Diagrams and the position of the reader

Use in the book

The first illustration juxtaposes the triangular Tradesman seen from above with progressively lowered views. It does not depict a Triangle changing into a Line. It trains the reader to separate an object's shape from the appearance available at a particular position. That distinction supports nearly every later demonstration.

The house plan in §22 makes a second kind of knowledge visible. A reader looking down can inspect the separate entrances for Men and Women and the north-facing roof at once. An ordinary inhabitant cannot occupy this survey position. The picture is therefore both a convenient explanation and an early grant of the access the Sphere later exercises. The reader possesses a view before confronting the moral question of what such access entitles anyone to do.

The recognition figures in §6234, developing §523’s account, show how rank is inferred from ends, angles, proximity and fading brightness. They combine geometry with trained social reading. Later figures help distinguish a whole Sphere from the changing circular sections intersecting a plane. The pedagogical lesson is not that images replace argument: without a rule connecting the image to the observer's situation, the diagram can be misread as evidence that the object itself grows and shrinks.

In §2223 the material basis of diagramming returns as a barrier. The Square cannot provide his Flatland readers with the diagrams he needs, because their tablets and figures are all seen along one line. This gives the title page's promise of illustrations a productive fictional strain. The book in our hands successfully offers a kind of display unavailable to the audience within the story. The memoir does not explain every step by which its manuscript and images reach us; that unexplained passage is part of the romance's address across dimensions.

The figures accordingly belong to the argument, not merely to decoration. Their usefulness depends on the very asymmetry that the dedication asks readers to recognize. Seeing a plan is an advantage; recognizing the limited conditions of that advantage is the harder achievement.

Places to read

1. § 12“Fig. 1 represents the Tradesman” assigns an observer's position to a diagram
2. § 22“in the annexed figure” makes household arrangements simultaneously available
3. § 6234“Recognition by Sight” depends on interpreting appearance rather than seeing a whole outline
4. § 212“mathematical tablets” are glossed by the narrator as Lines
5. § 2223“no diagrams but Lines” names the communicative obstacle

In this edition

The supplied edition states that it includes ten modern redrawings of Abbott's original figures. This lexicon discusses their functions and supplies section locators rather than reproducing a second set that could blur source and commentary. The initial perspective figure, house plan, recognition diagrams and dimensional demonstrations can be consulted in the supplied PDF or EPUB.