The two return to space and the Sphere introduces A Square to the concept of solids. He stacks many square cards on top of each other to demonstrate how two-dimensional figures can be built into a three-dimensional solid (and an illustration is attached). A Square says that the demonstration is painful to see, since it looks like an Irregular figure to him. The Sphere explains that A Square is not used to seeing light and shade and perspective, so he introduces A Square to the cube, a living being. After detailed explanation and tactile demonstrations, the concept is clear to A Square.
The Sphere again teaches A Square using analogies, and this scene itself is an allegory of the process of teaching and learning. The new knowledge initially is painful to A Square, since it is so unfamiliar to him. Yet this knowledge is crucial for him to learn in order to enlighten other Flatlanders who live under the Circles’ oppression. This is where light and shade, a symbol for knowledge in general, appears.
A Square states that this is climax of the story. He then proceeds to recount his fall. Although it is painful for him to recall, he hopes it will arouse a spirit of rebellion in his readers, who may be stuck in their own dimensions. While the Sphere is teaching him the conformations of other regular solids, A Square interrupts him, asking if he could expose his intestines. A Square argues that since the Sphere had revealed the insides of all things in Flatland, there must be a similar way of seeing into the insides of solids.
One main purpose of Flatland is clearly stated here: A Square hopes to enlighten his readers on the knowledge of higher dimensions in order to incite them to fight against oppression, such as the Circles’ dominance over Flatland or the aristocracy’s rule of Victorian England. Note how A Square thinks in analogy, and now starts to grow curious about ideas even beyond the Sphere’s teachings.
A Square posits the existence of a land of Four Dimensions. However, the Sphere denies its existence, and argues that it is simply impossible to think about such a world. In response, A Square further probes his teacher by arguing that just as the Sphere had proved to him the existence of a higher world, and as A Square had attempted to teach the Monarch of Lineland of two dimensions, there must other even higher worlds. He theorizes a structure with 16 terminal points and 8 bounding cubes, following the Sphere’s previous logic.
What makes A Square different from the Monarch of Lineland, who refused to think beyond his linear world, is that he is humble enough to accept what is proven to him, and then has the curiosity to actually seek out more forms of higher knowledge. However, when he asks the Sphere about this knowledge, his teacher acts no differently from the Monarch, and assumes that his three-dimensional world is all there is to the universe.
A Square asks the Sphere to confirm or deny his hypothesis. The Sphere admits that some of his countrymen have considered a fourth dimension, but have not adopted an official theory. Therefore, he ends the discussion. But A Square continues to theorize higher worlds, even those of five, six, seven, and eight dimensions. Angered at A Square’s unending questioning, the Sphere pushes A Square back to his Flatland home.
A Square shows the liberating and elevating effects of knowledge. After learning about the third dimension, A Square is eager to seek for higher knowledge of even more mysterious ideas and worlds. In fact, knowledge is also salvation, since it promises a better future in higher worlds. Despite being a figure of wisdom, even the Sphere reacts emotionally to A Square’s eagerness. Notably, Abbott was theorizing about multiple dimensions (including time) long before Einstein published his theory of relativity.