A Square
The book is narrated by A Square. Flatland has a social organization that relies upon a caste system for class division and A Square belongs to upper caste that includes professionals. The book is structured as a memoir after being released from prison that relates the story of his crime. His crime? Attempting to teach others about the existence of the third dimension.
The Sphere
The Sphere is a stranger who visits Flatland at the dawn of each new millennium. He is from Spaceland and his return to Flatland serves the purpose of seeking out a new representative to become the apostle spreading the Gospel of the Three Dimensions. A Square proves to be a good choice because of his willingness to believe, but the relationship is complicated when he theorizes the possibility of the existence of dimensions beyond the third, an idea which the Sphere rejects with increasingly irritability.
Monarch of the World (Lineland)
Though he refers to himself as Monarch of the World, A Square thinks him a poor, ignorant fool because believes that straight lines constitute the entirety of existence. Thus, he is the Monarch of the World only to those in Lineland satisfied with their unique place in the universe. He also grows irritable with A Square’s theorizing of that which may exist beyond his reach, but in this case anger fuels violence.
King of Pointland
The Sphere describes Pointland to A Square as lower depths of all existence and the Abyss of No Dimensions. The purview of the King is even more constrained than the Monarch as he views Pointland as not just the only world in existence, but himself as the entirety of Pointland. Thus, he is not just ruler of his world, but the world itself that he rules, solipsistic little guy that he is.
The Circles
Occupying the highest level of the caste system in Flatland are the Circles who are the high priests of society. Thanks to what A Square describes as the quite judicious use of the divinely inspired aristocratic constitution’s singularly important Law of Compensation, the Circles have historically been successful in beating down all attempts at insurrections seeking to seed sedition among the lower representatives.
