The second chapter explored yet another way to view the fourth dimension. But for me, the concepts discussed allowed me to explore the third dimension in ways that I have never thought of before. Splitting up the sides of a square into two components (a binomial) and then taking the square was a simple yet elegant way to show how a square is put together. Doing the same thing with a cube and a hypercube then comes naturally. Whereas before using Pascal's triangle only applied to abstract formulas in my thinking now I have a visual analogy to relate it to. I still have problems envisioning a hypercube, though. But now I have a better idea of its structure.
The fold-out models also put another spin on the 4d dilemma. The analogy of unfolding a fourth dimensional object into 3 space as we do a 3d object into a flat piece of paper is intriguing. Having a 3d thing in front of you and knowing that it can be folded up somehow so that it goes into the fourth dimension is kind of weird, not to mention unnerving. I was thinking about what would happen for curved 4 dimensional objects. Our maps of the world are always distorted because the world just won't go flat. Instead we have to use projections to highlight the relevant characteristics that we want from the globe. What sort of projections can we see from 4d space and what sort of distortions will occur? Are there any natural examples that can be witnessed (like in black hole or somthing) or is this sort of exploration limited only to computer models?
Finally, the concept of interdimensional objects blew me away. I mean a fractal looks flat, but it isn't?!!! What's inbetween? Interdimensional objects seem to be as elusive as those dimensions beyond the third, an area that we can't observe directly yet through careful thought its mysteries might be unlocked.