Higher Dimensions.
One of the things that interested me in the first chapter of Beyond the Third Dimension was the idea that we can see all of a two dimensional world simultaneously. Couldn't a fourth dimensional "person" see all sides of us in our three dimensional world simultaneously? Could she look at us and in the same glance see the front of our faces, the backs of our hair, the sides of our legs, and the bottoms of our feet. When I was a little child, I used to draw women who would wear necklaces. In my pictures, I would show the whole circle of the necklace, because I knew it was there. I believe that Pablo Picasso also played with this idea. I think that in some of his paintings, though I am not certain of an specific example, he would stretch out a figure so more of it could be seen. Could a fourth dimension person see inside of us? Is there any dimension that could into a solid object, like person, that has different parts inside?
Lower Dimensions.
What if the King of Lineland decided to write a book about the second dimension, and he created an analogy for his readers by describing the life of a point in Pointland who has an encounter with the second dimension. Part of his story includes the point's superior vision of the zero dimension. A number to the "0" power is one. What is the dimensional implication of that? A number to a negative power is one over that number? What are the dimensional implications of negative powers?
Hypercubes
I am definitely interested in building hypercubes and experimenting with different coloring. Some of my ideas for different coloring include: coloring one cube the way it is represented in Beyond the Third Dimension on page nine, coloring the cubes that are flattened so they become more obvious, coloring parallel line the same color (are different edges parallel on a real hypercube?) and experimenting with coloring the many faces.
I also have a question about the illustration on page ten. Why does the picture on the bottom have a pentagonal shape in the center rather than a hexagon? I think I am missing the concept there.