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Let’s imagine a world of only two dimensions, something like a huge thick piece of paper (actually, of course, with no thickness at all) inhabited by 2-dimensional men. Someone from our 3-dimensional world with the aid of a light bulb projects the image of a cube into the 2-dimensional world, much to the surprise of our 2-dimensional friends. What they see is not a cube as we know it, but two squares, one within the other, with lines connecting the corners of one to the other:
We could, for instance, picture it as a small room with the small square in the middle representing the far wall, the sides being the right and left walls, the top part the ceiling and the bottom the floor. But what are 2-dimensional "men" able to make of it? All they understand is two dimensions, no more and no less. All they see is one large 2-dimensional square enclosing a smaller square with the corners connected. And if they were confronted with a figure like this or they would merely see two squares intermeshed with the corners connected, or perhaps two parallelograms with corners connected: The idea of "depth" is inconceivable to him. He only understands length and width, two dimensions.
Just as the 2-dimensional man tried in vain to find a third line perpendicular to the cross in front of him - no matter which way he placed it, it would not work - we, too, try in vain to imagine a fourth unique line perpendicular to the lines described by the three edges of a cube. But the answer is a line entirely outside our 3-D world, just as the line extending up wards from the surface of the paper is entirely out of the paper.
He is inside of the same plane as the square. What he sees, then, is not actually a square at all! What he sees is a line segment - with "depth"! He is fully aware of this "depth" and understands it without any problems, since it is a part of his daily experience. But he cannot actually see this depth, this second dimension, because it is in the same direction as his eyesight. His eyes perceive a square not as we do: but as a line segment:
Now let’s have some fun with our 2-dimensional friends. They would like very much to see a 3-D cube face-to-face, but have no way of doing so unless we help them. The best we can do is to take a cube and to push it through their 2-D world for them to observe. What do they see? If we push it through, keeping it parallel with the plane of their 2-D environment, they first see a square which suddenly appears out of nowhere. As we continue to push it through, the square seems to remain motionless and unchanged. Then it disappears as suddenly and as mysteriously as it had appeared.
Even more interesting is to pass a sphere through their world for them to observe. What would they see this time? First a dot appears, which rapidly grows into a circle. The wider the circle becomes, the slower it grows. Having reached its maximum size, namely the diameter of the sphere itself, the circle slowly diminishes, shrinking more rapidly until it suddenly disappears altogether.
By analogy we may conclude that a similar spectacle would unfold before our eyes should we desire to see a 4-D supercube or 4-D supersphere pass through our space. In the case of a 4-D supercube, we would first see a cube appear out of thin air, hover a bit, and then instantaneously disappear. It has just passed 4-dimensionally from one side of our 3-D world to the other, just as the cube passed through the plane. No motion was visible, because the only moving that took place was in a direction not found in our world of three dimensions.
If we were able to warp our 2-D friend slightly and place him within the surface of a sphere, he would have nowhere to go but the surface of the sphere itself. No matter which direction he chooses to go, he ends up in the same spot. Being enclosed within the sphere’s surface, he is doomed to travel in endless circles, although he himself is not aware of any turning. Striking out in a direction which seems to him to be a straight line, he invariably comes back to his original starting place and is completely baffled as to how this could be.
Similarly, if one of us were "warped" 4-dimensionally and placed within the 3-D surface of a supersphere, he would find himself in exactly the same predicament as the 2-D man described above. Regardless of which direction he chooses to go, he finds himself mysteriously going in a circle. The space in which he is situated seems to be infinite in all directions, but is actually finite. In fact, we would even be able to measure its volume down to the last cubic inch. What to him seems to be an immeasurable expanse of space is in actuality a finite volume of space curved 4-dimensionally into itself, like the curve surface of a balloon.