Back in his Flatland home, A Square decides to hide his experiences from his Wife, so he reassures her with a fake story. Once alone in his room, he broods over everything he has learned from the Sphere and recites the phrase “Upward, yet not Northward” repeatedly until he falls asleep. During his sleep, he has another dream of visiting another foreign world. The Sphere takes him to Pointland, the Abyss of No dimensions, which is inhabited by the lone Monarch of Pointland.
In simple terms, the axiom “Upward, yet not Northward” describes the essence of the third dimension. But a deeper look reveals that it also speaks of the exalting effects of knowledge—which literally lifts those who have been enlightened upwards into better worlds with higher dimensions. The “yet” is particularly hopeful, since it suggests that knowledge can be eventually achieved, as long as one is humble and curious enough like A Square.
The Monarch of Pointland believes himself to be the entirety of his universe. He is incapable of conceiving of anything other than himself, since he does not know what length, breadth, and height are. A Square is stunned by the complacency of the Point and tries to make him realize his insignificance in the world. But the Point takes A Square’s words to be his own, and is awed at his “own” thinking.
The Monarch of Pointland offers a humorous caricature of the ignorant. Completely engrossed in his own wisdom and intelligence, the Monarch is literally unable to think of anything beyond himself. It is truly a bitter image of what it is like to live without the ability to empathize with others and seek knowledge outside one’s self.
A Square and the Sphere return to Flatland, and the Sphere inspires A Square to teach others of higher dimensions. He apologizes for his previous bout of anger at A Square’s request for knowledge of higher mysteries, and then teaches A Square how to construct extra-solids and double extra-solids, according to analogy.
Throughout the book, analogy is used as a teaching technique. Here it is used explicitly by the Sphere once more, to help A Square seek higher knowledge (i.e. of extra dimensions), which is liberating.