Chapter Five was a confusing one for me. The first surpise was that there are a finite number of regular polyhedra in 3-space. Although, I had never really thought about it, I would have intuitively guessed that there are an infinite number of such figures. When I look at the figures of the regular polyhedra, I can't help but think about soccer balls, Buckey Balls, etc... There is certainly a relationship there. The reason that a hexagon cannot be the basis for a 3-space polyhedron is that 3 hexagons completely fill the area around a point. However, If we are allowed to curve the hexagons, we can make a ball out of them. A flatlander on a soccer ball would only see "rows of columns" of hexagons going in all directions. To A Sqaure, a soccer ball is a plane of flat hexagons. To him, the hexagons are not curved. However, it is also true that A Square cannot recognize that the hexagons are connected in such a way as to form a "curved polyhedron. Well, we can't eat our cake and digest it, too.... We either look at a soccer ball from the perspective of A Square and say that it is a plane of hexagons, or we look at the ball from human perspective and say the hexagons create a 3D figure, but are actually curved.
The whole duality thing is neat... Correct me if I'm wrong, but it would seem that there is no two dimensional analogue. In two-space, it would seem that all regular polygons are self-dual. Why should duality become in issue in the third dimension. I see the pretty pictures and see that it is true; but doesn't duality call for some kind or irregularity. Are 3-space polyhedra less perfect than polygons?
The confusing part of the chapter lies in some of the diagrams. I'm not "seeing" the diargrams of the simplexes. I can follow the logic, but I cannot visualize the construction of the higher dimensional polyotopes. To me, the pictures of four tetrahedra around an edge and the picture of five tetrahedra around an edge look too similar.
"Corpus Hypercubicus".... I like that...
Some good exercises would be to play "The Greek Geometry Game", to build projections of higher dimensional polyotopes, and to trace and constructthe fold-outs of tetraherda, dodecahedra, and the like. This reminds me of something from "Shape of Space" that's pretty neat. It involved connecting lots of hexagons to form a hyperbolic plane. I really do not see why this should work. If someone else has the time to try it, check out the book and give it a shot.
P.S. Go there you will, yes.