After failing to enlighten his Grandson, A Square decides to write a treatise on the mysteries of Three Dimensions, which he believes will be more effective in teaching others. To avoid breaking the law, he writes for several months about a hypothetical “Thoughtland” that resembles the third dimension. However, he finds it difficult to draw helpful diagrams and to compose a clear enough treatise that would help others understanding his meaning.
A Square’s struggles in writing “Thoughtland” seem to be Abbott expressing his own troubles in composing Flatland. These struggles also closely resemble the troubles the Sphere had in explaining the third dimension to A Square, and A Square had with the Monarch of Lineland.
One day, 11 months after his return from Spaceland, A Square attempts to envision a cube, but fails at his first try. Although he eventually succeeds, he is not sure if he is actually correct, so he begins to feel sad. A Square is still confident and determined to devote his life to spreading the Gospel of the Third Dimension, however. Thus, many times A Square becomes so passionate about teaching others that he expresses dangerous thoughts about Spaceland.
A Square’s resolve can be seen as dangerous, particularly in a society as highly regulated Flatland. This could be a warning from Abbott about hastily attempting to overturn the existing social hierarchy without a strong plan.
After occasionally mentioning ideas of seeing the “interiors of things” and of the Third and Fourth Dimensions, A Square gives an entire account of his experiences in Spaceland at a meeting of his Local Speculative Society, in response to the Prefect’s assertion that God had limited the world to two dimensions. A Square is then arrested and taken to the Council for trial.
Because of A Square’s inability to control his passion for preaching the Gospel of Three Dimensions, he is, of course, immediately arrested by the Circles. This recalls stories of early Christians, and even Jesus himself, who were persecuted for preaching their beliefs.
The next morning, A Square offers his defense to the President of the Council, but he is sentenced to eternal imprisonment. A Square writes that it has been seven years since he was imprisoned. He occasionally sees his Brother (who was imprisoned earlier) and laments that his brother does not understand the concept of the third dimension, despite having witnessed the Sphere’s revelation.
A Square has been writing Flatland from prison. Despite the inspiring messages he has conveyed throughout the book, the stark reality of having been defeated and trapped by those in power is very bleak. However, this depressing ending may actually be more effective in inciting his readers to action in real life.
A Square expresses regret because he does not have any converts. Yet he writes this memoir-like story in the hopes that it may incite a group of rebels who seek higher dimensions. A Square ends the story on a defeated note, questioning whether the mysteries of the Third Dimension are simply the products of a wild imagination or dream.
This ending is unsettling, and has generated much speculation on the meaning of A Square’s self-doubt and grim conclusion. Although he claims that he writes this treatise in order to inspire his readers to fight against oppression, in the end, he wonders himself if all his experiences with the Sphere and Spaceland were simply a dream.