It's almost frustrating to read the first chapter. It is as if the author is teasing the reader. He tells us that a hypercube has sixteen corners and even shows us a picture of the hypercube. But what is a two-dimensional picture of a four-dimensional figure supposed to mean? Are we seeing the fourth-dimension? If seeing the "insides" of Linelanders" means that we are at least one dimension higher than Linelanders, and if seeing the "insides" of Flatlanders means that we are one dimension higher than Flatlanders, how does that translate to the fourth dimension? What would it mean so see all of the insides of a human all at once? A CAT-Scan shows us slices of our insides. What would it be like to see all of those slices at once in our field of vision. If I am a four dimensional object (perhaps with the fourth dimension zero... or not), what is the three dimensional representation of myself. It would seem that if my fourth dimension was zero, then the three-dimensional representation of my body would be exactly what I see when I look in the mirror. Even so, what would I look to like to a four-dimensional eye? Would a four-dimensional creature have trouble understanding how a mere three-dimensional creature can possibly exist?
I'm not exactly sure what is meant by the X-Ray analogy. Is an X-ray comparable to what a four-dimensional creature would see when looking at a human, or is an X-ray an example of projecting three dimensions into two? I could definitely think of it on both terms. Since we are seeing the whole of the inside of the salamander (and we can even extract depth from the picture), is it not analogous to a Spacelander seeing a Flatlander (except one dimension higher)? Also, I would even liken an X-ray picture to a two-dimensional image of a hypercube. Which one it is, or if the two are the same, I really can't decide as yet.
Possible exercise ideas:
1. Rotate yourself through the fourth dimension.
2. What are different ways in which multiple dimensions can be perceived from a two-dimensional rendering?
3. Predict the number of corners and edges in an n-dimensional cube.
4. How are dimensions like temperature and color different from the standard dimensions such as length, width, and height?
5. What can "perpendicular" mean beyond the third dimension?