A review of Flatland, by Edwin A. Abbott

I’m an amateur painter, and one of the things I learned early on is that it’s a real challenge trying to render in only two dimensions what we see in three. If it’s this difficult for me to create something worthwhile in two frozen space dimensions, how much more wonderful is it that God created in three? And not only that, but the three space dimensions are swept along inside this phenomenon of time, changing fluidly, such that there are really four dimensions to His creation. And not only that, but His “painting” exists not just for this one space-time perspective, but for every such point all throughout the universe. Even if I were Leonardo da Vinci instead of the duffer I am, my work would serve more as a reminder of the wonder of God’s creation, than a wonderful thing unto itself.

This rumination led me to contemplate heaven, as most worthwhile ruminations do. It’s a significant thing, going from two dimensions to three. Adding a new dimension doesn’t just change our perspective somewhat, nor merely expand an existing view. The difference is paradigmatic. The shift from square to cube, for example, is more than just introduction of a new shape. These are geometrical, not merely arithmetical, progressions.  

And that is where Flatland opens. The narrator (we never find out his name) explains that he lives in a two-dimensional world.   There are mysteries undiscovered in this world, such as where light comes from. But the mysteries remain unsolved. One living in a two-dimensional world is unable to perceive a reality in which there is a third spatial dimension. One day the narrator explains to his grandson how the area of a parallelogram is expressed by multiplying length by width. The grandson grasps that for a square, this is the same as raising the number representing the length of the side, to the second power. The grandson then impertinently asks whether raising a number to the third power might express another kind of physical object, and therefore another space dimension besides the usual two. The grandson is sent off in shame at asking something so absurd.

But then there is a vision of Lineland. As you would surmise, this is a land of but one dimension, rather than Flatland’s two.   This opens the mind of our narrator, because, seeing the diminution of reality as he drops from two to one dimension, he is now able to conceive the expansion of reality beyond what he presently perceives in his two-dimensional world.

And so it happens. Our protagonist is visited by someone from Spaceland. You’ve already guessed it: this is a being from three-dimensional spatial reality. A sphere, actually, who has to explain that he is not a circle, but rather an infinity of circles. By this explanation, it becomes possible to conceive another dimension which would change everything one knows about reality. The narrator of this quirky chronicle then goes himself to Spaceland, and his eyes are opened. He sees depth and width and height, and learns the solution to riddles that were impenetrable before – like where light comes from.

Alas, the Sphere rejects the narrator’s speculation, much as the narrator rejected his grandson, when the narrator suggests the possibility that there might be a realm with yet another, fourth dimension, which he dubs “Thoughtland.” But you, the reader, will discern by now that this extrapolation is inevitable and real. And though it is inevitable, it is imperceptible to us here and now, except conceptually; mentally — to the mind but not the senses — unseen yet “imaged” in the mind.

Like heaven appears to us now.

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