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.
In Flatland A Square explains the "art of feeling" as the primary way that lower-class and Female shapes recognize each other. Through this, Abbott satirizes the snobbery of the educational systems of his time. A Square also incorporates tactile imagery to convey his disdain for this method:
Feeling is, among our Women and lower classes—about our upper classes I shall speak presently—the principal test of recognition, at all events between strangers, and when the question is, not as to the individual, but as to the class [...] though we cannot see angles, we can infer them, and this with great precision. Our sense of touch, stimulated by necessity, and developed by long training, enables us to distinguish angles far more accurately than your sense of sight, when unaided by a rule or measure of angles.
The tactile imagery A Square uses to describe the "art of feeling" explains how touch is used as one way to distinguish the "real" shapes of other polygons in Flatland. Although A Square is dismissive of “Feeling” as primitive and uneducated, he also makes it clear that most shapes can instinctively do it very well. Interestingly, this is also a moment where the author deliberately alienates the reader from the narrator. It’s difficult enough to imagine “seeing” things in two dimensions, but here A Square tells the reader that their three-dimensional senses wouldn’t be of much use either. Although it seems like he is explaining an unfamiliar process so that they can understand, he is actually telling his audience in “the World of Space” that they couldn’t possibly understand it.
By stating that women and the lower classes primarily rely on feeling for recognition, A Square is implying that women and the “lower classes” are unable to think or “infer” in the same way as other shapes. They can only “Feel.” The narrator later goes on to explain the rarefied processes more socially elevated Shapes are taught to use to “see” each other. “Feeling” is regarded as being beneath them once they have learned these. This division of sensory perception along class and gender lines satirizes the accepted behaviors of the time, where intellectual capabilities and social status were often unjustly correlated by the ruling classes. The wealthy were often considered to be smarter and more educated regardless of the fact that the poor had far less access to education. It is also made clear that A Square thinks that “Females” and the “lower classes” are in fact universally only capable of “Feeling,” regardless of their individual characters or strengths. Thinking is arbitrarily reserved for Shapes with more sides, and "feeling" is for "Females" and Triangles.
When he is nostalgically reflecting on the times before color was made illegal in Flatland, A Square describes a colorful battle scene using rich visual imagery:
The sight of a line of battle of twenty thousand Isosceles suddenly facing about, and exchanging the sombre black of their bases for the orange and purple of the two sides including their acute angle; the militia of the Equilateral Triangles tricoloured in red, white, and blue; the mauve, ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidly rotating near their vermilion guns; the dashing and flashing of the five-coloured and six-coloured Pentagons and Hexagons
The use of visual imagery in this passage creates a bright, dynamic tableau for Abbott's reader. The visual imagery of Flatland is often quite difficult to follow, as everything is colorless and much of the story is imagined in two dimensions. This scene, although it's still "Flat," has a lot more visual cues the reader can build upon. Its diction is also even more startling when it's juxtaposed with the dull, grayish language of the "fog" of colorlessness in Flatland. Compared to the rest of the novel, this passage is a riot of color. The description of the battle preparation is intense, with colors and movements and sounds piling on one another in a cacophony. Words like "dashing" and "flashing" add a sense of motion and nervous energy to the scene, making the colors appear even more vibrant and lively.
