Polygon

Use in the book

Section 323 makes the multiplication of sides into a ladder of titles. Nobility begins with the Hexagon and rises towards a figure that can pass as circular. The nearer a body comes to an outline without conspicuous corners, the higher its station. Refinement becomes a literal reduction in the prominence of angles, though the society continues to count and rank those angles minutely.

This progression creates an epistemic irony. The more prestigious the figure, the harder it is to distinguish by touch from a slightly different neighbour. A Master of Arts may confuse ten and twelve sides; a Doctor of Science cannot confidently settle the difference between twenty and twenty-four. Rank multiplies distinctions beyond the powers of ordinary observation. Their social importance increases just as their physical visibility diminishes.

Sight recognition gives the wealthy a way of inhabiting this complexity. A gathering of moving Polygons demands time, money and long practice. The Square can explain its theory but cannot always comport himself comfortably in the crowd. His embarrassment shows that numerical knowledge alone is not social fluency. The ability to move correctly among one's betters has become a form of cultivated knowledge possessed most securely by those born among them.

The word many-sided ordinarily permits a metaphor of versatility or breadth. Flatland literalises the phrase, but does not guarantee that additional sides produce a generous understanding of other lives. Polygonal cultivation is closely attached to the protection of rank, and the education that perfects it can leave a failed pupil without a tolerable place in society. Complexity of figure is not equivalent to complexity of sympathy.

The class also occupies an unstable position between professional respectability and sacred prestige. A Square looks upward to it; the highest Polygons seek the courtesy of circularity. This relative standing helps explain the restlessness of the hierarchy. Each stage can regard itself as superior while remaining anxious about a boundary above it. The lexicon's geometrical definition provides an essential check on that anxiety: the impressive title never ceases to name a figure made of straight sides.

Places to read

1. §323“the honourable title of Polygonal, or many-sided”
2. §523“a ten-sided with a twelve-sided Polygon”
3. §523“the delicate touch of the highly organized Polygon”
4. §6234“a crowd of rotating Polygons of the higher classes”
5. §6234“to comport oneself with perfect propriety in Polygonal society”

In this edition

Note 37 relates the titles to aristocratic grades and the approach towards circularity; note 57 supplies the relevant angle values. Those editorial calculations explain why tactile discrimination becomes difficult. The political restriction of Polygonal to noble bodies is a feature of the narrative's usage rather than a rule of mathematics.

Polygonal

Many-sided, but specifically an honourable designation acquired by the upper ranks of Flatland’s nobility. The transition from a geometrical adjective to a title makes social prestige appear to follow naturally from increasing numbers of sides. Printed form: Polygonal (§ 323). See Polygon.

Perfect propriety in Polygonal society (contextual phrase)

“to comport oneself with perfect propriety in Polygonal society” (§ 6234)

The alliterative phrase joins respectable conduct to a particular bodily and social class. Comport oneself means behave or carry oneself; propriety requires knowing how to move among high-ranking figures without collisions or unwelcome requests to touch them. The Square concludes that one should ideally be a Polygon oneself. This is more than a joke about snobbery: prolonged schooling and habitual practice give the rich an informational advantage over a visitor who cannot interpret their changing outlines. The apparently effortless grace of superior breeding has expensive conditions. The narrator’s own difficulty is useful evidence, since he possesses education and respectability yet still feels outmatched. Membership in the fashionable circle depends on embodied training that formal knowledge alone cannot supply. See Recognition by sight, Polygon, and Education and schools.