Irony of ignorance
The inhabitants of each dimension live in blissful ignorance of the existence of a higher plane or even a lower one. The Square has the opportunity to explore the other three realms aside from his own, Flatland. The irony comes with the inability to convince the monarchs of the possibility of higher dimensions. Though they recognize the unique structure the square adopts in Lineland they cannot accept the prospect of another plane. Moreover, the Sphere exhibits the same ignorance when the Square suggests the concept of a fourth dimension.
Irony of misogyny
Flatland acts as an allegory for Victorian society with its patriarchal ideals that oppressed women. Akin to the Victorian era women in Flatland are viewed as second-class citizens and inferior because of their geometrical structure. The system had adopted special rules to govern the movement of women including the use of separate doors. Ironically the system is not based on superiority – as it seems on the surface – but rather on the fear of women. Initially, the single line structure that they possess used to impale and even kill the men therefore the laws were created to mitigate the risk.
Irony of anomaly
Flatland is governed by the rules of geometry in all spheres of society therefore citizens with irregular shapes are considered outcasts. As such they have to be euthanized or undergo corrective procedures to be regular polygons. Ironically, outside the dimension, every shape will be an anomaly in another plane as proven by the Square’s excursions. Accordingly, the basis behind discriminating the irregular polygons is not the natural order that runs the realms.
The irony of the status quo
The Square is a product of his environment like other inhabitants in all four dimensions. As the protagonist, he demonstrates the same limited perspective and narrow-mindedness until his outlook is challenged. With the newfound insight, he intends to disrupt the status quo by enlightening the inhabitants of his realm and the lower planes. Ironically some monarchs are adamant to maintain the status quo even though they believe in the prospect of other planes.
Irony of perspective
In explaining the existence of higher dimensions, the Square attempts to use words to describe the planes. The ideas are abstract knowledge that can barely be demonstrated in a realm without that perspective intellectually and dimensionally. Though they resort to physically illustrating the concepts without words it barely manages to persuade their audience. Therefore the predicament of their intended mission is ironic since the perspective in the lower planes is limited.
