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A being identifying himself with a shadow would think that the surface of the wall was all the space there was. The conception of a third dimension of space would be as foreign to him as that of a fourth dimension is to us.

By means of this retrogressive step of imagining a kind of existence in which experience is confined to less than our number of dimensions we are irresistibly led to propound the question: Is there a kind of existence dimensionally more ample than that of our experience?

In his illustration Plato avoids an error in which most modern representations of a plane world are involved.

He makes his beings real, not mere abstractions, such as geometrical beings in a plane. He places before us an imaginary scene in which real beings would have a conscious experience of a condition more limited than that of their actual existence. I will represent the same idea in a different manner, choosing my illustration so that it will give us the means of answering all questions that occur in the study of four-dimensional space, and will also lead us to an appreciation of the reasons for inferring its existence.

For this purpose imagine a globe to be cut in half, and of the half a thin slice to be taken. Imagine this thin circular disk to be placed against a great steel sheet over which it can slip perfectly freely; and suppose, moreover, that in virtue of some adhesion or attractive force the disk was held in contact with

the sheet so as never to leave it (Fig. 2). This disk can be considered to be a "plane world."

Since it is material it should exercise attraction, and we must imagine the beings who inhabit it as standing on the edge. Let such a being be represented by the small triangle in Fig. 3. The force of gravity due to the disk would give him the direction of "up and down" (one dimension). Movement along the rim of the disk would give him the direction of "away and near" (a second dimension).

Now in order to have a being with a two-dimensional experience only, we must suppose that no movement other than in these two dimensions comes within the cognizance of his senses. Let us therefore suppose that the matter of which the disk and his own body are composed has only a very slight extension away from the disk. Let this extension be so slight that it escapes his closest observation.

Thus this "plane being," if he looked at any object composed of his matter, say a square figure, as in Fig. 4, would, as far as his consciousness was evoked, only see the length of the side opposite him. Since it is a real object, in order to come within the sphere of his senses it must have extension away from the surface of the sheet. But this extension he would not recognize, and although the figure is real he would speak of it as though it were a geometrical square. He would believe that if this square were indefinitely extended it would fill up the whole of space. We see that it would only cover an infinite plane surface.

A being such as here described would lack a sense which we have, namely, a sense of "Right and Left." We see that a square figure made out of his thin matter has two faces, one opposed to the sheet, one turned away from it. But he