would not know of these two faces. He would think of the square figure as totally accounted for by its height and length. We recognize that a two-dimensional existence is an abstraction. To be real an object must have all three dimensions. Thus his conception of his matter is false. His words refer to real things, but the thoughts which he connects with those words are thoughts about abstractions. The square figure, as he thinks it is, is not a real object at all. I speak of the plane being as if he were real, but a little consideration will show that no structurally organized being of this kind could exist.
For instance, since the thickness of his matter is less than any discernible extension, any canal or channel of visible size would divide his body into two disconnected portions. No circulatory system or alimentary canal would therefore be possible in his case. In fact, if we admit the existence of an ascending scale of dimensions, a being which has extension of comparable magnitude in each of three dimensions, as we have, is the first of beings in this ascending scale that can possibly exist.
The mechanical possibilities of a plane being would be of a very limited range. Two such individuals meeting, one would be obliged to climb over the other in order to pass him.
The very difficulty of apprehending the extreme limitation of such a being shows how great the step would be which he must take in order to apprehend our possibilities.
In like manner, even to state the possibilities of a four-dimensional existence would appear naturally to us in our turn a task of great difficulty.
But the beauty of the illustration I have given is that it enables us to settle every point that comes up in the imaginary construction of a four-dimensional world, by attending to the corresponding step that the plane being would have to take in forming his imaginary construction of a three-dimensional world.
I would therefore ask the reader to try to imagine himself as a very flat being moving along the rim of the disk, looking only before him and up and down, only able to reach out in front and upwards, not laterally, and only to be able to approach and recede from the objects on his earth in one plane of motion.
If one of us were to try to explain to a plane being the nature of our existence, we should have to introduce some new words into his vocabulary, and these new words would not have reference to anything that the plane being could see or point to.
If there is a fourth dimension of space, we must necessarily introduce words which have no reference to anything within the range of our conscious experience.
There must be some reason why we do not move in this fourth dimension, as there is a reason why the plane being cannot move "right and left."
In order to introduce the ideas that we have to form in the most simple way, let us suppose that there is a substance (analogous to the sheet against which the plane being slips) along which we slip freely in every movement we make, and against which every portion of our matter slips in every movement it makes.
Consider for the sake of simplicity that a square figure of the plane being's matter is made up of a number of particles forming a single layer against the plane.
The plane being would have to admit that every particle in such a square figure of his matter was as close to the sheet as every other particle, and although the interior of the square would only be approached by him by breaking through the bounding-lines, still from each point of the interior a line could go "right and left" in an unknown direction that he could not possibly conceive.
Similarly, on an analogous supposition, if we look at a cube of matter which is perfectly bounded by its faces in every way in which we can approach it, we must admit that there is a direction going off from every particle in the cube, and that it is possible to draw a line from each particle in each of the two opposite ways in this unknown direction.
The cube would be perfectly free to move in either of these opposite ways in the unknown dimension unless by reason of some constraint.
Let us call the hypothetical substance which is next to every particle of our
