In chapter one, we took a look at a view of 2-dimensional space from the 3rd dimension. When speaking of 2-D objects in the 2nd dimension and 3-D objects in the third dimension, we discovered that no 2-D object can ever see a 3-D object fully. On the other hand, a 3-D object can see all 2-D objects in the 2nd dimension simultaneously, even those objects which may be hidden from the view of other 2-D objects. As I stated in chapter one, we will now investigate the implications the previous discussion has for objects in the 3rd and 4th dimensions.
Let us return to our friend the sphere. The sphere lives in the 3rd dimension. Quite naturally, you might expect the sphere can see other 3-D objects like other spheres, cones, tetrahedrons and other irregularly shaped 3-D objects. Also as discussed in chapter one, all of these 3-D objects have x-ray vision of their 2-D neighbors; that is, 3-D objects can see all 2-D objects living in the 2nd dimension simultaneously.
Can we now perceive of what an object lying outside the 3rd dimension, namely the 4th dimension, might look like from the 3rd dimension, assuming it could be perceived at all? If, for now, we assume that we can, then, if we follow the logic of our discussion in chapter one, our object in the 4th dimension could conceivably be wholly invisible to an object in the 3rd dimension. Suppose, for the sake of discussion, that our 4-D object, whatever it is, lies entirely outside the 3rd dimensional space of a sphere living in its 3-D world. Then, the sphere would have no vision of that 4-D object whatsoever. An object so positioned in the 4th dimension trying to communicate with the sphere might appear to the sphere to be a spirit, just as the sphere in chapter one appeared likewise to the circle in its 2-D world.
But what if the 4-D object were to pass through the sphere's 3-D space; what might it appear to the sphere? The answer comes easily from our discussion in chapter one: as the 4-D object passes through a 3-D object's 3-D space, it would take on a 3-D form. Is this surprising? No, since everything a 3-D object can see is either 3-dimensional or of lower dimensional form. However, as was the case with the circle and sphere in chapter one, the sphere in our present discussion never fully appreciates what it perceives of it's 4th dimensional interloper because it never sees it in its entirety.
Now, let us suppose the 4-D object, like its 3-D counterpart of chapter one, encroaches on its neighbor, the sphere, in the 3rd dimension, nudging it out of its world into the 4th dimension. The sphere, quite naturally, becomes a bone fide 4-D object and can thus see its 4-D friend for who he really is rather than a mere series of 3-D cross-sections of itself. And, continuing the logical progression from chapter one, in the process of jumping into the 4th dimension, the sphere has acquired the ability to see all of its 3-D friends simultaneously while not having that ability within the 3rd dimension. Likewise, none of the sphere's 3-D friends have the ability to see him now. The sphere has disappeared from sight of its former 3-D world.
While it may be difficult for the mind to grasp what a 4-D object might look like, it is safe to assume it either exists wholly outside our present 3rd dimension, or if it does contact our space, it appears to have a form which we, in our 3-dimensional space, can perceive it to be: a 3-D or lesser dimensional object.
What are the implications of this discussion on what many perceive the 4th dimension to be, namely time? This will be the subject of chapter three.