Flatland

Flatland

Let us imagine we inhabit a strange country where everyone is perfectly flat. Following Edwin Abbott, a Shakespearean scholar who lived in Victorian England, we call it Flatland. Some of us are squares, some are triangles, some have more complex shapes. We scurry about, in and out of our flat buildings, occupied with our flat businesses and dalliances.

Everyone in Flatland has width and length, but no height whatever. We know about left-right and forward-back, but have no hint, no trace of comprehension, about up-down – except for flat mathematicians. The say, “listen, it’s really very easy. Imagine left-right, imagine forward-back, ok, so far? Now imagine another dimension, at right angles to the other two. And we say: “What are you talking about? At right angles to the other two? There are only two dimensions. Point to that third dimension. Where is it?” So the mathematicians, disheartened amble off. Nobody listens to mathematicians.

Every square creature in Flatland sees another square as merely a short line segment, the sight of the square nearest to him. He can see the other side of the square only by taking a short walk. But the inside of a square is forever mysterious, unless some terrible accident or autopsy breaches the sides and exposes the interior parts.

On day a three-dimensional creature – shaped like an apple, say, comes upon Flatland, hovering above it. Observing a particularly and attractive and congenial-looking square entering its flat house, the apple decides, in a gesture of interdimensional amity, to say hello. “How are you? “ asks the visitor from the third dimension. “I am a visitor from the third dimension”. The wretched square looks about his closed house and sees no one. What is worse, to him it appears that the greeting, entering from above, is emanating from his own flat body, a voice from within. A little insanity, he perhaps reminds himself gamely, runs in the family.

The question of what lies beyond is meaningless. Flat creatures cannot, on their own, escape there two dimensions. Image: © Megan Jorgensen (Elena)

Exasperated at being judged a psychological aberration, the apple descends into Flatland. Now, a three-dimensional creature can exist, in Flatland, only partially; only a cross section can be seen, only the points of contact with the plain surface of Flatland. An apple slithering through Flatland would appear first as a point and then as progressively larger, roughly circular slices. The square sees a point appearing in a closed room in his two-dimensional room and slowly growing into a near circle. A creature of strange and changing shape has appeared from nowhere.

Rebuffed, unhappy at the obtuseness of the very flat, the apple bumps the square and sends him aloft, fluttering and spinning into that mysterious third dimension. At first the square can make no sense of what is happening; it is utterly outside his experience. But eventually he realizes that he is viewing Flatland from a peculiar vantage point “above”. He can see into closed rooms. He can see into his flat fellows. He is viewing his universe from a unique and devastating perspective. Traveling through another dimension provides, as an incidental benefit, a kind of X-ray vision. Eventually, like a falling leaf, our square slowly descends to the surface. From the point of view of his fellow Flatlanders, he has unaccountably disappeared from a closed room and then distressingly materialized from nowhere. “For heaven’s sake,” they say, “what happened to you?” – “I think”, he finds himself replying, “I was up”. They put him on his sides and comfort him. Delusions always run in his family.

Imagine the universe just like Flatland, except that unbeknownst to the inhabitants, their two-dimensions universe is curved enough through a third physical dimension. When the Flatlanders take short excursions, their universe looks flat enough. But if one of them takes a long enough walk along what seems to be a perfectly straight line, he uncovers a great mystery: although he had not reached a barrier and has never turned around, he has somehow come back to the place from which he started. His two-dimensional universe must have been warped, bent or curved through a mysterious third dimension. He cannot imagine this third dimension, but he can deduce it. Increase all dimensions in this story by one, and you have a situation that may apply to us.