Summary: A square from a world of two dimensions explains the customs of his land and his life-changing encounter with a sphere.
Review: My understanding is that Abbott meant this book to function as a satire of Victorian society—hence, the ridiculously sexist and classist nature of Flatland. However, though I found Flatland intriguing, I did not continue to read for the social commentary, but for the mathematical fun. From the descriptions of how shapes recognize one another in a two-dimensional world (everyone appears as a line segment) to the depiction of Pointland, Flatland brims with mathematical humor and wit.
The first half of the book deals with Flatland and its inhabitants, customs, and history. Though this proves vaguely interesting, the narrator skips over the types of questions I most wanted to learn about—how the shapes move without feet and build houses or write letters without hands. The repression of women and the follies of the aristocracy provide some scandalous material, but I had trouble buying it all without knowing how these shapes do anything without limbs. Some pertinent background information would have greatly helped my attention. As it was, I had to make a conscious effort to suspend my disbelief and I was never certain the history of the color rebellion was worth it.
The second half of the book really makes reading worthwhile. The descent of a sphere into Flatland introduces our narrator square to the concept of three-dimensions, though, of course, he initially finds this as difficult to grasp as we do the fourth-dimension. A hilarious give-and-take between the sphere and the square ensues. Once the square experiences three-dimensionality for himself, however, he can make the mental leap to fourth-, fifth-, sixth-, infinite dimensions. If this seems crazy, readers only have to think back to the square’s former ignorance and suddenly the world seems full of possibility.
Lovers of mathematics should not pass this book by; the slog through the history of Flatland is well worth it to arrive at an exploration of some great mathematical concepts. Abbott then turns mathematics into a fascinating, but troubling, commentary on society and its resistance to new ideas. An insightful, eye-opening, yet humorous book.