In the novel’s first chapter, A Square introduces the reader to the difficult concept of his two-dimensional world using a simile and visual imagery:
Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows—only hard and with luminous edges—and you will then have a pretty correct notion of my country and countrymen.
The simile Abbott uses here compares the inhabitants of Flatland to "shadows" moving around and through flat surfaces. Although it’s almost impossible to imagine a two-dimensional universe, it’s helpful to imagine the Shapes as being like shadows. It gives the reader a way to begin to imagine their flat, two-dimensional bodies that, like shadows, are both present and totally lacking visual mass and depth. It’s a concept with familiar elements, which A Square then expands on, saying the shapes are very much like shadows, only “hard and with luminous edges.” The simile is a simplistic way to help readers imagine the more complex elements of a world and beings that aren’t three-dimensional.
The visual imagery of a "vast sheet of paper" populated by geometric figures moving about also serves to help readers visualize A Square’s homeland. In the real world, we primarily interact with two-dimensional objects on screens and as representations drawn with lines on paper. This familiar imagery helps create a mental picture of a two-dimensional plane, even for a reader who might not understand the mathematical particularities of the term. However, the author acknowledges the limitations of this imagery in fully conveying the experience of living in Flatland. Although they can get an approximate sense of it, the idea of moving through a two-dimensional world remains difficult to fully imagine for readers accustomed to three-dimensional space. Trying to understand this difference also gives the reader some additional context for the confusion and discomfort A Square feels when he himself is introduced to one-dimensional and three-dimensional space by The Sphere.
A Square explains how willing he is to sacrifice his own safety and happiness to disseminate knowledge of the third dimension to the public. He uses an allusion and a simile to illustrate this for the reader:
[...] yet, like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity.
The allusion A Square makes here is to Prometheus, a figure in Greek mythology who stole fire from the gods and gave it to humanity. Prometheus was punished for eternity for this thievery for the greater good: in legend, he was chained to a rock and forced to endure as an eagle arrived to eat his liver every day. It would then grow back overnight, the cycle repeating endlessly. His story is often used as an analogy to describe someone who sacrifices or endangers themselves for the sake of human advancement. By likening himself to Prometheus, A Square aligns his mission to this legacy of necessary sacrifice. Even though he knows it is dangerous to disseminate the knowledge of other dimensions, he feels he must bring the truth about Spaceland to Flatland.
Toward the end of the novel, A Square describes his intense sense of urgency in recording his knowledge about Spaceland before he forgets or it starts to seem like a fantasy. As he does this, he uses a simile to convey the slippery character of his memories:
I immediately sent for my Grandson; for, to confess the truth, I felt that all that I had seen and heard was in some strange way slipping away from me, like the image of a half-grasped, tantalizing dream [...]
The simile the narrator uses here draws a comparison between A Square’s fading recollection of the wonders of Spaceland to "the image of a half-grasped, tantalizing dream." This comparison captures the feeling of something real but difficult to believe quickly becoming impossible to remember properly. Although A Square knows that what he saw with The Sphere was true, it seems so fantastical that he is worried he will lose track of it, much like the lingering fragments of a dream that fade upon waking. Though A Square has experienced Spaceland, Lineland, and Pointland to varying degrees, their memories quickly begin to feel dreamlike. He is concerned that his previously dogmatic views will begin to cloud his memory.
This moment points to the novel's exploration of the relationship between perception and reality. A Square's struggle to hold onto the knowledge of Spaceland underscores the idea that reality is often more convincing and easier to grasp than concepts outside one’s normal frame of reference. It would be easier for A Square to convince himself that his time with The Sphere was a dream than to hold onto it as reality.
A Square uses the Egyptian myth of the Sphinx to articulate his frustration at the difficulty of making the other Shapes believe his stories about Spaceland, employing both allusion and simile:
Heavily weighs on me at times the burdensome reflection that I cannot honestly say I am confident as to the exact shape of the once-seen, oft-regretted Cube; and in my nightly visions the mysterious precept, “Upward, not Northward,” haunts me like a soul-devouring Sphinx.
The Sphinx was a mythical creature from Ancient Egypt. It was known for posing impossible riddlers to the unlucky travelers who crossed its path, and devouring anyone who could not solve them. The simile A Square uses here refers to the difficulty of this riddle and the dangers the Sphinx posed to its victims.
The “mysterious precept” of moving in three dimensions haunts A Square. He knows he must find a way to tell people about his experiences with The Sphere, and to explain what the Cube looks like to the other polygons, but he fears the “soul-devouring” consequences of both doing so and failing to do so. The Sphinx would let people who could solve its riddle leave in peace, but the difficulty of answering its question accurately made it an almost impassable obstacle. Like the Sphinx, the obstacle of how to get his point across to the other denizens of Flatland blocks A Square at every turn, “haunting” him.
