Math

Flatland Is a 19th-Century Tale About Higher Dimensions

You live in a three-dimensional world, one where you're free to move forward and backward, right and left, and up and down. You're also familiar with lower dimensions: a drawing on a page is two-dimensional, for example, and a line could be considered one dimension. But what about a world with four dimensions? We can describe it with numbers, but as a concept, it's pretty hard to picture. That's just the challenge that Edwin Abbott Abbott took on in his 1884 novella "Flatland." It's about the difficulties in describing higher dimensions to beings in lower dimensions, with a sizable helping of political and social commentary to boot.

Slide To The Left, Slide To The Right

The narrator of the story is a square, a well-to-do mathematician with a line for a wife and four pentagon sons. (In this society, the more sides you have, the higher your status. All of the women are lines.) He lives in Flatland, a two-dimensional world where shapes move north/south and east/west, but have no concept of up/down.

One night, the square has a dream that he visited a one-dimensional world where only lines and points moved back and forth on a one-dimensional expanse. He tried in vain to explain the second dimension to the king of this world, who called his troops to advance a violent attack in retaliation for such ridiculous delusions when the square awoke from his dream.

One night after sending his grandson to bed because of a failed math lesson (the little hexagon suggested that if a point moving one way makes a line, and a line moving parallel to itself makes a square, a square must be able to move parallel to itself to make something else. "If you would talk less nonsense, you would remember more sense," the square retorted.), the square got an otherworldly visitor: a sphere from the third dimension.

Think Outside the Plane

At first, the square thought the sphere was just a circle. "I am indeed, in a certain sense a circle," it replied, "and a more perfect circle than any in Flatland, but to speak more accurately, I am many circles in one." The sphere declared that he came from Space, which confused the square — weren't they already in Space? The sphere scoffed.

"Define Space."

"Space, my Lord, is height and breadth indefinitely prolonged."

"Exactly: you see you do not even know what Space is. You think it is of two dimensions only; but I have come to announce to you a third — height, breadth, and length."

From here, the sphere tries over and over to describe three-dimensional space to this two-dimensional being.

"We began with a single point, which of course - being itself a point - has only one terminal point. One point produces a line with two terminal points. One line produces a square with four terminal points. Now you can give yourself the answer to your own question: 1, 2, 4, are evidently in geometrical progression. What is the next number?"

"Eight."

"Exactly. The one square produces a something-which-you-do-not-as-yet-know-a-name-for-but-which-we-call-a-cube with eight terminal points. Now are you convinced?"

After much debate, a few angry outbursts, and a dizzying journey to the third dimension, the square is finally convinced. He's so convinced, in fact, that he wonders aloud if there could also be a world with four dimensions. "There is no such land," the sphere replies. "The very idea of it is utterly inconceivable."

Just as True Today

Of course, scientists today regularly deal with the concept of four dimensions and beyond. And although we can use diagrams and animations to try and describe what those higher dimensions might be like, it remains just as puzzling to we citizens of Space as it was for the citizens of Flatland.

If you'd like to check out the whole novella — and delve into the surprisingly sharp political satire Abbott manages to infuse in a story about polygons — you can listen to the audiobook for free with a 30-day trial of Audible. Affiliate links help support Curiosity!

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Written by Ashley Hamer November 5, 2017