what limitations do we accept as natural that are actually arbitrary?
In a 2D world, you can't look "down" at shapes from above. You see everything edge-on, like looking at coins on a table at eye level. Every shape appears as a line segment. Flatlanders distinguish each other by subtle shading gradients and by "feeling" each other's angles.
A sphere from Spaceland (3D) visits A. Square to reveal the third dimension. When a sphere passes through Flatland, it appears first as a point, grows into a circle, then shrinks back to a point and vanishes. To a Flatlander, this looks like magic—a shape appearing from nowhere, changing size, then disappearing.
The sphere lifts A. Square out of his plane, allowing him to see Flatland from above for the first time. He sees inside closed rooms, inside other shapes—everything laid bare. It's a revelation that changes him forever.
If a 2D being cannot imagine 3D, and a 3D being showed them... what about us? Could there be a fourth spatial dimension we're equally blind to? The sphere refuses to consider this possibility, showing that even enlightened beings have limits to their imagination.
Flatland isn't just about geometry. It's about the limits of perception, the arrogance of certainty, and the possibility that reality is far stranger than we can conceive.