In Flatland, The Sphere's refusal to consider the idea of a fourth dimension when A Square tentatively asks about it is a typical example of situational irony. After A Square raises the issue, The Sphere thinks for a moment and then dismisses him:
Sphere (after a pause). It is reported so. But men are divided in opinion as to the facts. And even granting the facts, they explain them in different ways. And in any case, however great may be the number of different explanations, no one has adopted or suggested the theory of a Fourth Dimension. Therefore, pray have done with this trifling, and let us return to business.
The irony in this situation stems from The Sphere’s own role as a teacher. He has spent the most part of their interactions trying to explain the way the universe actually contains more than two dimensions to A Square. Although A Square was initially skeptical, the many examples The Sphere shows him have convinced him otherwise. However, despite his own position as a more cultured and enlightened figure attempting to broaden A Square's understanding, The Sphere dismisses the possibility of a dimension beyond his own comprehension as absurd. He believes in Flatland, Lineland, and Pointland because he has been there and seen them, but he cannot imagine anything else. The Sphere, who represents a higher level of understanding compared to A Square, falls prey to the same kind of closed-mindedness he is trying to overcome in A Square.
This interaction points to the novel's commentary on the nature of closed-mindedness and people’s tendencies to stick to inherited or accepted ideas. Through it, Abbott shows how even people (or Shapes) who are more intellectually advanced are not immune to the limitations of their own perspectives.
As they observe him, The Sphere's explanation to A Square about the Monarch of Pointland is a strong example of the novel’s use of situational irony. It’s a key moment of Abbot’s narrative on willful ignorance and the limitations of perspective, as The Sphere remarks:
“Behold yon miserable creature. That Point is a Being like ourselves, but [...] he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn this lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy. Now listen.”
The situational irony of this passage lies in The Sphere explaining to A Square that the Monarch of Pointland is willfully ignorant, blissfully unaware of dimensions beyond its own existence. The Monarch of Pointland is the ruler and sole inhabitant of a zero-dimensional version of reality. Because it only knows itself, it dismisses everything else. It’s entirely self-contented in its ignorance. This creature is an extreme version of the limited views A Square sees in Flatland, and Abbott saw in Victorian England.
However, that isn’t the passage’s only situational irony. A Square, while listening to this explanation, exhibits a smug dismissive attitude towards the Monarch's limited perspective. He thinks that he is immensely superior to the Monarch of Pointland, not realizing that his own understanding is similarly constrained compared to The Sphere's.
