Flatland is a two-dimensional world inhabited by geometric shapes. From a jail cell, the narrator, A Square, tells about the social organization in his country and being visited by the Sphere. In the second portion of the book, A Square recalls a dream that takes place in Lineland, a one-dimension world. After the dream, A Square talks about his wife and teaching his grandson about geometry. Then he speaks of his adventure to Spaceland with the Sphere, and after seeing what it was like, wanted to go tell the people of Flatland about it.
The Sphere decided to tell the community himself, and his almost-arrest conveyed the corruption present in the society because of the need for power by the circles. After being sent back to Flatland, A Square has another dream. The dream takes place in Pointland, the Abyss of No Dimensions. There he meets the Monarch of Pointland. After returning to Flatland, the Sphere encourages him to share his higher knowledge with the people of Flatland. After waking from his dream, he decides to teach his grandson about the theory of the third dimension.
But the Grand Council proclaimed that anyone who talks about the existence of other worlds will be punished. Because of this, his grandson becomes frightened and refuses to acknowledge the information his grandfather gave him. Discouraged by the failure to persuade his grandson, A Square begins to write a treatise about the Third Dimensions. Eventually, A Square is arrested for sharing his experiences in Spaceland and the idea of a third dimension at a town hall meeting. He then tells about being in jail for seven years as he writes Flatland. The society in Flatland is very strict and based on the number of sides a shape has.
Women are a single, straight line and aren’t even a part of the social pyramid present. The isosceles triangles are the lowest level polygon. They make up the working class and the soldiers. No matter how intelligent the triangles are, they are forced to in a factory or the military.
The equilateral triangle is only slightly higher than the isosceles because of the higher angle measure. The equilateral triangles are the middle class and serve as tradesmen and merchants. The next level consists of squares. Squares serve the purpose of lawyers and are the professional gentlemen of the social classes. The narrator of Flatland, A Romance of Many Dimensions, is a square.
The next part of the classes are the nobility. Nobility starts with six-sided shapes. As the number of sides increases from here, the nobility increases as well. The highest level possible are the circles. They are the priests and advisors of Flatland.