In Flatland, how is the shape of a two-dimensional child based on the shape of the parents?
The women of Flatland are straight lines, and so confer no genetic shape on their children. The men are two-dimensional objects with varied numbers of sides, and their children almost always have one more side than their fathers. This holds true only for equilateral shapes; non-equilateral shapes will usually have children of their same shape and side number. The narrator explains:
It is a Law of Nature with us that a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on.
(Abbot, Flatland, geom.uiuc.edu)
The more sides a person has, the higher his class ranking. The lowest class are non-equilateral shapes, who labor as workmen and soldiers. It is possible, although unlikely, that the child of a non-equilateral person will become equilateral, and the narrator hints that this is due to great skill, achievement, or intelligence, which causes a person to equalize in the length of their sides. After a sufficient number of generations, a child will be born with so many sides of small length that he is indistinguishable from a circle, and these are considered of the highest possible class in Flatland society.