A piece of paper on a smooth table affords a ready image of a two-dimensional existence. If we suppose the being represented by the piece of paper to have no knowledge of the thickness by which he projects above the surface of the table, it is obvious that he can have no knowledge of objects of a similar description, except by the contact with their edges. His body and the objects in his world have a thickness of which however, he has no consciousness. Since the direction stretching up from the table is unknown to him he will think of the objects of his world as extending in two dimensions only. Figures are to him completely bounded by their lines, just as solid objects are to us by their surfaces. He cannot conceive of approaching the centre of a circle, except by breaking through the circumference, for the circumference encloses the centre in the directions in which motion is possible to him. The plane surface over which he slips and with which he is always in contact will be unknown to him; there are no differences by which he can recognise its existence.
But for the purposes of our analogy this representation is deficient.
A being as thus described has nothing about him to push off from, the surface over which he slips affords no means by which he can move in one direction rather than another. Placed on a surface over which he slips freely, he is in a condition analogous to that in which we should be if we were suspended free in space. There is nothing which he can push off from in any direction known to him.
Let us therefore modify our representation. Let us suppose a vertical plane against which particles of thin matter slip, never leaving the surface. Let these particles possess an attractive force and cohere together into a disk; this disk will represent the globe of a plane being. He must be conceived as existing on the rim.
Fig. 4.
Let 1 represent this vertical disk of flat matter and 2 the plane being on it, standing upon its rim as we stand on the surface of our earth. The direction of the attractive force of his matter will give the creature a knowledge of up and down, determining for him one direction in his plane space. Also, since he can move along the surface of his earth, he will have the sense of a direction parallel to its surface, which we may call forwards and backwards.
He will have no sense of right and left—that is, of the direction which we recognise as extending out from the plane to our right and left.
The distinction of right and left is the one that we must suppose to be absent, in order to project ourselves into the condition of a plane being.
Let the reader imagine himself, as he looks along the plane, [fig. 4], to become more and more identified with the thin body on it, till he finally looks along parallel to the surface of the plane earth, and up and down, losing the sense of the direction which stretches right and left. This direction will be an unknown dimension to him.