All inhabitants of flatland are planes (except woman, who are lines), and the more edges a given inhabitant has, the higher its societal rank (circles, i.e. priests, are the most respected inhabitants). In order to recognize the rank of fellow inhabitants in two-dimensional space (which is difficult in 2D since all planes are represented as lines to observers residing in the plane), flatlanders have learned to use 3 different ways: 1) by sound 2) by feeling and 3) by sight (although recognition by sight only works in foggy areas of flatland). The parts of the book I liked the most are when A. Square visits lineland, talks with the (single) inhabitant of pointland and welcomes an inhabitant of spaceland into his home. It nicely illustrates the difficulties of speculating about n+1 dimensional spaces.
This is also the part of the book that remotely relates to the problems we are interested in our research group: Since we try to find ways to infer goals from intentional artifacts such as search queries, we face a problem similar to the problem A. Square is facing when he tries to understand spaceland: We have to speculate, infer and reason about the intentional dimension (the goals of web users) without having direct access to it (only through intentional artifacts).
What I find remarkable about the book is the fact that it makes a series of critical remarks relating to the treatment of woman, class warfare, the role of art and culture, tolerance and education in the 19th century (the book was written in 1884) without explicitly refering to any real world events or persons. Although many of the societal deficiencies of the 19th century have vanished in the 21st century western world, many of the issues are still prevalent – such as intolerance and inequality.
On the surface, “Flatland: A romance of many dimensions” is an interesting tale of a fictive world who is likely to entertain the mathematically inclined, for readers who are willing to interpret the story more deeply, the book offers a range of deep societal criticism and exciting food for thought.
Update: Here’s a link to a video on youtube that nicely illustrates a few basic ideas of the book: “Dr. Quantum visits Flatland“.