If, however, we take the other starting-point, and, not assuming any fixed fundamental cube, look only at the act of arrangement, the two Blocks, III. and IV., are found to be identical in every internal relationship.
For, taking the act of arrangement as the basis, if, when we have built up the Block IV., we look upon each of the cubes as an arrangement of the same kind as the whole, then the cube 1 in Diag. IV. is represented in Diag. VI. And it is evident that if Diag. IV. is built up out of cubes like Diag. VI., the small cubes, 1, 4, 7, lie in the same edge as the cubes 1, 4, 7, in Diag. IV. Thus it will be found for every relationship in Diag. III. there is an exactly similar relationship in Diag. IV.
In this case if, for the sake of material illustration, we use marked cubes, it seems that we must not suppose each particular cube to have a fixed marking of its own, but that we must suppose the markings to spring up on the sides of the cubes in accordance with the places into which they are put.
There is another manner of regarding the matter which may help to bring out the point at issue.
If we suppose that we are putting up the cubes in one room while another person is putting up cubes in an adjoining room; if we can tell him what we are doing, using the words right and left, he will be able to put up a block exactly like ours. But if we do not allow ourselves to use the words right and left, but speak to the other person as if he were simply an intelligence without having the same kind of bodily organization as ourselves, we should find that, supposing he could put up the block of cubes, it would be a mere matter of chance whether he had put up the block as we had put it, or whether he had put it up in an image way. And the same with regard to any shape. We could tell him that the cubes should be put together, and we could tell him the relationship which they should have with regard to one another; but the figure he put up would just as likely be an image of our shape as not.
And we could go on for ever building more and more complicated shapes and telling him to do the same, and no hitch or difficulty would come. But at the end all his shapes might be ours just reversed, as if seen in a mirror.