Flatland is very much a book of it's time, but it's still fairly interesting to read today.

The book is roughly broken into two halves: first, Abbott discusses the society of flatland, much of which alludes to various facets of Victorian society regarding the treatment of women, the class structure, the nobility, the difficulty with moving up the social ladder, and so on. I didn't find this section all that interesting as a matter of social commentary, but it does set up some interesting challenges of a two-dimensional world that can really only perceive a single dimension.

The second half is about the narrator's experiences interacting with other dimensions; first through a vision of "Lineland" which is a one-dimensional world, then by interacting with a three-dimensional being from "Spaceland," then some brief interactions dealing with "Thoughtland" and "Pointland" which are four- and zero-dimensional worlds, respectively. Pretty much every character is stubborn to the point of being stupid, but the way higher-dimensional beings perceive and try to explain themselves to the lower-dimensional beings is actually an interesting thought experiment. Mathematically, everything seems very simple, but the concept of an extra direction orthogonal to our entire existence is interesting to contemplate. We can envision "up" from the perspective of a 2D plane easily, but "extra-up" from the perspective of our 3D space is something we can conceptually accept but not necessarily perceive or visualize in any way.

What might be the most interesting part of this book is its age. Flatland was published in 1884, before any conception of quantum mechanics or Hilbert spaces, but posits that beings should be able to exist in an infinite number of dimensions, each one able to perceive and move through the entirety of the lower dimensions trivially compared to those lower-dimensional beings. It's something that we can talk about mathematically today, but before any of the actual physics of this was establish Abbott was talking about some of the consequences and difficulties with the way we can communicate with other-dimensional beings. The first half of the book is somewhat stuck in the time it was written, but the second half is remarkably ahead of its time.