In the second half of the novel, the Square describes a dream in which he visits Lineland and has an extended conversation with its king. He then describes a visit from a mysterious Sphere who has descended from Spaceland with the goal of enlightening the Square on the true nature of the physical world. He ends the novel by discussing the consequences of his visit to Spaceland.

On the second-to-last day of the 1999th year of “our era,” The Square stays up late studying geometry and has a dream in which he visits a place called Lineland (43). This is a place consisting of small straight lines and points all moving back and forth and chirping chaotically on a longer straight line. He tries to communicate with them, but they ignore him. Finally, one of the small lines introduces himself as “the Monarch of the world” (43). The Square has a hard time learning anything about Lineland, as the king assumes they accept all the same facts to be true. However, the Square finally determines that the king believes Lineland to constitute all of space: he has no conception of anything that exists outside of his straight line.