A Square, the narrator and protagonist, opens the book with Part I by introducing his readers to his world of Flatland, which he likens to a sheet of paper on which straight lines, triangles, squares, pentagons, and other figures roam about.
The narrator immediately starts the book with an analogy to help the reader compare the paper-like landscape of his world to what will later look a lot like Victorian Britain. This very mathematical world, governed by geometrical concepts and theories, shows that reason and logical thinking will be an important theme throughout the work.
A Square says that Flatlanders, lacking the ability to distinguish each other by sight, only see each other as straight lines, much in the way one sees the side of a penny from the edge of a table. He includes three figures that illustrate how a triangle appears from above, close to the level of a table, and at the level.
Abbott’s inclusion of actual images suggests that analogy may be limited if only conducted through words. The fact that Flatlanders all see each other as lines will later prove ironic, since they distinguish each other by more absurd means.