Flatland: The Dangers of Living One-Dimensionally

Flatland is a 2-in-1 mathematical fantasy and social satire. Keep reading to see how it illuminates the importance of perspective, event in contemporary times!

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Left: Flatland book cover. Right: Floating 2D shapes with faces.

Flatland was published in 1884 under the pseudonym “A. Square” and is narrated by a two-dimensional square of the same name. Written by Edwin A. Abbott, an English schoolmaster, the novella invites the reader into the inner workings of Flatland, a two-dimensional plane home to all sorts of shapes. While serving as a social satire on Victorian society, it also discusses interesting mathematical ideas about dimensions.

Flatland culture functions under a strict social hierarchy dictated by sides and angles. Those with the most sides and highest regularity are revered in society, while those with the least experience little personal rights and freedoms. One day A. Square is visited by a sphere, who unveils the existence of the third dimension and completely changes A. Square’s perception of reality.

Although this mathematical fantasy was first written shy of two centuries ago, I believe that its message is timeless. While I do realize that not every American aligns fully with the two party system, I do believe that our inability to curb divisiveness and enact genuine change is to do with our close-mindedness as a society. In this article, I invite you to take a look at the story of A. Square, and think about how his realizations might change the way you view our world now.

Introduction to Flatland

Flatland, above all, is about perspective. Furthermore, it is about the limitations of perception as it pertains to an individual perspective. The two dimensional people of Flatland cannot conceive of the existence of the third dimension, although it does exist, because their perceptive abilities in the second dimension are physically incapable of processing it. The inhabitants of this land are similarly unable to imagine a world where all shapes, or citizens, can be equal, because they’ve existed under such a strictly stratified social hierarchy for so long.

Flatland by Edwin A. Abbott, cream parchment background with geometrical drawing book cover.

In either case, we, as readers and inhabitants of a three-dimensional, culturally diverse world, can see how silly it is to deny the existence of the things that Flatlanders claim are impossible. More than that, we can see that, in their case, tradition does not dictate what objective reality is — something that the Victorians failed to realize and something we struggle to understand about ourselves even now. All of this is to say that just because we cannot comprehend something within the limits of our own perception does not mean that it does not exist.

Lower Dimensional Beings

In order to prepare A. Square to see and comprehend the third dimension, the sphere first shows him a vision of Lineland, a one dimensional land existing solely along an infinite line and inhabited by a succession of lines running from left to right. Due to the limit of their perception, the King of Lineland vehemently rejects A. Square and his ideas.

Diagram of Lineland via Flatland Novel. Diagram shows a 2D birdseye view of Lineland, made up of line men and women, featuring A. Square above the line.

After being enlightened regarding the existence of the third dimension, our protagonist has a dream of being in the Abyss of No Dimensions, which contains Pointland. Pointland is constituted by the existence of a single point. This dot, the monarch of Pointland, provides comic relief with his solipsistic mindset. Existing in a zero-dimensional world, he believes that he is the sole point of existence in the universe, even readily convincing himself that A. Square’s disembodied voice is simply his own echoing back at him.

Redefining the Impossible

Between the analogies of Lineland and Pointland, A. Square begins to understand that our perception limits our ability to fully know or understand every facet of our reality. Thus, the assertion that any person can know with full certainty the “proper way to be” is a faulty assertion. This applies not only to the existence of other dimensions, but the rigidity of class structure, and the general danger of having a closed mind. Our understanding of up and down, right and wrong, and which from what is all based on the perspective from which we are viewing those things.

Impossible, then, should not define what certainly cannot be done but what cannot be done with full certainty. To say that something is impossible is to assert that you are capable of discerning the objective capabilities and possibilities in all of the universe across all of time. One need not fuss with inter-dimensional realities to understand this point, as our contemporary world stands as a testament to all things that would have been thought impossible centuries or even mere decades ago. To say something is improbable given your set of knowledge, then, would be more compatible with the truth than to say it cannot exist whatsoever.

The Existence of Higher Dimensions

A. Square asks the sphere if there exists a fourth dimension, or even more. The sphere scoffs at his remarks and insists that such ideas are preposterous. The irony here is quite funny. The sphere’s defense of the unquestioned supremacy of the third dimension is entirely contradictory to all that he has taught A. Square.

While the sphere later realizes the hypocrisy of his judgment and rectifies this with A. Square, this open questioning and self-reflection generally opposes the actuality of much of human social behavior. We forget that knowing a lot does not mean that we know everything, and we forget that advancing wealth and technology do not constitute the best and most important aspects of society.

Understanding Our Own Limits

A. Square, as an academic, is indoctrinated by the idea that academia’s explanation for all of society and existence is the end-all-be-all explanation. This is something he must peel back by analogous experiences with lower beings before he can come to understand that he is a lower being himself. This scaffolded instruction is just as much for A. Square’s understanding as it is for the reader themselves, as reflected by the sphere’s shortcomings.

Flatland movie poster showing dimensional objects facing off over a glowing white cube in the ground.

As much as we can cite socioeconomic progress in England and elsewhere since the writing of this novel, it doesn’t mean that we have surpassed all unknowns or achieved perfect societies. Not only that but asserting one single path to or one single definition of progress contradicts the ideas put forth by the novel. The reality of the second dimension is different than the reality of the third dimension, and the way that the whole of reality is perceived is different and limited in both. The third dimension illuminates the unknowns of the second dimension, but neither dimension discerns all the secrets of the objective reality of all the dimensions.

Similarly, an awareness of our shortcomings as a society does not mean that we are perceiving the full extent of our defects. Our subsequent efforts to mend or mitigate these shortcomings through government or other forms of social agency are not necessarily the way that is objectively best to achieve that end. As there seems to be an infinite number of ways we can do wrong, so too there must be an infinite number of possible ways we can do right.

Flatland‘s Continuing Relevancy

When I read Flatland, I think of it in the context of today’s sociopolitical climate. Flatland highlights both the importance of and the obstacle to self awareness, and shows how a lack of full self awareness can lead to counterproductive policy and behavior. The binary perspective of the American political system, between its highly divided two parties, sometimes creates the sense that the aim of solidifying or tearing down tradition is more important than accounting for the true, likely far more nuanced feelings, of the general public.

Flatland invites its readers to take a look outside of themselves and fully defamiliarize with their own perspective in order to realize the arbitrariness of many societal traditions that uphold inequality. As we attempt to work towards progress of any kind, we should remember that we all share the ability to make mistakes and the inability to know all. Perhaps then, so humbled, we’d all be a lot nicer to each other.

I’ve read Flatland in both math and English classroom settings, and I always seemed to be the only one in class who really loved it. Although Abbott’s contemporaries slept on him, too, the novella turned out to receive extensive acclaim and praise for a reason! Even if you don’t like math, I would highly recommend this book — both for its witty and insightful commentary and as an exercise in opening your mind to things you may be closed off to.

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