Where, To Be A Square, Is An Honor

Flatland Penguin

Edwin A. Abbott’s “Flatland,” published in 1884, is an obscure Victorian gem that very few have heard of.

A rare mathematical fantasy, it describes a cramped world of two dimensions, whose inhabitants can’t conceive of a third dimension—and nor would they like to.

Its charming narrator, A. Square, takes readers on a tour of his straitjacketed country. All its citizens are geometric figures—triangles, squares, pentagons, hexagons, dodecagons, who reside in a plane that knows not the mildest thickness. There, they mill about freely, “but without the power of rising above or sinking below it.”

Its people, varied as they seem to us, are indistinguishable by sight to each other, for from their edgewise perspective—which is the only perspective possible—everyone looks alike. They all appear as straight lines, although some are more luminous and bolder than others.

Yet, for all the homogeneity, they can still ascertain the class of an approaching or a receding bar, by feeling his perimeter and angularity. The more sides a person has, the higher he is on the social ladder. His angle is a measure of his intellectual caliber. The brawnier the radian, the brainier is the individual.

The book is divided into two parts. The beginning section paints a portrait of a society, which we, today, would regard as shockingly elitist and painfully sexist.

Soldiers and the lowest classes are isosceles triangles. Having only two equal margins and a slender base, they almost don’t count as humans. Equilaterals make up the middle class. Professionals and gentlemen are squares or pentagons. The nobility begins at hexagons. Priests are circular beings, who run this realm.

Upward mobility is what everyone aspires for. As a rule, a son will have one more side than his father. Therefore, he’ll automatically climb the next higher rung in the social ladder.

The son of a square is a pentagon; the son of a pentagon, a hexagon; and so on.

Among the aristocracy, parental ambition is so strong that they have their infants fractured so as to increase their sides. Anyone born with an “irregularity” (a side too short or too long or crooked) must be made “regular,” failing which, the individual must be put in prison or killed.

This practice of repairing an “imperfection” has echoes of the controversial concept of eugenics, which aims to segregate the most genetically-gifted humans, and mate them to improve the stock.

A satire in the tradition of Johnathan Swift’s “Gulliver’s Travels,” its absurd portrayal of females, which derives from the plight of Victorian ladies, crackles with zany humor.

Women are straight lines. And as they have no pretensions to the gentlest of gradients, they’re perceived as blockheaded, with no knowledge even of their own wretched position.

When driven into a fit of rage, the members of the “thinner sex” can turn into deadly creatures, capable of puncturing an entire household.

A Woman is a needle; being, so to speak, all point, at least, at the two extremities.

As their sharp rear ends make them pedestrian hazards, they’re barred from walking or standing in a public place, without constantly swinging their posteriors, from right to left, so as to make their presences known to those behind.

However, if one were to adhere strictly to the rules of arithmetic, then the notion of their dimwittedness would, of course, fade, as straight lines do have angles, and a wide angle at that: 180 degrees. Which would, for argument’s sake, make them smarter than their own men.

Flatland is perpetually lit, by day and night, in all places equally, even though the Sun doesn’t reach it. Houses are five-sided polygons, with roofs facing the north to shield them from the rains that pour from that direction.

The concluding segment veers off in a markedly different direction.

In a paradoxical dream, our friend in Flatland has a vision of Lineland, and its pitifully benighted monarch, who believes that his one-dimensional, linear domain, which he calls his kingdom, is indeed, the whole of space.

In its inventiveness and wit, “Flatland” carries flavors of Lewis Carroll’s “Alice in Wonderland,” published 20 years earlier. In a voice reminiscent of the Queen of Hearts, the ruler of Lineland declares:

“Behold me—I am a Line, the longest in Lineland, over six inches of Space.”

“Of Length,” the narrator suggests.

“Fool,” says the ruler. “Space is Length. Interrupt me again, and I have done.”

When the narrator ventures to ask him how Linelanders—who can’t budge outside a straight line, and whose horizon is limited to a mere dot—can recognize one another, the  king of Lineland tells him they do so, by dual voices.

His wives, he says, are at a distance of “six thousand miles, seventy yards, two feet, eight inches,” one to the north; another, to the south. When he speaks to them in his frontal voice, followed closely by his hinder voice, from the time it takes for the latter to reach them, they can tell his “shape,” with no uncertainty, to be 6.457 inches.

When the narrator is visited by a three-dimensional Sphere from Spaceland, where you, I, and everyone we know lives, he’s flabbergasted.

The Sphere tells him: “Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself, upward.”

“What? Northward?” asks the narrator.

“No, not Northward; upward; out of Flatland altogether,” tells the Sphere.

Once he has a grasp of the new reality, he entreats his superior companion to lead him to the “blessed region of the Fourth Dimension.” The Sphere, who’d previously mocked at his narrow worldview replies: “There is no such land.” Even the very thought of it is “utterly inconceivable.”

If this story has a moral, then it’s that we must shake off our complacency, and awaken our consciousness to new possibilities.

Hovering above Flatland, the narrator is awestruck to see the interiors of objects—both animate and inanimate—whose exteriors he’d, so far, only deduced through calculations. He feels like god, for to be able to perceive everything is the attribute of god alone. Extending that analogy, it’d appear that the god of a three-dimensional being would be a four-dimensional organism.

In science, however, the fourth dimension doesn’t exist as another facet of space. Albert Einstein’s theory of relativity treats time as the fourth dimension.


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