Geometry (Symbol)
Geometrical shapes and angles play an integral role in establishing the social order of each dimension particularly Flatland. In this two-dimensional universe, castes are determined by the polygonal traits that an individual possesses at birth. Only men are polygons as all women are lines which is a commentary to the gender power imbalance in the patriarchal society. The more sides a polygon has the higher the social class therefore isosceles triangles and below the squares and pentagons in the social strata. Furthermore, polygons with irregular sides are considered degenerates, and their traits are usually corrected or undergo euthanasia. Geometry symbolizes the inherent differences in individuals that govern their fate in terms of social opportunities, respect, and acceptance.
Dimensions (Symbol)
The Square is exposed to three more dimensions apart from his own namely Pointland, Lineland, and Spaceland. He observes the myopic outlook that the inhabitants of less sophisticated spaces harbor in interpreting their reality. The different dimensions symbolize the paradigms and perceptions that govern a particular society in maintaining the status quo. Pointland and Lineland represent civilizations that have a limited view and knowledge of their reality hence their social order is fragmentary. The Square expands the horizon of his conscious mind by exploring the three-dimensional space more advanced than Flatland.
Light and Shadows (Symbol)
Spaceland is distinguishable from all the other dimensions because of the presence of light and shadows. The inhabitants of the other dimensions have no spatial view of their reality and only dwell through set rules. The light in Spaceland symbolizes the higher truth and knowledge that is lacking from these limited dimensions. The Square embraces the light by accepting and exploring more prospects of much-advanced spaces and perceptions. Therefore, the presence of shadows in the presence of light also alludes to the idea of manipulating the truth.
Skepticism (Motif)
The protagonist observes that his newfound knowledge and the existence of other dimensions are hardly received with enthusiasm. The monarchs from lower dimensions are skeptical of the insights that the Square attempts to convey to them. They remain in ignorance by choice despite the possibilities of other worlds, a reaction the Square notices in all realms. Even suggesting the possibility of a fourth dimension to the Sphere also is received with cynicism. The skepticism can be attributed to their inability to expand the horizons of the mind or the need to maintain the status quo due to fear of a paradigm shift.
Flatland (Allegory)
The incorporation of other dimensions in the narrative besides Flatland is to establish that it is a progressive yet parochial society. Basically, Flatland is an allegory of Victorian society since the people viewed themselves as forward-thinking but their morality was biased. The women in the two-dimensional space are lines contrary to their male counterparts who are polygons. This is a commentary on the gender inequality and female oppression that defined the patriarchal Victorian society. Moreover, the emphasis on social strata through geometrical shapes comments on the division between the aristocrats, bourgeoisie, and the lower-class.
