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By Root 527 0

to understand God is like a two-dimensional being trying to understand the third dimension. In his classic tale Flatland, Edwin Abbott describes such an existence, where a circle would only be perceived as a line. Watching a three-dimensional object such as a sphere pass through Flatland, a resident would see only a point and then a succession of circles growing larger at first and then smaller as it returns to a point before vanishing.

Abbott’s surrealistic story begins in a world of two dimensions, where the inhabitants—geometrical figures such as lines, triangles, squares, pentagons, hexagons, and circles—move left and right, forward or backward, but never “up or down.” Looking at a coin you can see the shapes within the circle, much like you could see the inhabitants of Flatland from Spaceland looking down; but if you turn the coin on its side, the interior disappears and you only see a straight line. This is what all geometrical shapes look like to Flatlanders.

One day a mathematician Square in Flatland encounters a stranger that mysteriously changes sizes from a point, to a small circle, to a big circle, back to a small circle, and finally vanishes altogether. Since Flatlanders do not arbitrarily grow and shrink in size, the Square is confused. The stranger explains that he is not a single circle changing sizes but “many circles in one,” and to prove his three-dimensional nature to the Square he employs logic and reason: “I am not a plane figure, but a solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of circles, of size varying from a point to a circle of thirteen inches in diameter, one placed on top of the other. When I cut through your plane as I am now doing, I made in your plane a section which you, very rightly, call a Circle.”

The Square still does not understand, so the stranger, a Sphere, turns from example to analogy:

Sphere: Tell me, Mr. Mathematician, if a Point moves Northward, and leaves a luminous wake, what name would you give to the wake?

Square: A straight line.

Sphere: And a straight line has how many extremities?

Square: Two.

Sphere: Now conceive the Northward straight line moving parallel to itself, East and West, so that every point in it leaves behind it the wake of a straight line. What name will you give to the figure thereby formed? We will suppose that it moves through a distance equal to the original straight line. What name, I say?

Square: A Square.

Sphere: And how many sides has a square? How many angles?

Square: Four sides and four angles.

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

The problem, of course, is that “upward” has no meaning for a two-dimensional being who has never experienced the third dimension of “height.” The Square is still confused, so the Sphere walks him through a clear-cut proof: If a point produces a line with two terminal points and a line produces a square with four terminal points, then the next number is 8, which the Sphere explains makes a cube—a six-sided square in Spaceland. This he further proves with logic: If a point has zero sides, a line two sides, a square four sides, then the next number is 6. “You see it all now, eh?” says the Sphere triumphantly. Not quite. For the dimension-challenged Square, reason is not revelation: “Monster, be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. Either thou or I must perish.”

With failed reason the Sphere, in a throe of frustration, reaches into Flatland and yanks the Square into Spaceland, whereupon he instantly transforms into a cube. Revelation! But then a thought occurs to the Cube. If the Sphere is many circles in one, there must be a higher dimension that “combines many spheres in one superior existence, surpassing even the solids of Spaceland … . [M]y lord has shown me the intestines of all my countrymen in the land of two dimensions by taking me with him into the land of three. What therefore more easy than now to take his servant on a second