Flatland is centered around an allegory: the shapes and structures of its geometrical world are mirrors for the rigid Victorian society in which Abbott lived. The representative work that shapes perform in Flatland extends into every part of the novel. The allegory that A Square's story contains allows the author to offer complex reflections on society, class structure, and gender roles without making overt statements. Just like in Victorian England, how a shape is seen by others has an enormous effect on how they are treated. Through its use of allegory, Flatland critiques this focus on perception and other aspects of the classism and sexism of Victorian society. However, it also asks the reader to think about the fact that England’s traditional values and class systems didn’t always embrace the rapidly changing world of the 19th century.
The society that A Square comes from is a highly structured one. All of the beings who live in it are geometric structures, and each “shape” represents a specific class. Circles—or priests—occupy the highest social niche. They represent the enormous power of the church and the aristocracy in the 1900s. Squares and other polygons make up the professional class. They are seen as steady and reliable, and usually given the respect accorded to professionals in the real world. The lower and working classes are the Triangles: the fewer sides a shape has and the sharper its angles, the lower its social status. This clear demarcation of societal roles is an analogy for the 19th century reality that one’s birth often predetermined one’s social status.
Moreover, female shapes in Flatland aren’t shapes at all but “lines,” and are considered the lowest in the social hierarchy. The wife of a Circle might be given more respect than that of a Triangle, but that is as far as women’s power is allowed to extend in Flatland. This representation is an allegorical criticism of the marginalization and underestimation women had to endure during the 1900s. Because lines have “sharp” ends, they are considered more dangerous and volatile than polygons. As the book was written in a time when the women’s rights movement was first gaining political power, this reflects the conservative Victorian fear of women's emancipation. Shapes are frightened of “Lines” because of the perceived threat they pose, just as the women’s suffrage movements of the Victorian period unnerved conservative politicians and lawmakers.
Flatland is a land of two dimensions: when A Square meets the Stranger, a three-dimensional being, he at first can’t understand what he is seeing. The novel uses the concept of dimensions as an allegory for understanding and discovery. A Square's journey from a two-dimensional to a three-dimensional understanding of the universe is the novel’s driving force. However, his ideas are not well-received by the other Shapes. He is unable to convince his fellow Flatlanders of the existence of a third dimension. This reflects the challenges that progressive, disruptive thinkers often face in a conservative society.
