The subject, however, abounds in abysses for stupidity to fall into, and I had to clamber out of each of them; so it took me several years before I got quite on the right tack. Then it was easy enough: any one in a few weeks could learn to conceive four-dimensional figures. Not only is it easy, but there are abundant traces that we do it continually without being aware of it. I am sure if the loveliness of the work while one is doing it, and the simplicity and self-evident nature of the results when obtained, were generally known, it would be a favourite amusement.
Now one of the first things that presented itself to my attention when I began to move the four-dimensional figures about was a fact which bore curious reference to my difficulty about the fundamental cube. If the reader remembers, it seemed to me as if the cube out of which the whole block of known cubes was built ought to be able to be inverted. That is to say, it seemed to me that there was a self-element present in my knowledge of the cubes. But in order to cast out that self-element the fundamental cube which lay at the basis of the whole block would have to be able to be inverted, or pulled through.
Now I found that when I took a four-dimensional figure which came into space by a cube—that is, a figure which rested on space by a cube, or one of whose sides was a cube—when I took a figure of that sort up in the fourth dimension and twisted it round and brought it down again, this cube would sometimes be inverted or pulled through—although I had done nothing to it, but had simply twisted the whole figure round without disturbing the arrangement of its parts.
Thus evidently to a higher child it would be no more difficult to invert or pull through a cube or a figure than it would be to me to twist one round.
Hence it was obvious that right and left was really a self-element in my block of cubes. I being in our space was under a certain limitation, and that limitation made me feel as if a right-handed arrangement was different from a left-handed arrangement.
A being who was not limited as I was would see that they were one and the same. Hence, in knowing the set of blocks it was necessary to cast out “right and left,” and the names had to be learnt over again in new positions.