"The change in our conceptions, which we make in passing from the shapes and motions in two dimensions to those in three, affords a pattern by which we can pass on still further to the conception of an existence in four-dimensional space."
Let us start then by imagining a very large, flat and perfectly smooth surface; such for instance as the top of a highly polished table or the surface of a sheet of still liquid.
We have seen that such a surface constitutes space of two dimensions, because through any point in it we can only draw two lines at right angles to one another. In order to draw a third such line we must get out of the surface altogether and draw the line perpendicular to it.
Next we must try to imagine that this surface is populated by a race of beings of an extraordinary thinness.
In order to grasp the analogy properly we must imagine them to be so constituted that they are incapable of realising any direction in space which does not lie in the aforementioned flat surface on which they live.
We can imagine this by supposing that their thickness, i.e.:—their extension in the third dimension perpendicular to their surface,—is so small as to be invisible to them and also that their "nerve endings" all lie on their periphery. This last is equivalent to saying that they have no "sense organs" facing the third dimension and that therefore they cannot receive impressions, or respond to any stimuli that come to them from that direction.
It follows, therefore, that unless they develope special sense organs which face the third dimension they will be acquainted only with such objects and events as lie, or take place, in their surface.
It is of course inconceivable that they should be truly "plane" beings in the mathematical sense and possess no thickness at all. But if we suppose that their thickness is of the same order as the diameter of a chemical "Atom"—that they are "one atom thick" so to speak,—the conditions laid down as to their limitation will be fulfilled.
Now we have supposed the flat surface in our analogy to be perfectly smooth in the true sense of the word. That is to say of such a nature as to offer no resistance whatever to the passage of objects over it.
This means that plane beings will not be sensible of any opposition to their movement as far as the surface is concerned. Also, as we have supposed that they have no nerve endings facing it, it follows that they cannot feel any pressure from it. In short they will be totally unaware of its existence.