A Square
Before his revelation, A Square is a relatively conservative, orthodox member of Flatland society who harbors many prejudices about natural order and the hierarchy of society. He believes that women have no intellectual capacity, for example, and he approves of killing irregular figures to stop them from breeding. As A Square journeys to Lineland, Spaceland, and Pointland, he comes to see that people often cannot conceive of worlds that challenge their own experiences or beliefs. Nevertheless, A Square treats his expanded view of the nature of the world with a religious reverence and works to spread his new understanding despite the threat of imprisonment.
Sphere
Sphere comes to Flatland to spread the Gospel of Three Dimensions. He selects A Square as his student and sends him dreams of Lineland and Pointland to help him understand how hard it is to explain things to a being determined to continue believing in its own view of the universe. Sphere reacts to A Square's inability to understand him with frustration and becomes genuinely angry when A Square asserts that there must be a fourth dimension. Though Sphere considers himself a higher order of being, his beliefs are still limited by his own experience of the world.